Existence of smooth solutions to the Landau-Fermi-Dirac equation with Coulomb potential
Abstract.
In this paper, we prove global-in-time existence and uniqueness of smooth solutions to the homogeneous Landau-Fermi-Dirac equation with Coulomb potential. The initial conditions are nonnegative, bounded and integrable. We also show that any weak solution converges towards the steady state given by the Fermi-Dirac statistics. Furthermore, the convergence is algebraic, provided that the initial datum is close to the steady state in a suitable weighted Lebesgue norm.
1. Introduction
We consider the homogeneous Landau-Fermi-Dirac equation with Coulomb potential
| (1.1) |
where is the standard projection matrix
The function models the distribution of velocities within a single species quantum gas. The particles considered here are fermions (e.g. electrons) interacting in a grazing collision regime [1]. The parameter quantifies the strength of the quantum effects of the system for the particular species considered and depends on Planck’s constant, the mass of the species, and the number of independent quantum weights of the species. In particular, we notice that in the case eq. (1.1) reduces to the classical Landau equation. The Pauli exclusion principle implies that satisfies the a priori bound
This bound is the key ingredient in our proof. See also [21] for a discussion on the Boltzmann equation with Fermi-Dirac statistic.
Equation (1.1) is well-understood for the cases of moderately soft and hard potentials, namely when the kernel is replaced by for . In [2], the authors consider the moderately soft potentials case () and show algebraic convergence of non degenerate solutions towards equilibrium for initial data satisfying a suitable non saturation condition. Existence and uniqueness of weak solution for hard potentials () are shown in [7], regularity and smoothing effects are studied in [15, 14], and exponential convergence towards equilibrium in [4]. In [3], the authors present fundamental properties of the entropy and entropy production functional for hard and moderately soft potentials. The existence of nondegenerate steady for any potential is shown in [8].
The Landau-Fermi-Dirac equation shares several properties with the classical Landau equation. Multiplying (1.1) by a test function , integrating by parts, and applying a straightforward symmetry argument, one obtains
| (1.2) |
Conservation of mass, momentum and energy conservation follows from (1.2) by choosing . A version of the H-theorem for (1.1) is also available: with
in (1.2), one obtains that
| (1.3) |
where
Eq. (1.3) is the entropy balance equation associated to (1.1), with being the (physical) Fermi-Dirac entropy functional. The only smooth function that nullifies the entropy production, , is the Fermi-Dirac equilibrium distribution
| (1.4) |
which is also the only smooth minimizer of under the constraints of given mass, momentum, and energy [8]. The constants , , and are determined by the mass, first and second moment of the initial data
There are other nonsmooth distributions of the form
with of a measurable subset, that satisfies (formally) and solves (1.1). These particular stationary solutions are called saturated Fermi-Dirac states. As such, any solution to (1.1) with general initial data could approach, as time grows, such saturated states. However, given an initial data with mass , momentum and energy , there exists only one value of , uniquely determined by , and , for which is an admissible stationary solution. For below such value, the only steady-state is .
Taking the formal limit in eq. (1.1), one obtains the classical Landau equation. Furthermore, as modulus a multiple of the mass :
The addition of a multiple of the mass to does not change the entropy balance equation (1.3) nor the form of the equilibrium distribution (1.4), thanks to the conservation of mass property. The equilibrium distribution also converges towards the classical Maxwellian distribution as . Finally, strictly related to the limits and is the fact that the relative entropy
converges to the relative entropy of the classical Landau equation
The next observation concerns the structure of the collision operator. For a smooth , the interaction term can be expressed as a second order elliptic nonlinear operator with non-local coefficients:
where the matrix is defined through the map
with
| (1.5) |
1.1. Main Results
Our first result concerns existence of smooth solutions to (1.1). Unlike in the case of the classical Landau equation, we are able to show global-in-time existence of smooth solutions for a general class of initial datum. Our regularity estimates depend on the quantum parameter. At the present moment, it seems out of reach to obtain similar results uniformly with respect to . Therefore, in the rest of the manuscript we set .
Theorem 1.1.
Suppose satisfies , , and . Then, there is a solution with such that , , for each , and for each , and ,
| (1.6) |
Moreover, has decreasing (Fermi-Dirac) entropy and satisfies conservation of mass, energy, and momentum.
If the initial data has moments with , the solution is unique.
By a simple time rescaling, we obtain global-in-time existence and uniqueness for any quantum parameter:
Corollary 1.2.
Fix and let satisfies , , and . Then, there is a unique with such that , , for each , and for each , and ,
Moreover, has decreasing (Fermi-Dirac) entropy and satisfies conservation of mass, energy, and momentum.
Theorem 1.1 is proved in several steps. First, we approximate the problem by discretizing the time variable and adding suitable regularizing terms. The approximating problem is well-posed thanks to suitable fixed point arguments. After, we use uniform and entropy inequalities to take limits as our regularizing terms vanish. A crucial ingredient is the uniform positive lower bound for the diffusion matrix , which follows from the boundedness of the second moment of and a uniform negative upper bound for the Fermi-Dirac entropy. This guarantees that equation (1.1) remains uniformly parabolic during the evolution of the system.
The weak solutions from Theorem 1.1 are, in fact, smooth solutions, provided the initial data has high enough moments:
Theorem 1.3.
Let be a weak solutions as in Theorem 1.1. If the initial data is, in addition, such that then .
The higher regularity of the solution is obtained thanks to parabolic regularity arguments, Morrey’s inequality and Schauder estimates. The parabolic regularity argument yields estimates for in for any . Via interpolation between Sobolev spaces, we obtain a bound for in a fractional Sobolev space. From here, we deduce, the Hölder continuity of via Morrey’s inequality. A standard parabolic bootstrap argument yields .
Our regularity results do not hold in the limit , since they heavily rely on the bound . For the classical Landau equation, the Cauchy problem has been understood only for weak solutions [35] [18] [6] [1] [29]. Recently, in [28] and [34], the authors showed that, for a short time, weak solutions become instantaneously regular and smooth. The long time asymptotic for weak solutions has been studied in [13] and [12]. However, the question of whether solutions stay smooth for all time or become unbounded after a finite time is still open. Recent research has produced several conditional results regarding this inquiry. These results show regularity properties of solutions that already possess some basic properties (yet to be verified). They include (i) conditional uniqueness [23] [16], and (ii) conditional smoothness for solutions in with [34] [28]. In a very recent manuscript [19], the authors studied behavior of solutions in the space . They show that for general initial data there exists a time after which the weak solutions belong to . This result agrees with the one in [25], in which the authors showed that the set of singular times for weak solutions has Hausdorff dimension at most . In [9], the authors show that self-similar blow-up of type I cannot occur for solutions to the Landau equation.
The second result of this paper concerns the convergence towards the steady state as the time approaches infinity. We show that the convergence is algebraic, provided that the initial datum is close to the steady state in a suitable weighted Lebesgue norm. Hereafter, we denote with the function defined in (1.4) with .
Theorem 1.4.
Given any initial datum , , such that , the solution to the initial value problem associated to (1.1) converges strongly in as to the Fermi-Dirac distribution with same mass, momentum and energy as .
Furthermore, there exists a constant such that, if
then
The unconditional convergence (without rate) towards the steady state is obtained from the entropy balance equation in the following way. We integrate the balance equation in time and use the ellipticity properties of the entropy dissipation to deduce that along a suitable sequence of time instants . The monotonicity in time of the relative entropy yields that strongly in as .
The algebraic convergence for initial data close to the steady state is achieved by linearizing (1.1) around . First, we show existence of a spectral gap for the linearized Landau-Fermi-Dirac operator between two different weighted Lebesgue space. Precisely, such relation has the structure
with not included in . This latter fact is the reason why we are not able to obtain exponential convergence towards equilibrium, but only algebraic. After, we derive a uniform bound for some moment of the solution to the linearized equation in a weighted Lebesgue space. In the last step, we bound the contributions of the nonlinear corrections, and derive a differential inequality for the weighted -norm of the perturbation
An elementary argument of ordinary differential equations’ theory yields algebraic convergence to zero with rate for , provided that, at initial time, the latter is small enough and .
1.2. Notations
Here we list some of the notation conventions adopted throughout the manuscript:
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Universal constants that may change from line to line are denoted or if the constant is allowed to depend on the quantities and .
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We write to mean there is a universal constant such that . Similarly, we write to mean and . If we write , the implicit constant is allowed to depend on .
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We write for and a Banach space to denote the space of strongly measurable -valued functions satisfying
When we write without specifying the measure space, we mean .
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We use the japanese bracket notation . Given , we denote with the space of functions that have the following norm
finite. We denote with the norm of .
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Given we denote with the conjugate exponent of , .
2. Coefficient Bounds
The following standard bounds will be used throughout our proofs.
Lemma 2.1.
Any such that , , and satisfies
where depends on and .
Proof.
We begin by quoting a known result (see [20] Lemma 6 and Proposition 4, or [34] Lemma 3.2 and 3.3) that says that for any nonnegative function with mass, second momentum and entropy bounded, for all :
where the constant depends only on the quantities
In light of this inequality, we need only show that
| (2.1) |
and that there exists a strictly positive constant such that
| (2.2) |
The proof of (2.1) and (2.2) can be found in [7] in Lemma 3.1. ∎
The previous lemma together with (1.3) show that, as long as the initial data has strictly negative entropy, our equation is uniformly parabolic, and saturated-Fermi-Dirac-distributions are not admissible solutions.
Lemma 2.2 (Upper Bound on ).
Proof.
For arbitrary,
provided and . Optimizing in yields and the bound,
for , and . Note that .
Finally, notice that
∎
Lemma 2.3 (Upper Bound on ).
3. Existence of bounded weak solutions
In order to find weak solutions to (1.1), we first introduce an extra dissipative term to counter the degenerate ellipticity of (see Lemma 2.1) and study the approximating problems
| (3.1) |
We will first prove there exist solutions to (3.1), then taking ,we recover global-in-time weak solutions to (1.1). To this end, we introduce an auxiliary equation,
| (3.2) | |||
obtained by dividing the time interval into subintervals, each of length , linearizing (3.1) around a measurable function , and adding an additional localizing term, . In the first step of our construction, we use the Lax-Milgram Theorem to find unique weak solutions to (3.2) and prove the following proposition:
Proposition 3.1.
Let with , be a measurable function, and . Then, there is a unique that satisfies
| (3.3) | ||||
for any .
For a fixed , Proposition 3.1 defines a solution operator to (3.2) via . In the second step of our construction, we seek solutions to the nonlinear system:
| (3.4) |
for any fixed . Equivalently, we seek a fixed point satisfying the bound . To this end, we show is continuous and compact, and the set is bounded in . Therefore, we apply the Schaeffer Fixed Point Theorem to conclude the following proposition:
Proposition 3.2.
Suppose with . Then, there is a family of functions such that and solve (3.4). That is, for , satisfies that for any ,
| (3.5) |
Furthermore, for each , and .
In the third step of our construction, we seek a weak solution to the auxiliary equation (3.1)on a time interval . To this end, we divide into pieces of size and from Proposition 3.2, for an initial datum , we may define
where solves (3.4) with parameters . We show propagation of moments and use a variant of the Aubin-Lions Lemma to conclude the following proposition:
Proposition 3.3.
Suppose , , and and . Then, for any , there is an with such that for each , , , , and satisfies (3.1) in the form,
| (3.6) |
for each and
| (3.7) |
for each . Furthermore, conserves mass and satisfies the bound
Finally, in the fourth step of our construction, we conclude the proof of Theorem 1.1. From Proposition 3.3, for an initial datum and a sequence , we obtain a family of solutions to the equation (3.1) with parameters on the interval . We show propagation of higher moments and an H-Theorem for the equation (3.1). Combined with Lemma 2.1, this implies a uniform lower bound on the coefficients , which is sufficient to gain compactness as .
3.1. Step 1: Existence and Uniqueness of Solutions to (3.2)
In this step, we use the Lax-Milgram Theorem to prove Proposition 3.1. We recall that in this step, we construct weak solutions to
| (3.8) | |||
where , , , , , and are fixed.
Proof of Proposition 3.1.
3.2. Step 2: Existence of Solutions to (3.4)
In this step, we use a fixed point argument to prove Proposition 3.2. We show that the nonlinear, semi-discretized equation,
has a solution provided is known and satisfies and . Moreover, we show these assumptions are propagated, so that for a fixed , we have the existence of a family for for any .
We begin by showing the existence of solutions to the nonlinear weak formulation (3.5) provided is known and satisfies and . To this end, we fix and define with , where is the unique solution to (3.3) given (and fixed ). We also fix to be . We would like to apply the Schaeffer Fixed Point Theorem [24, Theorem 11.3] to to conclude that there exists a fixed point for in . To apply Schaeffer’s Theorem we need to verify the following conditions:
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- •
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The map is compact. This is done in Lemma 3.6 via the compact embedding .
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To this end, we have our first a priori bound:
Lemma 3.4.
For with , let be the unique solution to (3.3). Then, and satisfies the estimate
| (3.9) |
Proof.
We note that the preceding lemma immediately implies the following result:
Lemma 3.5 (A priori bounds on approximate fixed points).
Let be the unique solution to (3.3) with with and . The map defined as is such that is a bounded subset of .
Proof.
Lemma 3.6 (Compactness).
For and as in Lemma 3.5, is pre-compact as a subset of for any .
Proof.
Fix any such . Then, we note that Lemma 3.4 guarantees that is bounded in , uniformly in measurable. We claim that embeds compactly in for provided .
Indeed, fix a sequence uniformly bounded in and , so that . Then, use Rellich-Kondrachev and a diagonalization argument to extract a subsequence for which in for every compact. We will show in . Fix . Then, decompose the norm into two parts via,
| (3.10) |
The first term converges to for any fixed . For the second, we interpolate between and and use the Sobolev embedding to guarantee the norm is uniformly bounded in . Thus,
| (3.11) | ||||
where , i.e. . So for and , this converges to as uniformly in . Thus, first pick sufficiently large that the second term of (3.10) is less than for all . Then, pick sufficiently large such that first term of (3.10) is less than . ∎
Lemma 3.7 (Continuity).
Let be defined as in Lemma 3.5. Suppose strongly in . Then, strongly in .
Proof.
Suppose in . Combining the a priori bound from Lemma 3.4 and compactness from Lemma 3.5, is uniformly bounded in and compact in . Therefore, by extracting subsequences, it suffices to show that if in and in and in , then . Finally, since Proposition 3.1 guarantees uniqueness of solutions to (3.3), it suffices to show
| (3.12) | ||||
Since solves (3.3) with coefficients , we know
The weak convergence in is sufficient to pass to the limit in each term, except in the term containing . For this term, we first observe that
Since the function is Lipschitz, we get
since in . Therefore, we obtain (3.12) and the proof is complete. ∎
The following lemma states that any fixed point of is also a solution to (3.5). Note, this is not immediate because (3.5) does not contain any positive part operators, while (3.3) does.
Lemma 3.8.
Suppose satisfies with and . Then, and consequently solves (3.5).
Proof.
The idea is to test the weak formulation (3.3) with and and show that both are identically :
Since each term is positive, all are and we conclude on , i.e. . Similarly,
Now, because , . Thus, each term is negative and we conclude . ∎
Next, we show the assumption that is propagated. That is, if , then and therefore, we may iterate the fixed point argument to construct a family solving (3.5).
Lemma 3.9.
Suppose satisfies with and . Then, satisfies the estimate
| (3.13) |
Proof.
Let be a cutoff function in such that
Then, we test (3.3) with to obtain
First, we claim that the right hand side converges to as . Indeed, we bound each term separately, beginning with as,
Next, we bound using and Lemma 2.3 to obtain
For , we integrate by parts to obtain
Now, vanishes by a similar estimate, using Lemma 2.2. Finally vanishes by the estimate
Thus, piecing together all the above estimates, we conclude that vanishes as . Second, taking sufficiently fast so that are increasing to , the monotone convergence theorem yields
By Lemma 3.8, and the proof is complete. ∎
Proof of Proposition 3.2
Fix as in the statement of Proposition 3.2. Suppose moreover that have been constructed so that and for and satisfies (3.5). We will now construct . Indeed, fix and the solution map to (3.3) with fixed.
As stated at the beginning of this step, the role of Lemma 3.4, 3.5, 3.6 and 3.7 is to verify the hypotheses of the Schaeffer Fixed Point Theorem for .
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Lemma 3.4 implies maps to itself;
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Lemma 3.5 implies that approximate fixed points of are bounded in ;
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Lemma 3.6 implies is a compact map;
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Lemma 3.7 implies is a continuous (nonlinear) map.
Therefore, the Schaeffer Fixed Point Theorem (see [24, Theorem 11.3] for a precise statement) yields a (not necessarily unique) fixed point of the map . Because , . As , solves
However, since by Lemma 3.8, , and we may remove the positive parts to conclude solves the desired weak formulation, namely (3.5). Finally, Lemma 3.9 implies . By induction, the proof is complete.
3.3. Step 3: Existence of Solutions to (3.1)
In this step we construct weak solutions to the nonlinear, continuous time equation,
| (3.14) |
on an arbitrary fixed time interval for any fixed and for fixed initial data , where and . We first prove uniform in (the time mesh) and (the strength of the added localization) estimates on solutions to equation (3.4). For all , let . Define the piecewise interpolant of as
| (3.15) |
and the backward finite difference operator as
We also introduce the shift operator
With this new notation, we can rewrite (3.5) as
| (3.16) | ||||
where
For strong compactness, we need propagation of moments (shown in Lemma 3.11) in the form of
and a variation of the Aubin-Lions Lemma for piecewise constant functions, which requires an estimate (shown in Lemma 3.12) of the form
We begin with and estimates, which are continuous-time analogous of Lemma 3.9 and Lemma 3.4, respectively.
Lemma 3.10 ( and Estimates).
Suppose and . Then, the following estimates hold:
| (3.17) |
and
| (3.18) |
Proof.
Inequality (3.18) follows by iterating Lemma 3.9. Next, we estimate the norm of by testing (3.5) with , using Young’s inequality and to obtain
For the last integral, we integrate by parts, using and get
Since , . Therefore, using , the interpolation inequality , and Young’s inequality, we have
Using Lemma 3.9, we obtain
which implies, recursively,
Taking and recalling the definition of in (3.15) finish the proof of the lemma. ∎
Lemma 3.11 (Propagation of Moments).
Suppose and . Then, the following estimates hold:
| (3.19) |
and
| (3.20) |
where the implicit constants are independent of , , and .
Proof.
Let be as in Lemma 3.9. Then, we test (3.5) with to obtain
We bound using to obtain
For we use Lemma 2.3, , and :
For , we integrate by parts twice to get
Lemma 2.2 and yields
Lemma 2.3 and yield
Combining all above estimates we obtain
Now, taking , recalling the definition of in (3.15) and letting , the monotone convergence theorem implies (3.19).
The proof of (3.20) proceeds nearly identically after testing with . ∎
The bounds in Lemmas 3.10 and 3.11 are sufficient for weak or weak star compactness. For strong compactness, we will use the version of the Aubin-Lions Lemma for piecewise constant functions [22, Theorem 1].
Lemma 3.12.
For any , , and , for defined above with ,
| (3.21) |
Moreover, the family is compact in , provided .
Proof.
Let us define the triple , and for a fixed . Following the proof of Lemma 3.6, the embedding is compact for . Certainly, continuously for this range of . Moreover, we have shown in Lemmas 3.10 and 3.11,
| (3.22) |
where the constant on the right hand side is independent of and . To obtain (3.21) we first consider
For , thanks to Lemma 2.2, one gets
and, using Lemma 2.3,
Finally,
and since ,
using Lemma 3.11. We note , , and are uniformly bounded in and (but not in ) by Lemmas 3.10 and 3.11. Thus (3.21) follows.
Let , fix some , and be the corresponding sequence of piecewise constant solutions to (3.16). Thanks to the estimates from Lemma 3.10, Lemma 3.11, and Lemma 3.12, we may assume that converges to , as , in the following topologies:
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Weak star in ,
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Weakly in ,
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Weak star in ,
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Strongly in for and .
Moreover, by taking a further subsequence, we will also have that pointwise almost everywhere. Therefore, thanks to Fatou’s lemma
All these convergences are enough to pass to the limit in (3.16). We briefly highlight the convergence in the nonlinear terms. Let us first consider . We have
We estimate using Hölder’s inequality and :
thanks to the strong convergence, and, similarly, using Lemma 2.3,
Next, we handle the nonlinear term involving , which we decompose as
The term convergence to zero thanks to the weak convergence of in and Lemma 2.2. For , we use Hölder’s inequality, Lemma 2.2, estimate (3.17) and the strong convergence in to obtain , since
We treat the left hand side of (3.16) by integrating by parts,
For sufficiently large,
as is compactly supported in . Moreover, for , so that
Since is smooth, the right hand side converges to as . Finally, since is smooth and are uniformly bounded in , we have
| (3.23) |
This concludes the proof of (3.6). Lemma 3.12 implies that, for some ,
for every . Hence, (3.23) yields . The distributional formulation implies
for each . Now, fix and let such that . Then, substituting into the above weak formulation, and passing to the limit , we obtain (3.3).
3.4. Step 4: Proof of Theorem 1.1
We conclude the proof of Theorem 1.1 by showing compactness in for solutions to (3.1). We already have uniform in bounds of the form,
| (3.24) |
Thus, to gain strong compactness as , we will show (in Lemma 3.15) the estimate
| (3.25) |
by leveraging the degenerate dissipation present in (1.1) (see Lemma 2.1), which up to this point, we have neglected. However, we do not have control over and therefore, we also show propagation of higher moments in Lemma 3.13.
To this end, we recall the dependence of our solutions on the parameter . Throughout this section, we will write to denote the solution to (3.1) on constructed in Proposition 3.3 with parameter . Let us begin with a propagation of higher moments estimate that is uniform in :
Lemma 3.13.
Suppose for some and . Then, the family satisfies the uniform in estimate,
| (3.26) |
Proof.
We note that the propagation of moments for follows directly from Proposition 3.3. We will prove the rest of the them by induction on the integer part of . Indeed, fix some and suppose that (3.26) holds for any . Then, we will test (3.3) with where is as in Lemma 3.9. We obtain
We estimate the right hand side by decomposing into multiple parts:
For , after integrating by parts, thanks to Lemma 2.2 and we obtain
where in the last line we used the induction hypothesis. For , we use Lemma 2.2 and to obtain
Similarly, for , we use Lemma 2.3 and to obtain
Finally, for , integration by parts yields
Combining all above estimates, we prove (3.26) for any . The proof is complete. ∎
The following lemma, combined with Lemma 2.1, gives a quantitative lower bound on the ellipticity of . This will allow us to gain some control over uniformly in .
Lemma 3.14.
Suppose , , , and . Then, has decreasing entropy, i.e. for almost every ,
| (3.27) |
Moreover, the dissipative coefficients are bounded uniformly from below:
| (3.28) |
Proof.
By Lemma 2.1, (3.28) is a consequence of (3.27) and
| (3.29) |
The energy bound (3.29) is shown in Lemma 3.13. It remains to estimate the entropy and obtain (3.27). We test (3.3) with
We have
We now take . For the left hand side, we use conservation of mass from Proposition 3.3 to obtain:
By the monotone convergence theorem,
Next, for , we decompose further as
For , we integrate by parts to obtain
For , we use with and as to obtain
Finally, we note for that
Thus, combining our estimates, we have shown
We conclude by noticing that
∎
The next lemma contains the coercive estimate we need to pass to the limit .
Lemma 3.15 ( Estimate).
Suppose with and . Then, the family satisfies the estimate
Proof.
We test (3.3), with where is a cutoff function as in Lemma 3.9. We obtain,
We expand the right hand side as
We bound for using the propagation of moments from Lemma 3.13 and the upper bounds on the coefficients and from Lemma 2.2 and 2.3. We will lower bound using Lemma 3.14. We begin to bound by decomposing further:
For , integration by parts, Lemma 2.2, and imply
For , obtain by Lemma 2.2, and ,
Piecing together, we obtain
For , we directly use the estimates from Lemma 3.13, 2.2, and 2.3 and to obtain
Next, for , we integrate by parts and recall to decompose further:
We bound using Lemma 2.3 and , to obtain
For , we bound using so that
Hence,
Using , integration by parts yields
Finally, we note and by Lemma 3.14,
Summarizing, we obtain
Letting , applying the monotone convergence theorem, and taking a supremum over yield the desired bound on .
Next, we test (3.3) with an arbitrary test function and, by duality, obtain a bound on . In particular, we have
Since
and
we conclude
This completes the proof. ∎
In the next lemma we state a weighted estimate, proved via a slight modification to the -estimate in Lemma 3.15.
Lemma 3.16.
Proof of Theorem 1.1
Fix , with , and and fix some sequence and let be the solutions with constructed in Proposition 3.3. Then, the uniform-in- estimates from Lemma 3.13 and Lemma 3.15 together with Aubin-Lions Lemma imply that we may assume that for some limit in the following topologies:
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Weak star in ,
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Weakly in ,
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Weak star in ,
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Strongly in for any and .
Furthermore, we may also assume pointwise almost everywhere on . Therefore, by Fatou’s lemma, it follows that for almost every ,
| (3.30) |
Note also that the weak star convergence in is sufficient to guarantee .
Next, since each solves (3.1) on the time interval , for any , we have
We conclude the proof of existence by following the same steps as in the proof of Proposition 3.3.
To show uniqueness of solution, we assume by contradiction that there exist two solutions and . Their difference
is identically zero at and solves the following weak formulation:
We consider for some , and get
The term is estimated with Young’s inequality:
Similarly,
using the bound from below for and the bound from above in Lemma 2.2:
Hölder and Hardy-Littlewood-Sobolev inequalities applied to lead to
using Lemma 3.16 to bound the moments of (uniformly in time). Similarly,
We briefly show how the term is uniformly bounded. Let be large enough. For , Hölder inequality yields:
with and . The choice of so that leads to the desired estimate.
Combining the estimates for and choosing small enough, one gets
Since thanks to Lemma 3.16, Gronwall’s inequality implies that for all . This concludes the proof of the theorem.
4. Regularity of Weak Solutions
In this section, we prove Theorem 1.3. Throughout this section we consider initial data such that , for a general , and . The exact value of needed for Theorem 1.3 is determined in Lemma 4.3.
As a first step, we use Lemma 3.16 to show that weak solutions of (1.1) instantaneously regularize and belong to weighted and weighted for any .
Proof.
Fix arbitrary. We first recall the notation for divided differences,
for which the following discrete product formula holds,
We test (1.6) with and obtain
On the left hand side, we perform a discrete integration by parts:
We decompose the right hand side as
For we use Lemma 2.1 to obtain
Next, for any , we upper bound using Young’s inequality and ,
In the same way, we bound . We bound using Young’s inequality and for by Calderon-Zygmund (see Chapter 9.4 in [24]),
For , we bound , using Young’s inequality and Lemma 2.3, as
Again, we bound in a similar manner to and . For , we use , Lemma 2.2, and (via a simple modification to Proposition IX.9(iii) in [10]), to obtain
Next, for , we integrate by parts and use Lemma 2.2 and to obtain
We bound in a similar manner to and . For , we use Young’s inequality and Calderon-Zygmund, to obtain
For , we use from Lemma 2.3 to obtain
Finally, is bounded similarly to and . Thus, we have shown (using once more that ),
where the implicit constants depend only on and . Now, taking and taking , , we see is weakly differentiable and
Next, taking a supremum over in and an average over , and applying Lemma 3.16, we get
This concludes the proof of the lemma. ∎
Next, we show how to control the -regularity of :
Lemma 4.2.
Proof.
Thanks to Lemma 4.1, we can take
as test function for (1.6), and obtain
On the left hand side, we perform one discrete integration by parts and one standard integration by parts and get
We also perform discrete and standard integration by parts to decompose the right hand side as
where denotes the error terms, which originate from the discrepancy between the product rules for and . These terms are bounded identically to the others and so we omit the bound on . For , our coercive term, we use Lemma 2.1 to obtain
For , when two derivatives land on the kernel , we use Young’s inequality, Hölder’s inequality in space, the Sobolev embedding , and Calderon-Zygmund, to obtain for any , the estimate
Similarly, for , when two derivatives land on the kernel , we use Young’s inequality, Hölder’s inequality, Calderon-Zygmund, Lebesgue interpolation, the Sobolev embedding , and Lemma 3.16 to obtain for any , the estimate
To bound the remaining terms and , we modify the arguments from Lemma 4.1 in a similar fashion, using the additional tool of the Sobolev embedding as necessary, to obtain
Thus, taking sufficiently small and taking the limit , we conclude that is weakly differentiable and we obtain
Taking a supremum over and an average over , and applying Lemma 4.1, we obtain
∎
Remark 1.
From Lemma 4.2, one can continue to bootstrap spatial regularity, and obtain the corresponding higher regularity estimates, that provided , for each , , and moreover,
If is rapidly decaying, i.e. for each , then is Schwartz class in space. That is, for each .
Instead of bootstrapping spatial regularity and deducing the corresponding time regularity from the equation, we use Lemma 4.2 to conclude Hölder regularity of . Combined with the parabolic divergence structure of (1.1), we deduce spatial and temporal regularity simultaneously via classical Schauder estimates. As the initial step, we have the following lemma:
Proof.
By Lemma 4.2, we conclude for each . Therefore, by a duality argument, belongs to for . By a (real) interpolation of the Sobolev spaces and , we obtain for strictly less, but as close as one wishes, than , and strictly less, but as close as one wishes, than , for any (see Theorem 3.1 in [5]). Hence, choosing , Morrey’s inequality implies , for some . ∎
Now, we are ready to apply a standard bootstrapping argument and conclude is smooth:
Proof of Theorem 1.3
By Lemma 4.3, we conclude for some and solves the divergence form parabolic equation,
| (4.1) |
in the weak sense. Hence, Lemma 4.7 in [27] shows that and belong to . Thus, satisfies a divergence-form parabolic equation with Hölder continuous coefficients. By Theorem 12.1 from Chapter 3 in [31], we conclude . Bootstrapping the argument, we obtain higher regularity of the coefficients and , from which follows, as desired.
5. Long time behavior
In this section we prove Theorem. 1.4. Without loss of generality, we can assume that . We first rewrite the initial value problem associated to (1.1) in the following compact form
| (5.1) |
where the Landau-Fermi-Dirac operator is defined by
| (5.2) | ||||
and the quantities , are defined in (1.5).
We first show unconditional convergence without rate towards the steady state for (1.1), which is the first part of Theorem 1.4.
Proposition 5.1 (Convergence to the steady state).
Given any initial datum , , such that , the solution to the (5.1) tends to the Fermi-Dirac distribution with same mass, momentum and energy as when .
Proof.
We recall that satisfies a uniform in time bound in , and therefore . In what follows we will often make use of this relation without mentioning it.
Integrating the entropy balance equation in time yields
with
Since , there exists a sequence such that as . Define . Given the lower bound for we deduce
Therefore is bounded in . However is bounded in , so the product is bounded in . Furthermore is bounded in . We deduce via Sobolev embedding that is relatively compact in , and more in general (via the bounds and the bound on the second moment of ) in for every . Let us denote with its limit. We have that weakly in . This is enough to deduce via a generalized Fatou argument [11, Lemma A.4] that
with
and is arbitrary. Via monotone convergence we deduce
It follows that . Since we know that , it follows [8] that . This means that strongly in for . In particular the relative Fermi-Dirac entropy as . On the other hand, we know that is non-increasing, so it must hold . This easily implies the strong convergence as in . This finishes the proof of the Lemma. ∎
Our next goal is to prove exponential convergence of the solution to (1.1) towards the steady state in case the initial datum is close enough to in the norm . This is in the second part of Theorem 1.4. We linearize our equation around the steady state . We will work in weighted Lebesgue spaces with weight defined by
| (5.3) |
where is the Fermi-Dirac distribution defined in (1.4). Writing
| (5.4) |
it defines the linearized operator and the quadratic and cubic perturbations , , respectively.
Via straightforward computations [4] one finds
| (5.5) | ||||
| (5.6) | ||||
| (5.7) |
Define the spaces
and recall that .
Our goal is to prove a spectral gap estimate for the linearized operator . We will apply [17, Lemma 10]. In order to do so, we adapt the latter result’s framework and therefore define for the following Hilbert spaces
Clearly with continuous embedding.
We split then the linearized operator into two contributions, in the following fashion:
| (5.8) | ||||
| (5.9) | ||||
| (5.10) | ||||
where is an arbitrary constant, to be specified later. We also recall the definition of the Maxwellian :
and point out that (via direct computations).
We prove now the following coercivity estimate for .
Lemma 5.2.
is bounded and for every , provided that is large enough.
Proof.
From (5.9) and the definition (1.5) of it follows, via an integration by parts,
| (5.11) | ||||
for . Being symmetric and positive definite for , Cauchy-Schwartz yields
Therefore
Via a duality argument it follows that is bounded as an operator .
Choosing in (5.11) yields
| (5.12) |
The last integral can be estimated via Cauchy-Schwartz:
| (5.13) |
Let us focus on the first integral on the right-hand side of (5.13). Lemma 2.1 and the fact that lead to
For every , since is uniformly positive on (with an dependent lower bound), it follows via (the standard) Sobolev’s embedding and Poincaré’s Lemma
From (5.13) and the above inequality we deduce
| (5.14) |
Let us now consider
Young’s inequality with the convex conjugated functions , (with arbitrary and fixed small enough such that ) leads to
By defining and rescaling , the above inequality can be rewritten as
By employing the log-Sobolev’s inequality with Gaussian weight [26] one obtains
Replacing with and choosing the minimum point of , one finds
| (5.15) |
Lemma 2.1, relation and (5.15) yield
| (5.16) |
At this point, (5.14) and (5.16) yield
which implies, given that ,
| (5.17) | ||||
Choosing , we absorb the third integral on the right-hand side of (5.17) via , yielding
| (5.18) | ||||
By interpolating between and and applying Young’s inequality one finds
Therefore, for large enough, it holds . We conclude
This finishes the proof of the Lemma. ∎
Concerning , we are going to prove the following result:
Lemma 5.3.
For it holds
| (5.19) |
with
Furthermore is a compact operator and the following bound holds for
| (5.20) |
Proof.
Let now be a bounded sequence in . For , , we have
Hölder’s inequality yields
| (5.21) |
In a similar way, one shows
| (5.22) |
This means that , are bounded in for . Let us now consider, for arbitrary,
Given that , from [10, Corollary 4.28] it follows that is relatively compact in for every measurable set with finite measure. A Cantor diagonal argument yields the existence of a subsequence of (not relabeled) such that is strongly convergent in for every . Given that , it is easily seen that
| (5.23) |
On the other hand, Young’s inequality for convolutions yields
Since then as , while for . From this fact and (5.23) we obtain
| (5.24) |
In a similar way one shows that
| (5.25) |
Let us now deal with . One can prove, via a similar argument as the one employed to show (5.23), that (up to subsequences)
| (5.26) |
On the other hand, for ,
so Hardy-Littlewood-Sobolev’s inequality yields
This means that
Putting the above relation and (5.26) together yields the strong convergence of in . Finally, is obviously relatively compact in . Thus is a compact operator for every . Bound (5.20) is a straightforward byproduct of the previous computations and of estimates (5.21), (5.22). This finishes the proof of the Lemma. ∎
We now want to prove the following theorem:
Theorem 5.4 (Spectral gap for ).
There exists a constant such that
| (5.27) |
Proof.
Remark 2.
Next, we show some bounds for and . Define preliminarily for and arbitrary measurable function
The following result holds.
Lemma 5.5 (Bounds for ).
For every , , and every ,
| (5.28) | ||||
| (5.29) |
with , , for every such that , , , .
Proof.
The upper bound in (5.28) is already known since can be estimated from below via the Maxwell-Boltzmann distribution. Therefore we only prove the lower bound.
We first observe that it is enough to prove the statement for and , since via Cauchy-Schwartz and Young’s inequality it holds (remember that is symmetric and positive definite)
Let now deal with the case . We start by considering . It holds
Let us now consider
Since
Hölder inequality yields
It follows
Let us now consider, for , ,
It follows
Hence (5.28) holds.
Let us now prove (5.29). It holds (via Young’s inequality for convolutions)
while
This finishes the proof of the Lemma. ∎
The next lemma deals with the nonlinear contributions and .
Lemma 5.6 (Bounds for nonlinear terms).
For every , , it holds
| (5.30) | ||||
| (5.31) | ||||
for every .
Proof.
Let us first consider the contribution of the quadratic terms.
that can be rewritten as
For every , thanks to (5.28), we get
while Cauchy-Schwartz and Young’s inequality lead to
However, it is easy to see (via direct computation) that
so, using (5.28), we obtain
which implies
Applying (5.28) once again leads to
Let us now consider, for arbitrary ,
It is quite easy to see that
while, on the other hand,
Since , it follows
Let us now deal with . Young’s inequality yields
From (5.28), (5.29) it follows
Finally, let us consider, for a generic ,
where the last inequality holds because . It follows via Cauchy-Schwartz inequality
Gagliardo-Nirenberg inequality leads to
Choosing yields
From (5.28) we conclude
Since , the terms can be estimated in a similar way as the terms . Therefore we deduce that (5.30) holds.
Next, we deal with the contributions from the cubic terms:
From (5.28) and relation it follows
The term can be estimated like to obtain
The term can be estimated like to obtain
but, given that , it follows
Finally, since , the terms can be estimated in a similar way as the terms . Therefore we deduce that (5.31) holds. This finishes the proof of the Lemma. ∎
We are now ready to prove the conditional algebraic convergence result, thereby concluding the proof of Thr. 1.4.
Lemma 5.7 (Algebraic rate of convergence for initial data close to equilibrium).
There exists a constant such that, if , and if for some , then
Proof.
From (1.1), (5.4) it follows that the perturbation satisfies the equation
| (5.32) |
Testing the above equation against in the sense of yields
From (5.27), (5.30), (5.31) it follows that a suitable constant exists such that
We are now going to prove that
| (5.33) |
Indeed, the left-hand side of (5.33) is a sum of terms having the form
If then from the property and the fact that is bounded in for every it follows immediately that
so via Young’s inequality
If , it suffices to notice that
Since , Jensen’s inequality yields
Therefore (5.33) holds. We therefore conclude that, for some suitable constant and ,
for every . We point out that
The maximum of is achieved for . Choosing in this way yields
for and
Since by assumption on the initial data, we deduce that for all . It follows that, for some ,
| (5.34) |
Integrating (5.34) in time yields
| (5.35) |
We will now show that . We proceed iteratively, proving that
| (5.36) |
for . We argue by induction on . Estimate (5.35) and the assumption on the initial datum imply that (5.36) holds for . Let us now assume that (5.36) holds for , generic. By testing (5.32) against in the sense of , exploiting Lemma 5.2 and bound (5.20) and proceeding like in the proof of (5.34) one finds
| (5.37) | ||||
for some , . By integrating (5.37) in time we get
| (5.38) | ||||
From the assumption that for as well as the inductive hypothesis it follows that the right-hand side of (5.38) is finite, meaning that (5.36) holds for . Via the induction principle we deduce that (5.36) holds for . Choosing in (5.38) and exploiting (5.36) for yields (5.36) for . In particular
Therefore via Hölder’s inequality
so from (5.34) it follows
This (via Gronwall’s inequality) finishes the proof of the lemma, and of Theorem 1.4. ∎
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