Nonlinear stability of compressible vortex sheets in three-dimensional elastodynamics
Abstract.
We investigate the nonlinear stability and existence of compressible vortex sheet solutions for three-dimensional isentropic elastic flows. This problem involves a nonlinear hyperbolic system with a characteristic free boundary. Compared to the two-dimensional case, the additional spatial dimension introduces intricate frequency interactions between elasticity and velocity, significantly complicating the stability analysis. Building upon previous results on the weakly linear stability of elastic vortex sheets [19], we perform a detailed study of the roots of the Lopatinski determinant and identify a geometric stability condition associated with the deformation gradient.
To address the challenges of the variable-coefficient linearized problem, we employ an upper triangularization technique that isolates the outgoing modes into a closed system, where they appear only at the leading order. This enables us to derive energy estimates despite derivative loss. The major novelty of our approach includes the following two key aspects: (1) For the three-dimensional compressible Euler vortex sheets, the front symbol exhibits degenerate ellipticity in certain frequency directions, which makes it challenging to ensure the front’s regularity using standard energy estimates. Our analysis reveals that the non-parallel structure of the deformation gradient tensor plays a crucial role in recovering ellipticity in the front symbol, thereby enhancing the regularity of the free interface. (2) Another significant challenge in three dimensions arises from the strong degeneracy caused by the collision of repeated roots and poles. Unlike in two dimensions, where such interactions are absent, we encounter a co-dimension one set in frequency space where a double root coincides with a double pole. To resolve this, we refine Coulombel’s diagonalization framework [21] and construct a suitable transformation that reduces the degeneracy order of the Lopatinski matrix, enabling the use of localized Grding-type estimates to control the characteristic components. Finally, we employ a Nash-Moser iteration scheme to establish the local existence and nonlinear stability of vortex sheets under small initial perturbations, showing stability within a subsonic regime.
Key words and phrases:
Vortex sheets, Elastodynamics, Contact discontinuities, Linear stability, Nonlinear stability, Para-differential calculus, Nash-Moser iteration2010 Mathematics Subject Classification:
35Q51, 35Q35, 74F10, 76E17, 76N991. Introduction
Vortex sheets are interfaces between two inviscid incompressible or compressible flows, characterized by a contact discontinuity in the fluid velocity. Across these interfaces, the tangential velocity field exhibits a jump discontinuity, while the normal component of the flow velocity remains continuous. Vortex sheets arise from various physical phenomena in fluid mechanics, including oceanography, plasma physics, astrophysics, elastodynamic and aerodynamics. In compressible flows, they are one of the fundamental wave types, along with shock waves and rarefaction waves, in multi-dimensional (M-D) hyperbolic systems of conservation laws. Studying the existence and stability of compressible vortex sheets can give a better understanding of M-D Riemann problems and the behavior of entropy solutions; see Chen-Feldman [11] and Dafermos [26].
In this paper, we focus on studying the vortex sheets in 3D compressible inviscid flows in elastodynamics: (cf. [29, 33] for the physical background):
| (1.1) |
where denotes the density, the velocity, is the th column of deformation gradient and for the pressure with a smooth strictly increasing function on We also introduce Mach number where
| (1.2) |
The fluids occupy in 3D space and the interfaces are 2D embedded inside the fluid.
Note that by taking divergence of the third equations in (1.1), we end up with
In column-wise components, we can write the intrinsic property (involution condition for the elastic flow, refer to [26]) as follows:
| (1.3) |
This intrinsic property holds at any time throughout the flow if it is initially satisfied.
1.1. History Review
The study of compressible vortex sheets has a long and rich history, originating from the seminal works of Miles [41, 42] and Fejer-Miles [27]. These early studies established that in the two-dimensional compressible Euler flow, vortex sheets exhibit violent instability when the Mach number , an effect analogous to the Kelvin–Helmholtz instability in incompressible fluids. Later, Artola–Majda [5, 3, 4] investigated the interaction between vortex sheets and highly oscillatory waves, demonstrating that global-in-time nonlinear instability persists for , making global existence results challenging in the multidimensional settings.
A major breakthrough in the mathematical analysis of compressible vortex sheets came from Coulombel and Secchi [24, 25], who employed microlocal analysis and the Nash-Moser iteration technique to establish the local-in-time nonlinear stability of 2D compressible vortex sheets under small perturbations. Their results were restricted to the supersonic regime , relying on stability conditions akin to those used for shock waves by Majda [37, 36] and Coulombel [20, 21]. Extensions of these results to non-isentropic Euler flows [23, 45, 46] further demonstrated how entropy variations influence stability. More recently, research has been carried out to steady three-dimensional compressible vortex sheets [53, 55, 56] and relativistic vortex sheets [12], providing insights into broader applications and mathematical structures.
In three-dimensional flows, the dynamics become substantially more intricate. Miles [42] observed that disturbances propagating at large angles to the undisturbed flow amplify instability, and Serre [50] later demonstrated through normal mode analysis that 3D compressible vortex sheets remain unstable for all Mach numbers, mirroring the Kelvin–Helmholtz instability in incompressible fluids. This suggests that additional physical mechanisms – such as external forces, surface tension, or viscosity – are required to stabilize the interface.
Magnetohydrodynamics (MHD) provides one such stabilizing effect. Chen–Wang [10] and Trakhinin [52] independently proved that non-parallel magnetic fields stabilize compressible current-vortex sheets, a result that was also obtained for 2D MHD current-vortex sheets [54, 44]. Another stabilizing mechanism arises from viscoelastic effects; numerical simulations and theoretical studies [6, 31] suggest that viscoelasticity counteracts vortex sheet instabilities. Huilgol [31, 32] studied vortex sheet formation in viscoelastic fluids, showing that unsteady shearing motions can induce vortex sheet structures, while Hu–Wang [30] investigated singularity formation in viscoelastic flows. The stabilization by surface tension was confirmed in the work of Stevens [51], where the local existence and structural stability for 3D compressible Euler vortex sheets were obtained.
Significant effort has been devoted to understanding the influence of elasticity on vortex sheet stability. Linear and nonlinear stability for 2D compressible elastic vortex sheets has been rigorously established in [14, 15, 16, 17, 18]. More recently, Chen–Huang–Wang–Yuan [19] investigated the weakly linear stability of 3D compressible elastic vortex sheets, deriving necessary and sufficient conditions for stability through spectral analysis and a priori estimates.
1.2. Classical Challenges and Resolutions in 2D
One of the fundamental difficulties in analyzing vortex sheets stems from the characteristic nature of the free boundary, which limits control over the trace of characteristic components [24, 34, 38, 7]. Specifically, the failure of the uniform Kreiss–Lopatinski (UKL) condition leads to a loss of tangential derivatives in estimating the solutions in terms of source terms in the linearized problem [24], making standard approaches insufficient for proving stability results. Additionally, the presence of elasticity complicates the root structure of the Lopatinski determinant, making it difficult to directly apply classical Kreiss symmetrization techniques.
A recent development in overcoming these difficulties was introduced in [14] in the study of 2D rectilinear compressible elastic vortex sheets, where the authors proposed an upper triangularization technique to isolate outgoing modes from the system at all points in the frequency space. This method effectively separates the system into a closed form where the outgoing modes can be proved to be zero, simplifying the analysis to estimating only the incoming modes, which can be derived directly from the Lopatinski determinant. As a result, the linear stability was achieved. This approach was further extended by Chen–Huang–Wang–Yuan [19] to analyze 3D linear stability for rectilinear compressible elastic fluids, providing a crucial step toward the nonlinear dynamics.
Linearizing around a non-constant background state introduces spatially dependent coefficients in the system, leading to additional complications in controlling the behavior of solutions. One convenient way to derive energy-type estimates is to use paradifferential calculus of Bony [8]. A particularly difficult issue in the paralinearization approach is that the Lopatinski determinant may vanish at certain frequencies (called roots). Coulombel [20, 21] and Coulombel–Secchi [24] developed a bicharacteristic extension method to construct weight functions that mitigate this degeneracy. However, this method relies on the assumption that the leading order symbol matrix for the paralinearized system of the non-characteristic form remains diagonalizable along bicharacteristic curves. This assumption fails when the roots coincide with the points where the system cannot be reduced to a non-characteristic form – these points are referred to as poles. This breakdown has been a significant obstacle in applying classical energy methods.
To overcome this, a new approach was designed in [15], based on a refined upper triangularization of the para-linearized system. Instead of relying on bicharacteristic curves, this method constructs weight functions that depend solely on the background state variables. This avoids discrepancies between bicharacteristic extensions and pole distributions, allowing for a more robust stability analysis. A key advantage of this approach is that it provides a framework to ensure improved regularity of the outgoing mode even in the variable-coefficient case, which plays a crucial role in compensating the loss of higher regularity for the characteristic components near the poles. This is particularly important when extending stability results to nonlinear settings, where controlling derivative loss is essential for proving local well-posedness [16].
1.3. New Challenges from Dimension Increase and Resolutions in 3D
In addition to the above challenges, the transition from two-dimensional to fully three-dimensional vortex sheets introduces new analytical and structural challenges. The additional spatial dimension not only increases the degrees of freedom in frequency space but also creates more potential instability directions, making frequency interactions significantly more intricate. Unlike in 2D, where instabilities are constrained to a plane, 3D vortex sheets exhibit a broader range of possible resonance mechanisms. However, in the case of elastic vortex sheets, the additional elasticity components in 3D play a stabilizing role by restricting the growth of unstable perturbations, provided that certain geometric conditions on the deformation gradient are satisfied.
1.3.1. Enhanced ellipticity from elasticity
As is mentioned in Section 1.2, the energy estimates suffer a loss of tangential derivative due to the failure of the UKL condition. Since the wave front appears only in the boundary conditions of the vortex sheet system, one key strategy is to ensure that instabilities can only arise from the traces of solutions to the interior dynamical system rather than from the front symbol. In other words, the boundary conditions for the front need to satisfy certain ellipticity condition.
For 2D compressible Euler vortex sheets, this ellipticity is achieved because the front symbol is homogeneous and does not vanish on the closed hemisphere in the frequency space; see [24, Lemma 4.1]. This property allows for the recovery of one derivative in the regularity estimates of the vortex sheet front. Furthermore, it enables the elimination of the front from the system, reducing the problem to a standard boundary value problem with a symbolic boundary condition, similar to the case of shock waves [37]. The introduction of elasticity preserves the essential algebraic structure, ensuring that the ellipticity condition and subsequent reduction of the system remain valid [14].
However, increasing the spatial dimension introduces additional tangential components, significantly complicating frequency interactions and resonance effects. In particular, for the 3D compressible Euler vortex sheets, the front symbol exhibits degenerate ellipticity along certain frequency directions, which makes it more difficult to control the regularity of the front using standard energy estimates. A key discovery in our analysis is that elasticity provides an ellipticity enhancement mechanism that counteracts this degeneracy. More precisely, we show that if the deformation gradient on the free boundary satisfies the geometric condition
| (1.4) |
where and denote the first two rows of the deformation gradient (see (2.18)), then boundary ellipticity is restored, allowing for the recovery of one derivative for the free interface regularity; see (3.3). Another fundamental difference between the 2D and 3D elastic vortex sheet problems lies in the elimination of the front from the boundary conditions. In two dimensions, there exists a natural projection that removes the front-related terms from the boundary conditions, leaving the remaining system non-singular and ensuring that the normal component of the unknown function can be controlled by its non-characteristic part. In contrast, in 3D, no obvious projection structure is available due to the increased complexity of frequency interactions. Nevertheless, by exploiting the non-parallel structural property of elasticity in (1.4), we construct a suitable projection matrix that facilitates the elimination of the front, thus preserving the essential structure needed for stability estimates; see (3.26).
1.3.2. Resolving higher-order singularities in the Lopatinski condition
As is explained in Section 1.2, the presence of roots of the Lopatinski determinant leads to a loss of derivatives in the energy estimates, while at each pole, the para-linearized system cannot be reduced to a non-characteristic form. In the case of 2D elastic vortex sheets, the method developed in [15] effectively treats the scenario where a simple root coincides with a simple pole.
In 3D, the situation is considerably more complex. Specifically, there exists a co-dimension one set in frequency space where a double pole, arising from the left and right states of the two-phase system, collides with a double root. Since the system cannot be directly transformed into a non-characteristic form, we employ the refined upper triangularization technique of [15] to separate the outgoing modes into a closed form, allowing us to derive improved regularity estimates. These estimates are then used to analyze the coupling between the characteristic part and the outgoing mode, ultimately enabling control of the characteristic components. The final step is to estimate the (boundary trace of) incoming modes in terms of the outgoing modes and source terms.
A crucial factor in determining whether the boundary estimates can be closed is the behavior of the restriction of the boundary symbol to the stable subspace of the linearized system, which corresponds to the Lopatinski matrix . As noted earlier, is not invertible, and direct computation reveals that has a one-dimensional kernel at the roots. This singularity leads to the failure of UKL, resulting in derivative loss. The important work of Coulombel [21] on weakly stable shock waves developed a framework to handle cases where the boundary symbol vanishes at first order at the roots. This technique has been successfully applied to the study of 2D compressible Euler vortex sheets [24], 2D compressible elastic vortex sheets [15, 16], and 2D relativistic vortex sheets [13]. However, for 3D elastic vortex sheets, a double root may appear, leading to a higher-order degeneracy where vanishes at second order at the double root. To resolve this issue, we extend Coulombel’s argument by constructing two invertible mappings and near the double root, which are symbols of type (degree 0 and regularity 2; see Definition 3.1). Under these transformations, the Lopatinski matrix is transformed to
where are real-valued scalar symbols. The second-order vanishing of is captured by the symbol . We show that this construction ensures , which brings the problem into the framework of [21], allowing us to use localized Grding’s inequality to compensate for the loss of derivatives; see Section 3.6. We would like to comment that this diagonalization of into and the associated reduction in the degree of the double roots of the Lopatinski determinant are expected to be useful for other models with similar algebraic structures, such as contact discontinuities in relativistic vortex sheets, non-isentropic Euler equations, and multidimensional shock waves in non-characteristic free boundary problems.
The rest of the paper is organized as follows. In Section 2, we formulate 3D nonlinear problem of vortex sheets, fix the free boundary, linearize the system around a given constant solution, introduce the function spaces and useful lemmas, and state our main result, Theorem 2.1. In Section 3, we introduce the effective linear problem and its formulation with variable coefficients. In Section 4, we prove a well-posedness result of the effective linear problem in the usual Sobolev space with large enough. In Section 5, we transform the original nonlinear problem into the case with zero initial data. We construct approximate solutions to incorporate the initial data into the interior equations. The necessary compatibility conditions are imposed on the initial data for the construction of smooth approximate solutions. Finally, we show the existence and stability results of solutions to the reduced problem and conclude the main result in Section 6 by using Nash-Moser iteration.
2. Formulation, Notations and Main Result
In this section, we will derive the governing dynamics of vortex sheets from the elastic equation (1.1), linearize them around a planar vortex sheet, and state our main result.
2.1. Statement for the Vortex Sheet Problem
Recall the definition of vortex sheet solutions for (1.1). Let be a solution to system (1.1) which is piecewise smooth on the both sides of a smooth hypersurface
Denote for the partial derivatives, normal vector on and
where The solution satisfies the Rankine-Hugoniot jump relations at each point on
| (2.1) |
where we write as the jump of the quantity across the hypersurface For a vortex sheet (contact discontinuity), we require
| (2.2) |
Therefore the jump conditions reduce to
| (2.3) |
It is crucial that in the derivation of Rankine-Hugoniot condition, we need to regard
as an intrinsic property. Therefore, we also have
| (2.4) |
To flatten and fix the free boundary we need to introduce the function to set the variable transformation as follows. We first consider the class of functions such that Then we define
| (2.5) |
for In the following argument, we drop the index for notation simplicity. Define Inspired by [24, 28], it is natural to require satisfying the eikonal equation
| (2.6) |
when , and
| (2.7) |
for some constant .
Through this variable transformation, equations (1.1) become
| (2.8) |
for with free boundary where
| (2.9) |
It is noted that this choice simplify the expression of the nonlinear problem in the fixed domain and guarantees the constant rank property of boundary matrix in the whole domain.
It is obvious that system of conservation laws (1.1) admits trivial vortex sheets solutions consisting of two constant states separated by a planar front as follows:
| (2.10) |
Every planar elastic vortex sheet (namely piecewise-constant vortex sheet) is of this form through the Galilean transformation. For simplicity we assume that for
Then we need to solve the following initial-boundary value problem for in a fixed domain:
| (2.11a) | ||||
| (2.11b) | ||||
| (2.11c) | ||||
where we have dropped the index for convenience, and are given by
| (2.12) | ||||
| (2.13) |
with
By (2.4) and (2.6), we obtain that the boundary matrix of problem (2.11), i.e.,
has constant rank on if and only if
| (2.14) |
In the new variables, (1.3) become
| (2.15) |
where we denote the partial differentials with respect to the lifting function by
| (2.16) |
The following proposition shows that identities (2.14)–(2.15) are involutions for vortex sheet problem (2.6)–(2.11). The proof follows from a straightforward computation and hence is omitted.
2.2. Main Result and Discussion
In the straightened variables, the piecewise constant vortex sheet (2.10) corresponds to the stationary solution of (2.11a)-(2.11c) and (2.14),(2.15) as follows:
| (2.17) |
For we denote
| (2.18) |
From (2.17) we know that . We further define the vector projections (see Fig. 1)
| (2.19) |
In order to prove the nonlinear stability of elastic vortex sheets, we only need to show the existence of solutions to problem (2.6)–(2.11) on account of transform (2.5). The main result of this paper is stated as follows:
Theorem 2.1.
Let and be an integer. Suppose that the background state (2.17) satisfies and the following stability conditions:
| (2.20) |
and
| (2.21) |
where is defined in (3.39). Suppose further that the initial data and satisfy constraints (2.14)–(2.15) and the compatibility conditions up to order (cf. Definition 5.1), and that has a compact support. Then there exists a positive constant such that, if
then problem (2.6)–(2.11) admits a solution on the time interval satisfying
2.3. Functional Spaces
Now we introduce some necessary functional spaces, , weighted Sobolev spaces in preparation for our main theorem. Let denote the distributions and define
for with equivalent norms
respectively, where
We define the norm
with being the Fourier transform of with respect to Setting we see that and are equivalent, denoted by Now, we can define the space endowed with the norm
We also have
It is easy to see that when and ( for simplicity) is the usual norm of We denote when applying it to functions of For multi-index , we define For , we denote .
Moreover,
is introduced with the norm
Similarly, the space and its norm are defined. Furthermore, we abbreviate to , which is equipped with the norm
and .
We use the following notation: () if () holds uniformly for some positive constant that is independent of
In the following, we present the Moser-type calculus inequalities in weighted Sobolev spaces, which will be used in proving the higher-order tame estimates and convergence of the Nash–Moser iterative scheme.
Lemma 2.1.
Let , , , and . Let denote a –function defined in a neighborhood of the origin.
(a) If and , then
| (2.22) | ||||
| (2.23) |
(b) If , then
| (2.24) |
Moreover, if , then
| (2.25) |
Here (for ) are multi-indices, represents the standard commutator, and the increasing function is independent of , , , and . The same conclusions remain valid when is replaced by .
3. Variable Coefficient Linearized Problem
In this section we introduce the effective linear problem and its formulation with variable coefficients. We first write (2.11a) as
| (3.1) |
for Using Rankine-Hugoniot conditions, we derive that
| (3.2) |
where at We also have that
Now, we consider the following background states:
| (3.3) |
where and represent the states and changes of variables on each side of vortex sheets separately. are constants. and are functions which denotes the perturbation around the constant states. We assume the perturbation of the background states satisfying
| (3.4) |
Here is a positive constant. and have compact support in the domain
We also require the perturbed states (3.3) to satisfy the Rankine-Hugoniot conditions:
| (3.5) |
on where
Now, we assume that the following conditions on the perturbed states (3.3) holds:
| (3.6) |
| (3.7) |
for all and some positive constant We also assume from initial data that
| (3.8) |
Now, we linearize the (3.1) around the basic states (3.3) and denote by the perturbation of the states Then, the linearized equations are
for We define the first-order linear operator
and introduce the Alinhac’s“good unknown” [1]:
Then, we can rewrite the above equations as
where
Neglecting the zero-th order terms of and considering the following equations:
Since we have and Now, we linearize the boundary conditions around the same perturbed states and obtain that
at We also have that
We write the above system of equations into the following enlarged form:
where
Using the Alinhac’s “good unknown” [1], we obtain that
| (3.9) |
Therefore, we have the following linearized problem:
| (3.10) |
It is noted that the boundary condition in (3.10) does not contain the tangential components of We write the components of that are contained in (3.10) by as
| (3.11) |
where
We obtain the main theorem for the variable coefficients.
Theorem 3.1.
Suppose that the background solution defined by (3.3) satisfies and
| (3.12) |
and
| (3.13) |
where is defined in (3.39); moreover, the perturbation and have compact support, and in (3.4) is small enough. Then, there are two constants and which are determined by particular solution, such that for all and and all the following estimate holds:
where
3.1. Reduction of the System
In this section, we will transform the system (3.10) into an ODEs. This is achieved through linear transformation on the unknown variables and a paralinearization on the equations of the transformed variables. For the system (3.10), we can find the symmetrizer
We multiply to the interior equations of (3.10) and then integrate by parts to obtain the following Lemma 3.1.
Lemma 3.1.
There are two positive constants and such that for any the following estimate holds:
First, we transform the linearized problem (3.10) into a problem with a constant and diagonal boundary matrix. This is essential since the boundary matrix has constant rank on the whole half-plane
We consider the following transformation
where Then, we can obtain
Multiplying the above by the following matrix:
| (3.14) |
we obtain that
| (3.15) |
The interior equations of (3.10) for the new unknowns are
| (3.16) |
where
| (3.17) |
We consider the weighted unknown and rewrite (3.16) as
where and Then, we obtain the equivalent form of the boundary condition (3.10):
| (3.18) |
Note that and . It follows that
where is the first column of Then, we have and and
We first denote the “noncharacteristic” components of which will be used in Section 3.3:
| (3.19) |
Then, we denote the normal components of by checking the matrix coefficient in front of in the boundary conditions,
| (3.20) |
Thus, we write the boundary conditions
By defining and the system (3.18) can be re-written as
| (3.21) |
We are going to perform the paralinearization of the interior equations and the boundary conditions for (3.21).
3.2. Some Results on Paradifferential Calculus
For the sake of self-containedness, we present some necessary definitions and results concerning paradifferential calculus with parameters, as utilized in this paper. For rigourous proofs, refer to [7, Appendix C] and the references therein.
Definition 3.1.
Let and we define the following notions:
-
(i)
A function is called a paradifferential symbol of degree and regularity if is in and satisfies
for all where and is a constant.
-
(ii)
The set of paradifferential symbols of degree and regularity is denoted by
-
(iii)
A family of operators is said to be of order if, for all and there exists a constant such that
for all A generic family of such operators is denoted by
-
(iv)
For define the operator by
for all (the Schwartz class).
-
(v)
To any symbol we associate a family of paradifferential operators defined by
where is the inverse Fourier transform of The function is defined as
where and being a -function on satisfying
We have the following properties for the paradifferential calculus:
Lemma 3.2.
The following statements hold:
-
(i)
If and then
-
(ii)
If and then
-
(iii)
If then is of order In particular, if and is independent of then
-
(iv)
If and then the family is of order and the family is of order
-
(v)
If and then is of order
-
(vi)
Grding’s inequality: If is a square matrix symbol satisfying
for some constant then there exists such that
-
(vii)
Microlocalized Grding’s inequality: Let be a square matrix symbol and If there exist a scalar real symbol and a constant such that and
for all then there exist and such that
for all Here, for any complex square matrix denotes its conjugate transpose.
Similar to the constant coefficient case, the key idea in proving Theorem 3.1 is to transform the variable-coefficient linear problem (3.21) into an ODE. However, instead of applying a Fourier transform as in the constant-coefficient case, we employ paralinearization for (3.21). In the following section, we derive the para-linearized form of (3.21) and estimate the errors introduced by replacing the original system with its para-linearized counterpart.
3.3. Paralinearization
For the frequency space defined in the variable coefficient case:
where represents the Fourier variables with respect to separately. Using the homogeneity structure of the system, we will focus on the following unit hemisphere in the frequency space
This argument will later be extended to the entire frequency space For simplicity, we omit the tilde notation in the system. We denote
Using Lemma 3.2 (i)-(iii), we perform paralinearization and obtain that
where are positive constants. Then, we consider the coefficients of
We denote and
where are some positive constants. Combing the above estimates, we have
| (3.22) |
Next, for the interior differential equations, using Lemma 3.2 (i)-(ii), we have
Similar estimates also holds for . Hence, we obtain that
| (3.23) |
Now, we start to derive the specific expression for the paralinearized system. We write
Using the fact that we have
| (3.24) |
for some positive constant Then, using Grding’s inequality (Lemma 3.2 (vi)), we obtain that
for all where depends only on Using the properties of paradifferential operators (Lemma 3.2 (iv)), we obtain that where is an operator of order 1, then we have
for all Using (3.22) and Lemma 3.2, we have
| (3.25) |
Now, we want to analyze the part of the boundary condition where the front function is not involved. We write
It worth to point out that and could equal to 0 for 3D elastic flow, due to the possible frequency interactions. However, in 2D elastic flow, it is natural to obtain the ellipticity for the front symbol and no such kind of degeneracy of and happen. Now, we define the following orthogonal projector matrix:
| (3.26) |
for , and extend it as a homogeneous mapping of degree of 0 with respect to It is easy to check that
Since at least one of the is non-zero as long as by using the fact that . Without loss of generality, we can assume then we can define
such that where is a fixed positive constant and is a given function. For the analysis is similar and omitted it. Simple calculation yields that
for . It is easily seen that is homogenous of degree with respect to
We denote the last six rows in by Then we can separate the . We have
where an invertible matrix defined in the whole domain and is homogeneous of degree 0. For simplicity, we write
We can check that
where is the conjugate transpose of .
Then, using (3.3), we have
Using Lemma 3.2 and (3.22), we can also obtain the estimate for the front function one has
Taking sufficiently large, we obtain that
| (3.28) |
Then, collecting (3.25) and (3.28), we obtain that
Therefore, we obtain the estimate of and by estimating the source terms and the non-characteristic components
Denote that
| (3.29) |
Now, we need to estimate we need to use the other part of boundary conditions,
where
| (3.30) |
It is homogeneous of degree 0 with respect to We can define the following para-linearized system as
| (3.31) |
We remain to prove the following estimate:
| (3.32) |
First, we have
where we have denotes the first two rows of Then, we have
Moreover,
Therefore, from (3.32), one has
3.4. Microlocalization
For simplicity, we concentrate our analysis on the unit hemisphere
3.4.1. Poles
Considering the following differential equation:
| (3.33) |
where are defined in (3.14), (3.15), (3.17), and (3.30). Denote
| (3.34) |
Now, we consider the algebraic equations for , where
| (3.35) |
where
Similar equations also hold for Similar to the constant coefficient case, if we can solve by Then, using differential equations (3.33), we can obtain the differential equations only involve and
where
Similar arguments hold for Then, for the points in frequency space that cannot reduce the system into the non-characteristic form are the points:
| (3.36) |
where and are defined in (3.29) and (3.34) respectively. These points are exactly the poles of the system (3.33) and we denote
3.4.2. Roots of the Lopatinski determinant
Now, we derive the Lopatinski determinant. We write the eigenvalue of with a real negative real part by which satisfies
The corresponding eigenvector is
where
The case is similar for Denote the eigenvalue of with a real negative real part by which satisfies
The corresponding eigenvector is
where
The above eigenvalues and eigenvectors are well-defined and smooth on the whole space using similar calculation in the constant coefficient case. Hence, the Lopatinski determinant given below is well-defined for all the points in the frequency space,
| (3.37) |
It is homogeneous of degree 0 with respect to where
It is noted that the last two factors in (3.37) are never equal to 0, and the first two factors correspond to the roots respectively. Therefore, we need to discuss the roots of the third and fourth factors. All the coefficients in the factors of the Lopatinski determinant are continuous with respect to the background state and These factors reduce to the corresponding factors in the constant coefficient case, if the perturbation in (3.3) is zero. Assuming that in (3.4) is small enough and using a continuity argument, we obtain that the number of the roots in the third and fourth factors in (3.37) is the same as the number of the roots in the corresponding factors of the constant coefficient case. Hence, there are two roots in the third factors and we denote and
Remark 3.1.
The terms on the right-hand side of , , is small under the small perturbation assumption. Same argument holds for
It is easy to see that and are real and depend continuously on the background states and The roots of the Lopatinski determinants can be written by the following set according to the information provided by the boundary:
| (3.38) |
where
on We can extend the set and defined by the data of the boundary into the interior of the domain The coefficients in for can be defined by continuity of and on the background state and Denote the extended set
Similar to the poles, the roots of can be viewed as the strip in the frequency space parameterized by It originates from the boundary and propagates into the interior domain Moreover, we need that the roots of the eigenvalues and do not coincide with the poles and the roots of the Lopatinski determinant.
For simplicity, we write
and define
Now, we consider the following frequency sets:
-
(1)
-
(2)
-
(3)
.
Here, denotes the roots of the first factor in and the roots of the first two factors of in ; represents the last two factors of in ; and represents the roots of second factors in
Note that and As mentioned before, we write
which clearly does not intersect with and , i.e.,
Under the assumption (2.21), where
| (3.39) |
we obtain that
Notice that
We can introduce
Then, one has
From this it follows that
Moreover, the condition
guarantees that
Note that
holds with no restriction on the background solutions. Hence, except for the special case in which there are always interaction between the poles (roots), any of rest of two strips in do not intersect with each other in the whole domain unless they are identical.
3.5. Estimates in Each Case
In this section, we derive the estimates for each case and obtain the desired estimate for the paralinearized system. The relation among corresponds to a strip on with fixed Now, we focus on the situation where theses strips do not intersect with each other and construct neighbourhoods around them, except for Case 1, in which the poles and roots always intersect.
Up to shrinking these neighborhood, these neighborhood do not intersect with each other and do not contain any point in . Denote, on
Remark 3.2.
Due to the the stability condition imposed on the background solutions for non-parallel elastic deformation gradients, certain neighborhoods remain disjoint. For example, the neighborhood of , denoted as , cannot intersect with the neighborhood of , denoted as . This separation indicates the stabilizing effect of elasticity, which is consistent with the linear analysis of constant coefficients, see [19].
3.6. Case 1: Points in
We consider the kind of frequencies that are both poles and roots of the Lopatinski determinant. Consider as an example, since the other cases can be discussed similarly.
Different from 2D case, not only contains the poles of the equations , but also contains the poles for in (3.33). Hence we derive the estimates for . The estimates for will follow the same way. Introducing the smooth cut-off function whose range is On the support of is contained in and equals to on a smaller neighborhood of the strip satisfying We can extend by homogeneity of degree 0 with respect to into the whole domain . We know that for any integer Define
From (3.33), we have
Then
| (3.40) |
where and whose support is where . Consider two cut-off functions and in such that both of the supports are in
where on and on Now we multiply (3.40) by and obtain that
| (3.41) |
Here the support of is in which is the subset of Now we can uppertriangularize the first symbol on Define the transformation matrix on
Thus is homogeneous of degree with respect to The third and fourth columns of the third row are from the eigenvector , and , are chosen to make the third column of zero except for the third and fourth rows.
It is noted that , can be solved at all points in In the following, we introduce to exclude the frequencies at which degenerates. To ensure the invertibility of , we can take for simplicity. So In order to uppertriangularize the first order operator in we need to construct in
where which equals the determinant of Here are defined to be homogenous of degree 0. Then we have
It follows that We can finally get the first-order symbol for the uppertriangularization.
where
Here, whose exact expression is not important for our analysis.
The following Lemma 3.3, which plays a key role in paradifferential calculus, can be proved similarly to the approach in [20].
Lemma 3.3.
With the appropriate choice of and in there is a symbol in satisfying such that
is a symbol in on moreover,
where is defined as
for any symbols and
Now we set
Define and we obtain
is supported where and it is disjoint with the support of Then, from asymptotic expansion of the symbols, we have
Using (3.41), we have
Then, we obtain that
From Lemma 3.2 and Lemma 3.3, we know
| (3.42) |
Since the support of the Fourier transform of is in the support of we have
| (3.43) |
where is the same as except replacing in each element by It equals to on the support of is an extension of with to the whole space. Moreover, we see that in , where is defined as
| (3.44) |
This suggests that we can extend as to the whole space. For simplicity, we will write instead of in later arguments.
Denote . From the fourth equation in (3.43) we find
where
Consider two symmetrizers and where is defined in by (3.38) and extended to by homogeneity of degree 1. Thus , and
| (3.45) |
For the first term on the right-hand side of (3.45), using Lemma 3.2, we have
From the extension of we obtain that
for some positive depending on the background states. Using Grding’s inequality (Lemma 3.2(vi)), we have
Using Lemma 3.2 (iii)-(iv), for the rest of the terms on the right-hand side of (3.45),
where is taken to be small enough.
Note that
| (3.46) |
For the first three terms on the right-hand side of (3.46),
The terms can be treated similarly. Summing up (3.45) and (3.46) and integrating with respect to we have
Using the fact that
we can obtain that
| (3.47) |
In the following we apply the second symmetrizer to have
Hence, we get
| (3.48) |
Then, we consider 1st, 2nd, 5th, 6th, 8th, 9th, 11th, 12th of the system (3.43) which can be written as
| (3.49) |
where is an matrix symbol and belongs to is an matrix symbol and belongs to and is an matrix symbol given as follows
Here, we denote that
Now, we apply the symbol to (3.49) with the adjoint of Hence, we obtain that
where and From the definition of cut-off function, is non-zero in the support of Hence, we have
Since we have
Here, the support of is disjoint with ’s. holds at the frequency points where (also possibly ) in the support of We can write and hence
| (3.50) |
Applying the symmetrizer we obtain that
| (3.51) |
Now, using Lemma 3.2 (iii)-(iv), we estimate the above terms one by one,
For the first term on the left-hand side of (3.51), we obtain that
Then, we have
It is note that
| (3.52) |
For the second term on the right-hand side of (3.52), we have
The first term on the right-hand side of (3.52) can be written into
| (3.53) |
The first and third terms can be estimated by Cauchy-Schwartz inequality, for the second term
| (3.54) |
where is the extension of to the whole space with for some fixed positive constant For the second term on the right-hand side of (3.54),
where and are only supported on the support of which is disjoint with the support of Hence, we obtain that
Then, we obtain that
| (3.55) |
Note that we have
| (3.56) |
| (3.57) |
for
Similar to the outgoing mode we apply symmetrizer to (3.50),
In the above, the first term can be written as
For we can split into two terms:
We can estimate that
Thus we obtain
| (3.58) |
for by taking small enough.
For in (3.43), we have
Following the same estimates for with we have
| (3.59) |
and
| (3.60) |
for . Note from (3.57)–(3.58) and (3.59)–(3.60) that the estimates for the terms with are exactly the same.
For the incoming mode of (3.43),
| (3.61) |
First, we apply symmetrizer to (3.61),
Similar to the case for the outgoing modes, we obtain
Then, we have
| (3.62) |
Now, we apply symmetrizer to obtain that
It is noted that
and
We can obtain that
Hence it follows that
| (3.63) |
Considering (3.47),(3.48),(3.57),(3.58),(3.62),(3.63), dividing them by the appropriate power of we obtain that
For ,
Summing up the above estimates and taking sufficiently large, we have for and ,
| (3.64) |
We remark that the extra degree of freedom can cause complicated interaction between the poles of and Hence, is also the pole of the differential equation for in (3.33), as long as This is a key point in 3D analysis, since it is possible that the poles for the two equations coincide. In a similar way as before, we obtain for and that
| (3.65) |
where and is the transformation matrix for which is defined in a similarly way as for We write and for the outgoing modes and and for the incoming modes. Then we have
So the last step is to use the boundary conditions in (3.31) to estimate the terms , , , and . Notice that
Therefore we only need to estimate the boundary terms and . The goal is to use the boundary conditions (3.31) to prove the following estimate:
| (3.66) |
Let us rewrite the Lopatinski matrix as
| (3.67) |
We calculate its determinant at to satisfy
in a neighborhood of and in a neighborhood of Similar to the constant-coefficient case [19, Lemma 3.6], let us assume without loss of generality that .
Define the following matrices in a suitably small neighborhood of :
| (3.68) |
It is easily seen that Shrinking further if necessary, we have
| (3.69) |
with being some real-valued scalar symbols, whose explicit forms are not important for our analysis. We now fix the four cut-off functions and such that
Following the argument of [20, Section 3.4.3], we can obtain the following estimate by using the localized Grding’s inequality:
| (3.70) |
Now, we use the special structure of to obtain a lower bound for the term on the left-hand side of (3.70). Define
| (3.71) |
From (3.69), we have
| (3.72) |
where and is a scalar real symbol in . Applying the localized Grding’s inequality (see Lemma 3.2 (vii)), we have
| (3.73) |
for sufficiently large Similarly, we can obtain that, for sufficiently large
Inserting the above two estimates into (3.72), we have
| (3.74) |
Using the fact that we obtain that
Then, we have
Using the ellipticity of on the support of and that , we apply the localized Grding’s inequality (Lemma 3.2 (vii)) to obtain that, for sufficiently large
Then, for sufficiently large we have
| (3.75) |
Similarly, we can prove
| (3.76) |
Combining (3.70),(3.74)-(3.76), and taking sufficiently large, we derive (3.66), which is crucial.
3.7. Case 2: Points in
We need to estimate the part of corresponding to For simplicity, we discuss the differential equations for in The remaining neighborhood of and the discussion for we obtain the same estimates. Now consider the cut-off function in for any integer whose support on is contained in and is equal to in a smaller neighborhood of the strip where Denote
Hence, we obtain that
Here, is in , bounded and supported only in the set where Then, we take two cut-off functions and in the class for any integer Both of the functions are supported in on the support of and on the support of Similar to the previous discussion, after applying the cut-off symbol, we can find transformation matrices and and symmetrizers and to obtain
where is the same as in (3.42) in previous case, and are invertible symbols in and Define
After same argument for and we obtain that
| (3.79) |
In we have and in we have . Denote
It follows that
Applying the two symmetrizers and , we have
The roots of the Lopatinski determinant do not coincide with the poles of the differential equations. Thus we can estimate for in the same way. Multiplying (3.79) with some appropriate choosen matrix symbol in we obtain that
| (3.80) |
Here, on the support of We can extend into a new symbol satisfying Applying the two symmetrizers and we have
Combining the above estimates and dividing by to an appropriate power and then taking large enough, we obtain that
| (3.81) |
For we have
| (3.82) |
Similar to Case 1, the boundary terms in (3.33) can be used to estimate and and and Using (3.81) and (3.82), we have
| (3.83) |
where
3.8. Case 3: Points in
In this section, we discuss the poles that are not the roots of Lopatinski determinant. Our discussion focuses on the neighborhoods and As an example, consider . This neighborhood contains a strip in the frequency space where Here represents a pole of the differential equations for but not for In this case, the equation for can be reduced to a non-characteristic one. The arguments apply to both and , provided that the points where are excluded from the neighborhood. For simplicity, we restrict our discussion to the case of First, we introduce the cut-off functions and which are defined in the pole case. Using the matrices and and symmetrizers and along with appropriate adjustments to and , we derive the following equation:
| (3.84) |
where
The symbols in this equation are consistent with those introduced in the previous cases. The equation for yields
Consider the symmetrizer we obtain that
Taking small enough, we have
For in (3.84), applying the symmetrizer we have
For the incoming mode we take the symmetrizer
Combining all the estimates above, we obtain that
For , we obtain that
Now, we estimate and . These terms can be controlled by Using the boundary conditions (3.33) and using the fact that the Lopatinski determinant has a positive lower bound in the open neighourhood , we have
Putting together, we have
| (3.85) |
where
3.9. Other Case
The remaining points are those where the Lopatinski determinant is non-zero, allowing the system to be reduced into a non-characteristic form. In this case, a Kreiss’s symmetrizer can be constructed. This corresponds to the good frequency case in [24]. Consider the cut-off symbol in for any integer where is the sum of four cut-off functions for four neighborhood is the sum of two cut-off functions for two neighborhood and and is the sum of two cut-off functions for the two neighborhood is also the cut-off function which is 0 near the roots of the Lopatinski determinant and We can construct an open neighborhood that contains the support of but does not contain a small neighborhood of and Denote that
Following the approach in [24], we can eliminate all components of in the kernel of This leads to a differential equation for of the form
| (3.86) |
where and which are supported in the place where Using (3.31), similar to [20, 24], we have the following estimate
| (3.87) |
3.10. Proof of Theorem 3.1
Proof.
We now summarize all the estimates from the four cases discussed above. Taking sufficiently large and summing up (3.78), (3.83), (3.85),(3.87),we have that the left-hand side of the sum is bounded by
The support of is contained in the following set:
Note that . Then when , or equals We also have that vanishes only at some points where or Thus has a lower bound on the support of and we write
where all have block diagonal structures in Let and denote the corresponding cut-off functions in Case 1 and 2. So the term can be absorbed by
This term can also be controlled by the left-hand side of the sum of (3.78),(3.83),(3.85),(3.87). Hence it follows that
which completes the proof of Theorem 3.1. ∎
4. Well-posedness of the Linearized Problem
In this section we analyze the linearized problem for (2.11) and establish the well-posedness of solutions in the standard Sobolev spaces for all integers .
4.1. Variable Coefficient Linearized Problem
We begin by linearizing problem (2.11) around a given basic state . We suppose that
| (4.1) | ||||
| (4.2) |
for and , where and are positive constants, and is the background state given by (2.17). Moreover, we assume the basic state satisfies (2.6), (2.11b), and (2.14), i.e.,
| (4.3a) | |||||
| (4.3b) | |||||
| (4.3c) | |||||
| (4.3d) | |||||
| (4.3e) | |||||
for some constant . Constraints (4.3b) and (4.3c) ensure that the rank of the boundary matrix for the linearized problem remains constant on the domain . Denote , , , and for simplicity.
We need to absorb (4.3c) into (4.3e) and write the boundary into an enlarged form and later will analyze separately for the Nash-Moser iteration. The linearized operators can be defined as follows:
| (4.4) | ||||
| (4.5) |
where , and , and are defined separately by
| (4.6) | ||||
| (4.7) |
and
| (4.8) |
Motivated by Alinhac [1], we get
where are the “good unknowns”:
| (4.9) |
We now consider effective linear system:
| (4.10a) | |||||
| (4.10b) | |||||
| (4.10c) | |||||
where , , , and are given in (2.12), (4.6), (4.7) and (4.1), separately, , and
| (4.11) |
Here, are two smooth functions of that vanish at the origin, is a smooth function of trace while is a smooth vector-function of , which also vanishes at the origin. Additionally, the matrix is a smooth matrix function of . It is important to note that the boundary condition (4.10b) depends on the traces of solely through , where
| (4.12) |
To transform the linearized problem (4.10) into a form with a constant diagonal boundary matrix, we introduce the following matrices:
| (4.13) |
and
| (4.14) |
where and is the sound speed given in (1.2). Then it follows from constraints (4.3b) and (4.3c) that
Using the new variables
| (4.15) |
the problem (4.10) can be equivalently reformulated as
| (4.16a) | |||||
| (4.16b) | |||||
| (4.16c) | |||||
where
In (4.16b), the coefficients and are defined by (4.7) and (4.11) respectively. The matrix is given by
| (4.17) |
and represents the non-characteristic part of with . It is evident that and are smooth functions of , are smooth matrix-functions of , and is a smooth matrix-function of .
We are now prepared to state the following theorem. The proof of the theorem will comprise the rest of the section.
Theorem 4.1.
Let and with being fixed. Suppose that the background state (2.17) satisfies (3.12) and (3.13), and that belong to for all , and satisfy (4.1)–(4.3) and
| (4.18) |
Suppose further that the source terms vanish in the past. Then there exist constants and such that, if and , the problem (4.10) has a unique solution vanishing in the past and satisfying the tame estimates
| (4.19) |
4.2. Well-Posedness in
We recall the following a priori energy estimate for the linearized problem (4.10), as derived in Theorem 3.1.
Theorem 4.2.
System (4.10a) is symmetrizable hyperbolic, with coefficients satisfying the regularity assumptions by Coulombel [22]. Consequently, it is necessary to construct a dual problem that satisfies an appropriate energy estimate. To this end, we define
We use (4.3b) and (4.3c) to calculate
where is given in (4.1). Following [39, Section 3.2], we define the dual problem for (4.10) as:
where , are given in (4.7), (4.11) respectively. , and are defined as follows:
and the symbol represents the divergence operator in . are the adjoints of , respectively.
Using the same analysis as in [25, Section 3.4], we can obtain the well-posedness of the linearized problem (4.10) in .
Theorem 4.3.
For the reformulated problem (4.16), Theorem 4.3 implies that
| (4.22) |
For any nonnegative integer , a generic smooth matrix-valued function of is denoted by , and we write to denote such a function that vanishes at the origin. For example, the equations for in (4.10a) can be rewritten as:
| (4.23) |
The precise forms of and may vary from line to line.
4.3. Tangential Derivatives
The following lemma provides an estimate for the tangential derivatives:
Lemma 4.1.
Proof.
We will follow the approach of [25, Proposition 1] to consider the enlarged system. However, for estimating the source terms, we will use the Moser-type calculus inequalities (2.22)–(2.25) instead of the Gagliardo–Nirenberg and Hölder inequalities used in [25, Proposition 1].
Let with . Let with so that is a tangential derivative satisfying . We then apply the operator to (4.16a) to obtain
| (4.25) |
where
Similarly, from (4.16b) we have
| (4.26) |
where
Since the terms involving tangential derivatives of order in (4.25) do not solely contain , as in [25, Proposition 1], we write an enlarged system that accounts for all the tangential derivatives of order . This allows us to apply the a priori estimate of Theorem 4.2. It is important to note that the last term on the left-hand side of (4.25) cannot be treated simply as source terms due to the loss of derivatives in (4.20). We define
and from (4.25)–(4.26), we obtain the following system:
| (4.27a) | |||
| (4.27b) | |||
where , , and are block diagonal matrices with blocks , , and , respectively. Matrices belong to . The source terms and consist of and for all with , respectively. The enlarged problem (4.27) satisfies an energy estimate similar to (4.22), i.e.,
| (4.28) |
Now, by using Moser-type calculus inequalities (2.22)–(2.25), we estimate the source terms and First, we have from definition,
For , we obtain
| (4.29) |
Applying Moser-type calculus inequality (2.22) yields that
| (4.30) |
where with . Moreover, we have
Combining (4.29) and (4.30), we have
| (4.31) |
For with , similar to (4.31), we use (2.22) to derive
| (4.32) |
Combining (4.3), (4.31), and (4.32) leads to
| (4.33) |
4.4. Normal Derivatives of the Noncharacteristic Variables
Following [16], we compensate for the loss of normal derivatives by utilizing the estimates of the linearized divergences and vorticities. From equation (4.16a), we obtain:
| (4.35) |
This leads to
It follows from (2.22)–(2.23) that
and
It is easy to check that
Using the estimate above, we get
| (4.36) |
Next, we introduce the linearized divergences and vorticities, whose estimates allow us to recover the normal derivatives of the characteristic variables
| (4.37) |
according to the transformation given in (4.15).
4.5. Divergences
Inspired by the involutions in (2.15), we introduce the linearized divergences for as follows:
| (4.38) |
where the partial derivatives (with ) are defined in (2.16). We now present the following estimates for and .
Lemma 4.2 (Estimates for the divergences).
Proof.
The equations for in (4.10a) can be written as
| (4.40) |
Using equations (4.23) and (4.40), we apply the operator and use
to obtain that
| (4.41) |
Applying operator with to (4.41) yields
Multiplying the last identity by and integrating over , we have
| (4.42) |
for sufficiently large, where we have used the constraints (4.3b) and
Using Moser-type calculus inequality (2.24), we obtain
| (4.43) |
4.6. Vorticities
The linearized vorticities for the velocities and the linearized vorticities for the columns of the deformation gradient, are defined as follows:
| (4.45) | |||
| (4.46) | |||
| (4.47) | |||
| (4.48) | |||
| (4.49) | |||
| (4.50) |
for . The estimates of , for are provided by the following lemma.
Lemma 4.3 (Estimates for the vorticities).
Proof.
The equations for and in (4.10a) are given by
| (4.52) |
which leads to the transport equation
| (4.53) |
and similarly form (4.40), we have
| (4.54) |
Next, apply the operator with to (4.53) (resp. (4.54)) and multiply the resulting identity by (resp. ) to obtain
| (4.55) |
It follows from the constraints in (4.3c) that
We now integrate the identity (4.55) over and perform a similar analysis as for in Lemma 4.2 to obtain the desired estimates (4.51). The proof of the lemma is thus complete. ∎
4.7. Proof of Theorem 4.1
Thanks to Lemmas 4.2 and 4.3, we can derive the estimates for the normal derivative of characteristic variables defined by (4.37). More precisely, in view of (4.37), (4.45) and (4.46), and (2.16), we obtain
which implies that
| (4.56) |
Similarly, it follows from (4.38) and (4.48) that
| (4.57) |
for . Using identities (4.56)–(4.57), we apply Moser-type calculus inequalities (2.22)–(2.25) and use (4.36), (4.39), and (4.51) to obtain that
| (4.58) |
holds for .
Using identities (4.35), (4.56), and (4.57), we can combine estimates (4.38) and (4.51) to prove (4.58) by finite induction in . Since
we utilize (4.24) and (4.58) to obtain
| (4.59) |
for sufficiently large.
Theorem 4.3 establishes the well-posedness of the effective linear problem (4.10) for the source terms vanishing in the past. Building on the results in [47, 9], we can use the tame estimate (4.59) to reformulate Theorem 4.3 as a well-posdness statement for (4.10) in . Specifically, as shown in Theorem 4.1, there exists a unique solution , which vanishes in the past and satisfies (4.59) for all .
5. Compatibility Conditions and Approximate Solutions
To apply Theorem 4.1 in the general setting, we follow the approach in [25], and transform the original nonlinear problem (2.6)–(2.11) into a form with zero initial data. To achieve this, we introduce approximate solutions that incorporate the initial data into the interior equations. The construction of smooth approximate solutions imposes necessary compatibility conditions on the initial data.
5.1. Compatibility Conditions
Let with . Assume that the initial data satisfy and , and has the following compact support,
| (5.1) |
Using the trace theorem, we can construct satisfying
| (5.2) | |||
| (5.3) |
Define , which represents the initial data for the problem (2.6),
| (5.4) |
By (5.3) and the Sobolev embedding theorem, we have
| (5.5) |
for sufficiently small in .
Let the perturbation be denoted by , and define the traces of the -th order time derivatives at as follows:
| (5.6) |
Note that and .
Let , then the first equation in (2.6) and the equation (2.11a) can be written as
| (5.7) |
where and are two functions vanishing at the origin. Next, we apply to (5.7), take the initial traces, and use the generalized Faà di Bruno’s formula (see [43, Theorem 2.1]) to obtain
| (5.8) | ||||
| (5.9) |
where denote the traces . This leads to the following lemma (cf. [39, Lemma 4.2.1]).
Lemma 5.1.
To guarantee the smoothness of the approximate solutions, the initial data must satisfy the following compatibility conditions.
5.2. Approximate Solutions
Following the approach in [25], we now introduce approximate solutions that satisfy the problem (2.6)–(2.11) in the sense of Taylor’s expansions at .
Lemma 5.2.
Let with . Assume that and satisfy (5.1), and that initial data and are compatible up to order . If and are sufficiently small, then there exist functions , , and such that , , , and
| (5.14a) | |||||
| (5.14b) | |||||
| (5.14c) | |||||
| (5.14d) | |||||
| (5.14e) | |||||
| (5.14f) | |||||
Moreover, we have
| (5.15) | ||||
| (5.16) |
where we write as a generic function that tends to zero as its argument tends to zero.
Proof.
The proof is divided into four steps.
Step 1. First we consider and such that the following conditions are satisfied:
where , , and are constructed in Lemma 5.1. Utilizing the compatibility conditions (5.12)–(5.13), we apply the lifting result from [35, Theorem 2.3] to select and such that
and
Furthermore, , and can be chosen to satisfy (5.15), since have a compact support.
Step 2. Now, we define
Thus, we deduce that the functions satisfy (5.15), and (5.14b), (5.14d), and (5.14e) hold.
Step 3. Since and are already specified, we take , for , as the unique solution of the transport equation
| (5.17) |
supplemented with the initial data:
| (5.18) |
From equations (5.11) and (5.18), it follows that the constraints (5.14f) are satisfied at the initial time. Consequently, as in the proof of Proposition 2.1, we deduce that (5.14f) holds for all .
We define and for simplicity. The vector constructed in Lemma 5.2 serves as an approximate solution to (2.6)–(2.11). From (5.14d) and (5.15), it is clear that is supported within the region . Applying (5.16) and the Sobolev embedding theorem yields the following estimate:
for any integer .
Next, we rewrite the system (2.6)–(2.11) as a problem with zero initial data. Define the function as follows: for , and for . Thus, and , as implied by (5.14a), (5.15), and . Using Moser-type calculus inequalities and (5.16), we obtain:
| (5.19) |
Finally, based on (5.14), the solution to the original problem (2.6)–(2.11) on is expressed as , where and solve the following problem:
| (5.20) |
Thus, solving the problem (5.20) on completes the problem.
6. Nash–Moser Iteration
In this section, we analyze the problem (5.20) through a suitable modification of the Nash–Moser iteration scheme. First, we outline the iterative scheme for problem (5.20) and present the corresponding inductive hypothesis. We then complete the proof of Theorem 2.1 by demonstrating that the inductive hypothesis holds for all integers. It is worth noting that this section follows closely the standard procedure outlined in [25, 16], also see [2, 12, 48].
6.1. Iterative Scheme
We start by recalling the following result from [25, Proposition 4].
Proposition 6.1.
Let , , and with . Then there exists a family of smoothing operators such that
where for . These operators satisfy the following estimates:
| (6.1a) | |||||
| (6.1b) | |||||
| (6.1c) | |||||
and
| (6.2) |
where and are integers, and . In particular, if on , then on . Furthermore, smoothing operators can also be constructed for functions defined on (denoted by for simplicity), which satisfy the inequalities (6.1), with norms .
The following lemma establishes a lifting operator that will be used in constructing the iterative scheme and the modified state (see [28, Chapter 5] and [25] for the proof).
Lemma 6.1.
Let , , and . Then there exists a continuous operator mapping to satisfying when for all .
Following [25, 12], we now describe the iteration scheme for problem (5.20). Let be a given integer.
We begin by setting and assume that is given and satisfies
| (6.3) |
Next, we define the iteration scheme:
| (6.4) |
where the increments , , and are determined by the problem
| (6.5) |
Here, the operators and are defined in (4.10a) and (4.10b), respectively. The pair represents a modified state such that satisfies constraints (4.2)–(4.3). The source term will be determined later on. For a detailed construction of the modified state, see Section 6.3. Following (4.9), we write
| (6.6) |
Then, we set and for sufficiently large . Let be given and vanish in the past for . The terms and are determined by the equations
| (6.7) |
where
| (6.8) |
Here, are the smoothing operators from Proposition 6.1, with defined as
| (6.9) |
As a consequence, we can apply Theorem 4.1 to solve for problem (6.5).
Noticing (6.6), we need to construct functions and such that . From the boundary conditions in (6.5) (cf. (4.7), (4.1), and (4.11)), we obtain that satisfies
where we define
for simplifying the presentation. In accordance with the identities above, we take and as the solutions to the transport equations
| (6.10) | ||||
| (6.11) |
Here, is the lifting operator from Lemma 6.1, and the source terms will be chosen via a decomposition of the operator , as defined in (5.20).
Finally, we set , and assume that are given and vanish in the past for . Under these conditions, we determine and using the equations
| (6.12a) | |||
| (6.12b) | |||
where
| (6.13) |
and for . As in [28], we can show that the traces of on vanish. Consequently, it follows that , for and . These are the unique smooth solutions satisfying transport equations (6.10)–(6.11). Hence, can be derived from (6.6) and can be derived from (6.4).
From (6.8)–(6.7) and (6.12)–(6.13), it suffices to define the error terms , , and . To this end, by an analogous argument in [25, 12], we decompose
| (6.14) |
and
| (6.15) |
where
| (6.16) |
and
Take
| (6.17) |
As for error term , we decompose
| (6.18) |
and set
| (6.19) |
where
It follows from (5.14b) that
Then we derive from (6.10)–(6.11) and (6.18) that
Thus, by , one has
| (6.20) |
Moreover, from (6.5) and (6.15), we have
| (6.21) |
Denote by the th component of the vector for From (5.20) and (2.13),
| (6.22) |
Using (6.21), we have
| (6.23) |
| (6.24) |
and similarly,
| (6.25) |
From (6.14) and (6.21), together with (6.5) and (6.7), one has
| (6.26) | ||||
| (6.27) |
Substituting (6.12) into (6.24)–(6.25) and using (6.22), we get
| (6.28) |
From as , we conclude that if the error terms tend to zero, then
thus, the solution to (5.20) can be obtained formally.
In order to estimate the error terms, we need to introduce the inductive hypothesis as follows. Let us take an integer , a small number , and another integer , which will be determined later. Suppose that we have the estimate
| (6.29) |
then our inductive hypothesis consists of the following four parts:
where is given in (6.9) and decreases to zero with
| (6.30) |
We will show that for sufficiently small and , and for sufficiently large , is true and implies , thus is true for all , which will allow us to prove Theorem 2.1 completely.
Now we assume that holds, hence have the following estimates as in [25, Lemmas 6–7].
Lemma 6.2.
If is sufficiently large, then
| (6.31) | ||||
| (6.32) |
for , and . Furthermore,
| (6.33) |
for , and .
6.2. Estimate of the Quadratic and First Substitution Error Terms
First, we rewrite quadratic error terms , , and , from (6.14), (6.15), and (6.18) respectively, as follows:
where , , and are the second derivatives of operators , , and respectively. More precisely, we define
where operators and are given in (4.4)–(4.5), and is defined by
In fact, in our case, we have the following:
| (6.34) | ||||
| (6.35) |
A straightforward computation with an application of the Moser-type calculus inequality (2.22) yields the next proposition (see [25, Proposition 5]).
Proposition 6.2.
Let and with . If belongs to for all and satisfies for some positive constant , then there exist two constants and , independent of and , such that, if and , then
and
where and for , symbol denotes the trace of on , and represents the background state defined by (2.17).
In light of (6.29)–(6.31) and the assumption , as showin in [25, Lemma 8] or [12, Lemma 8.3], we can apply Proposition 6.2, the Sobolev embedding theorem, and the trace estimate to derive the following estimate.
Lemma 6.3.
If , then there exist suitably small and sufficiently large such that
for , and , where .
For the first substitution error terms , , and defined in (6.14), (6.15), and (6.18), as in [25, Lemma 9] or [12, Lemma 8.4], we can apply Proposition 6.2 and use (6.29), (6.32)–(6.33), hypothesis (), and the trace theorem to derive the next lemma.
Lemma 6.4.
If , then there exist suitably small and large enough such that
for where
6.3. Construction and Estimate of the Modified State
To control the remaining error terms, we construct and analyze the modified state as described in the following lemma.
Lemma 6.5.
Proof.
We divide the proof into four steps.
Step 1. It follows from (6.2)–(6.3) that , and hence we can define , , and by (6.36)–(6.37). Thanks to (5.14d), constraint (4.3d) holds for . As in [25, Proposition 7], we define
so that on , and constraints (4.3b), (4.3e) hold for
Step 2. Utilizing (6.4), the trace theorem, and the hypothesis , we obtain
| (6.39) |
Then we use Lemma 6.1, (6.2), and (6.39) to obtain
| (6.40) |
Step 3. Using (6.36), we have
| (6.41) |
Decomposing
the Moser-type calculus inequality (2.22), hypothesis (), and (6.31) implies
which together with (6.1a) yields that
| (6.42) |
The rest of terms on the right-hand side of (6.41) consist entirely of commutators. Let us detail the estimate of . Using (2.22), the Sobolev embedding theorem, (6.1a), (6.29), and (6.33), we obtain
Similarly, we obtain that
If , it follows from (6.1b) and (6.31)–(6.32) that
Similarly, we have
Applying the same analysis to the other commutators in (6.41) and using (6.42), we obtain
| (6.43) |
Step 4. Now, we construct and estimate following the approach of Secchi–Trakhinin [49, Proposition 28]. As outlined in Step 1, the functions and have already been specified. Next, we define as the unique solution, vanishing in the past, of the linear equations
| (6.44) |
where represents the component of operator corresponding to , defined as:
| (6.45) |
Since satisfies (4.3b), the equations (6.44) do not require additional boundary condition.
To estimate , we apply standard energy method. From (6.44), we deduce:
| (6.46) |
where
and From (6.36), we compute
Applying Moser-type calculus inequality (2.22), the Sobolev embedding theorem, (6.36)–(6.37), (6.43), (6.29), and (6.33), we obtain
| (6.47) |
For , we follow the same strategy used to estimate in Step 3 and obtain:
| (6.48) |
Regarding , from (6.1a), (5.17), and the inductive hypothesis , we find
for . Using (6.1a), (2.22), hypothesis , and (6.31) yields
for . Combining these estimates with (6.47)–(6.48), we have
| (6.49) |
Applying a standard energy argument to equations (6.46) and using estimate (6.49), we conclude
| (6.50) |
Finally, estimate (6.38) follows from (6.37), (6.40), (6.43), and (6.50). This completes the proof. ∎
6.4. Estimate of the Second Substitution and Last Error Terms
The following lemma provides the estimates for the second substitution error terms , , and , as defined in (6.14), (6.15), and (6.18), respectively.
Lemma 6.6.
If , then there exist suitably small and large enough such that
for where .
Proof.
From (6.34) and (6.36)–(6.37), we have
Using (6.35)–(6.37), we deduce that . Thanks to (6.36), the error term can be rewritten as
Apply the Sobolev embedding theorem, (6.29), (6.33), and (6.38), we find that
allowing us to use Proposition 6.2 for suitably small. Furthermore, from (6.29)–(6.31) and (6.38), we have
for . Using Proposition 6.2, hypothesis , and (6.38), we obtain the estimate for term thereby completing the proof of lemma. ∎
For the last error term (6.16),
we first observe that
as deduced from (5.14c), (6.36), and (6.33) for sufficiently small . Consequently, we arrive at the following lemma, analogous to [25, Lemma 8.6] or [12, Lemma 12]. The proof is omitted here for brevity.
Lemma 6.7.
If and , then there exist suitably small and large enough such that
| (6.51) |
for , where
6.5. Proof of Theorem 2.1
We first prove the following lemma for accumulated error terms , , and that are defined in (6.8) and (6.13).
Lemma 6.8.
If and , then there exist suitably small and large enough such that
| (6.55) | ||||
| (6.56) |
Proof.
Using the lemma above, we have the estimates for , , and .
Lemma 6.9.
If and , then there exist suitably small and large enough such that
| (6.57) | ||||
| (6.58) |
for , and
| (6.59) |
Proof.
In the following lemma, we obtain the estimate of differences , and by using the tame estimate (4.19). See [25, Lemma 16] or [12, Lemma 8.10] for the proof.
Lemma 6.10.
Let and . If and are suitably small and is large enough, then
| (6.60) |
Lemma 6.10 establishes the first part of the hypothesis . The following lemma addresses the remaining components of .
Lemma 6.11.
Let and . If and are suitably small and is large enough, then
| (6.61) | |||||
| (6.62) | |||||
| (6.63) | |||||
We refer to [25, Lemmas 17–18] or [12, Lemma 8.11] for the proof of Lemma 6.11. Let us assume , , and sufficiently small. Additionally, let be large enough to satisfy the assumptions of Lemmas 6.10–6.11, from which the inductive hypothesis follows. Then, as shown in [25, Lemma 19] or [12, Lemma 8.12], it can be proved that the hypothesis holds.
Lemma 6.12.
If is small enough, then the hypothesis holds.
With this, we are ready to complete the proof of Theorem 2.1. Our proof closely follows the idea in [25, 12], but is included here for completeness.
Proof of Theorem 2.1. Let and . Under the assumptions of Theorem 2.1, the initial data are compatible up to order . If is sufficiently small in , where , then the assumption (6.29) and all the requirements of Lemmas 6.10–6.12 are satisfied, due to (5.16) and (5.19). Thus, holds for all . From the estimate
we conclude that converges to some in , and converges to some in . Taking the limit in (6.61)–(6.62) for , and in (6.63), we obtain that satisfies (5.20). Consequently, is a solution to (2.6)–(2.11) in . The proof of Theorem 2.1 is complete. ∎
Acknowledgments
R. M. Chen was supported in part by the NSF grant DMS-2205910. F. Huang was supported in part by National Key R&D Program of China, grant No. 2021YFA1000800, and the National Natural Sciences Foundation of China, grant No. 12288201. D. Wang was supported in part by NSF grants DMS-1907519 and DMS-2219384. D. Yuan was supported by EPSRC grant EP/V051121/1, Fundamental Research Funds for the Central Universities No. 2233100021 and No. 2233300008.
References
- [1] S. Alinhac, Existence d’ondes de raréfaction pour des systèmes quasi-linéaires hyperboliques multidimensionnels, Comm. Partial Differential Equations, 14 (1989), pp. 173–230.
- [2] S. Alinhac and P. Gérard, Opérateurs pseudo-différentiels et théorème de Nash-Moser, Savoirs Actuels. [Current Scholarship], InterEditions, Paris; Éditions du Centre National de la Recherche Scientifique (CNRS), Meudon, 1991.
- [3] M. Artola and A. J. Majda, Nonlinear development of instabilities in supersonic vortex sheets. I. The basic kink modes, Phys. D, 28 (1987), pp. 253–281.
- [4] , Nonlinear development of instabilities in supersonic vortex sheets. II. Resonant interaction among kink modes, SIAM J. Appl. Math., 49 (1989), pp. 1310–1349.
- [5] , Nonlinear kink modes for supersonic vortex sheets, Phys. Fluids A, 1 (1989), pp. 583–596.
- [6] J. Azaiez and G. M. Homsy, Linear stability of free shear flow of viscoelastic liquids, J. Fluid Mech., 268 (1994), pp. 37–69.
- [7] S. Benzoni-Gavage and D. Serre, Multi-dimensional hyperbolic partial differential equations: First-order Systems and Applications, OUP Oxford, 2006.
- [8] J.-M. Bony, Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Ann. Sci. École Norm. Sup. (4), 14 (1981), pp. 209–246.
- [9] J. Chazarain and A. Piriou, Introduction to the theory of linear partial differential equations, vol. 14 of Studies in Mathematics and its Applications, North-Holland Publishing Co., Amsterdam-New York, 1982. Translated from the French.
- [10] G.-Q. Chen and Y.-G. Wang, Existence and stability of compressible current-vortex sheets in three-dimensional magnetohydrodynamics, Arch. Ration. Mech. Anal., 187 (2008), pp. 369–408.
- [11] G.-Q. G. Chen and M. Feldman, The mathematics of shock reflection-diffraction and von Neumann’s conjectures, vol. 197 of Annals of Mathematics Studies, Princeton University Press, Princeton, NJ, 2018.
- [12] G.-Q. G. Chen, P. Secchi, and T. Wang, Nonlinear stability of relativistic vortex sheets in three-dimensional Minkowski spacetime, Arch. Ration. Mech. Anal., 232 (2019), pp. 591–695.
- [13] , Nonlinear stability of relativistic vortex sheets in three-dimensional Minkowski spacetime, Arch. Ration. Mech. Anal., 232 (2019), pp. 591–695.
- [14] R. M. Chen, J. Hu, and D. Wang, Linear stability of compressible vortex sheets in two-dimensional elastodynamics, Adv. Math., 311 (2017), pp. 18–60.
- [15] , Linear stability of compressible vortex sheets in 2D elastodynamics: variable coefficients, Math. Ann., 376 (2020), pp. 863–912.
- [16] R. M. Chen, J. Hu, D. Wang, T. Wang, and D. Yuan, Nonlinear stability and existence of compressible vortex sheets in 2D elastodynamics, J. Differential Equations, 269 (2020), pp. 6899–6940.
- [17] R. M. Chen, F. Huang, D. Wang, and D. Yuan, On the stability of two-dimensional nonisentropic elastic vortex sheets, Commun. Pure Appl. Anal., 20 (2021), pp. 2519–2533.
- [18] , On the vortex sheets of compressible flows, Commun. Appl. Math. Comput., 5 (2023), pp. 967–986.
- [19] , Stabilization effect of elasticity on three-dimensional compressible vortex sheets, J. Math. Pures Appl. (9), 172 (2023), pp. 105–138.
- [20] J.-F. c. Coulombel, Weak stability of nonuniformly stable multidimensional shocks, SIAM J. Math. Anal., 34 (2002), pp. 142–172.
- [21] , Weakly stable multidimensional shocks, Ann. Inst. H. Poincaré C Anal. Non Linéaire, 21 (2004), pp. 401–443.
- [22] , Well-posedness of hyperbolic initial boundary value problems, J. Math. Pures Appl. (9), 84 (2005), pp. 786–818.
- [23] J.-F. c. Coulombel and A. Morando, Stability of contact discontinuities for the nonisentropic Euler equations, Ann. Univ. Ferrara Sez. VII (N.S.), 50 (2004), pp. 79–90.
- [24] J.-F. c. Coulombel and P. Secchi, The stability of compressible vortex sheets in two space dimensions, Indiana Univ. Math. J., 53 (2004), pp. 941–1012.
- [25] , Nonlinear compressible vortex sheets in two space dimensions, Ann. Sci. Éc. Norm. Supér. (4), 41 (2008), pp. 85–139.
- [26] C. M. Dafermos, Hyperbolic conservation laws in continuum physics, vol. 325 of Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 2000.
- [27] J. A. Fejer and J. W. Miles, On the stability of a plane vortex sheet with respect to three-dimensional disturbances, J. Fluid Mech., 15 (1963), pp. 335–336.
- [28] J. Francheteau and G. Métivier, Existence de chocs faibles pour des systèmes quasi-linéaires hyperboliques multidimensionnels, Astérisque, (2000), pp. viii+198.
- [29] M. E. Gurtin, An introduction to continuum mechanics, vol. 158 of Mathematics in Science and Engineering, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981.
- [30] X. Hu and D. Wang, Formation of singularity for compressible viscoelasticity, Acta Math. Sci. Ser. B (Engl. Ed.), 32 (2012), pp. 109–128.
- [31] R. Huilgol, Propagation of a vortex sheet in viscoelastic liquids—the rayleigh problem, Journal of Non-Newtonian Fluid Mechanics, 8 (1981), pp. 337–347.
- [32] R. R. Huilgol and G. C. Georgiou, Fluid mechanics of viscoplasticity, Springer, Cham, second ed., [2022] ©2022.
- [33] D. D. Joseph, Fluid dynamics of viscoelastic liquids, vol. 84 of Applied Mathematical Sciences, Springer-Verlag, New York, 1990.
- [34] P. D. Lax and R. S. Phillips, Local boundary conditions for dissipative symmetric linear differential operators, Comm. Pure Appl. Math., 13 (1960), pp. 427–455.
- [35] J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications. Vol. II, vol. Band 182 of Die Grundlehren der mathematischen Wissenschaften, Springer-Verlag, New York-Heidelberg, 1972. Translated from the French by P. Kenneth.
- [36] A. Majda, The existence of multidimensional shock fronts, Mem. Amer. Math. Soc., 43 (1983), pp. v+93.
- [37] , The stability of multidimensional shock fronts, Mem. Amer. Math. Soc., 41 (1983), pp. iv+95.
- [38] A. Majda and S. Osher, Initial-boundary value problems for hyperbolic equations with uniformly characteristic boundary, Comm. Pure Appl. Math., 28 (1975), pp. 607–675.
- [39] G. Métivier, Stability of multidimensional shocks, in Advances in the theory of shock waves, vol. 47 of Progr. Nonlinear Differential Equations Appl., Birkhäuser Boston, Boston, MA, 2001, pp. 25–103.
- [40] G. Métivier and K. Zumbrun, Large viscous boundary layers for noncharacteristic nonlinear hyperbolic problems, Mem. Amer. Math. Soc., 175 (2005), pp. vi+107.
- [41] J. W. Miles, On the reflection of sound at an interface of relative motion, J. Acoust. Soc. Amer., 29 (1957), pp. 226–228.
- [42] , On the disturbed motion of a plane vortex sheet, J. Fluid Mech., 4 (1958), pp. 538–552.
- [43] R. L. Mishkov, Generalization of the formula of Faa di Bruno for a composite function with a vector argument, Int. J. Math. Math. Sci., 24 (2000), pp. 481–491.
- [44] A. Morando, P. Secchi, P. Trebeschi, and D. Yuan, Nonlinear stability and existence of two-dimensional compressible current-vortex sheets, Arch. Ration. Mech. Anal., 247 (2023), pp. Paper No. 50, 83.
- [45] A. Morando and P. Trebeschi, Two-dimensional vortex sheets for the nonisentropic Euler equations: linear stability, J. Hyperbolic Differ. Equ., 5 (2008), pp. 487–518.
- [46] A. Morando, P. Trebeschi, and T. Wang, Two-dimensional vortex sheets for the nonisentropic Euler equations: nonlinear stability, J. Differential Equations, 266 (2019), pp. 5397–5430.
- [47] J. B. Rauch and F. J. Massey, III, Differentiability of solutions to hyperbolic initial-boundary value problems, Trans. Amer. Math. Soc., 189 (1974), pp. 303–318.
- [48] P. Secchi, On the Nash-Moser iteration technique, in Recent developments of mathematical fluid mechanics, Adv. Math. Fluid Mech., Birkhäuser/Springer, Basel, 2016, pp. 443–457.
- [49] P. Secchi and Y. Trakhinin, Well-posedness of the plasma-vacuum interface problem, Nonlinearity, 27 (2014), pp. 105–169.
- [50] D. Serre, Systems of conservation laws. 2, Cambridge University Press, Cambridge, 2000. Geometric structures, oscillations, and initial-boundary value problems, Translated from the 1996 French original by I. N. Sneddon.
- [51] B. Stevens, Short-time structural stability of compressible vortex sheets with surface tension, Arch. Ration. Mech. Anal., 222 (2016), pp. 603–730.
- [52] Y. Trakhinin, The existence of current-vortex sheets in ideal compressible magnetohydrodynamics, Arch. Ration. Mech. Anal., 191 (2009), pp. 245–310.
- [53] Y.-G. Wang and F. Yu, Stability of contact discontinuities in three-dimensional compressible steady flows, J. Differential Equations, 255 (2013), pp. 1278–1356.
- [54] , Stabilization effect of magnetic fields on two-dimensional compressible current-vortex sheets, Arch. Ration. Mech. Anal., 208 (2013), pp. 341–389.
- [55] , Structural stability of supersonic contact discontinuities in three-dimensional compressible steady flows, SIAM J. Math. Anal., 47 (2015), pp. 1291–1329.
- [56] Y.-G. Wang and H. Yuan, Weak stability of transonic contact discontinuities in three-dimensional steady non-isentropic compressible Euler flows, Z. Angew. Math. Phys., 66 (2015), pp. 341–388.