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On the mass transfer in the 3D Pitaevskii model

Juhi Jang
Department of Mathematics
University of Southern California
Los Angeles, CA 90089, USA [email protected] ,  Pranava Chaitanya Jayanti
Department of Mathematics
University of Southern California
Los Angeles, CA 90089, USA [email protected]  and  Igor Kukavica
Department of Mathematics
University of Southern California
Los Angeles, CA 90089, USA [email protected]
Abstract.

We examine a micro-scale model of superfluidity derived by Pitaevskii [Pit59] which describes the interacting dynamics between superfluid He-4 and its normal fluid phase. This system consists of the nonlinear Schrödinger equation and the incompressible, inhomogeneous Navier-Stokes equations, coupled to each other via a bidirectional nonlinear relaxation mechanism. The coupling permits mass/momentum/energy transfer between the phases, and accounts for the conversion of superfluid into normal fluid. We prove the existence of global weak solutions in 𝕋3\mathbb{T}^{3} for a power-type nonlinearity, beginning from small initial data. The main challenge is to control the inter-phase mass transfer in order to ensure the strict positivity of the normal fluid density, while obtaining time-independent a priori estimates.

Key words and phrases:
Superfluids; Pitaevskii model; Navier-Stokes equation; Nonlinear Schrödinger equation; Global weak solutions; Existence; 3D

1. Introduction and mathematical model

In this article, we present a rigorous analysis of the Pitaevskii model (a micro-scale description) of superfluidity  [Pit59] in three dimensions. The system consists of a superfluid phase and a normal fluid phase, described by modified versions of the nonlinear Schrödinger equation (NLS) and the Navier-Stokes equations (NSE), respectively. This is one of many different theories proposed to explain and quantify the underlying mechanisms of superfluidity. For more details, see  [BDV01, Vin06, PL11, BBP14, JT21, Jay22] and references therein. The Pitaevskii model, specifically, works well in the context of small length scales (≪\ll inter-vortex spacing), and has previously been explored in  [JT22a, JT22b, JJK23]. A similar model has also been numerically simulated in  [BSZ+23]. The superfluid phase is represented by a complex-valued wavefunction ψ\psi, while the normal fluid is characterized by its density ρ\rho, velocity uu, and pressure qq. The form of the Pitaevskii model used here is as follows (with the prefix “c” in the equation labels signifying that the equations are coupled):

∂tψ+λ​B​ψ\displaystyle\partial_{t}\psi+\lambda B\psi =−12​i​Δ​ψ+μi​|ψ|p​ψ\displaystyle=-\frac{1}{2i}\Delta\psi+\frac{\mu}{i}\lvert\psi\rvert^{p}\psi (c-NLS)
B=12​(−i​∇−u)2+μ​|ψ|p\displaystyle B=\frac{1}{2}\left(-i\nabla-u\right)^{2}+\mu\lvert\psi\rvert^{p} =−12​Δ+12​|u|2+i​u⋅∇+μ​|ψ|p\displaystyle=-\frac{1}{2}\Delta+\frac{1}{2}\lvert u\rvert^{2}+iu\cdot\nabla+\mu\lvert\psi\rvert^{p} (CPL)
∂tρ+∇⋅(ρ​u)\displaystyle\partial_{t}\rho+\nabla\cdot(\rho u) =2​λ​Re(ψ¯​B​ψ)\displaystyle=2\lambda\real(\bar{\psi}B\psi) (c-CON)
∂t(ρ​u)+∇⋅(ρ​u⊗u)+∇q−ν​Δ​u+α​ρ​u\displaystyle\partial_{t}(\rho u)+\nabla\cdot(\rho u\otimes u)+\nabla q-\nu\Delta u+\alpha\rho u =−2λIm(∇ψ¯Bψ)+λ∇Im(ψ¯Bψ)+μ2∇|ψ|p+2\displaystyle=-2\lambda\imaginary(\nabla\bar{\psi}B\psi)+\lambda\nabla\imaginary(\bar{\psi}B\psi)+\frac{\mu}{2}\nabla\lvert\psi\rvert^{p+2} (c-NSE)
∇⋅u\displaystyle\nabla\cdot u =0.\displaystyle=0. (DIV)

The strength of the superfluid’s scattering self-interactions is measured by μ>0\mu>0, while ν>0\nu>0 and α>0\alpha>0 denote the viscosity and drag coefficient of the normal fluid. The quantum scattering is a power-type nonlinearity, with exponent p∈[1,∞)p\in[1,\infty). The interactions between the two phases are mediated by the nonlinear coupling operator BB, and the strength of the coupling is quantified by the positive constant λ\lambda. The structure of this coupling permits bidirectional mass/momentum/energy transfer between the two phases, which results in a relaxation mechanism for the otherwise non-dissipative NLS. Indeed, the coupling gives  (c-NLS) a parabolic flavor. These equations are supplemented with the initial conditions

ψ​(0,x)=ψ0​(x),u​(0,x)=u0​(x),ρ​(0,x)=ρ0​(x)a.e.​x∈𝕋3.\psi(0,x)=\psi_{0}(x),\quad u(0,x)=u_{0}(x),\quad\rho(0,x)=\rho_{0}(x)\quad\text{a.e.}\ x\in\mathbb{T}^{3}. (INI)

We use periodic boundary conditions, i.e., we are working in a 3-dimensional torus [0,1]3[0,1]^{3}. The above equations are a slight modification of Pitaevskii’s original work  [Pit59], which were valid for any type of scattering interactions and for a compressible, heat-conducting normal fluid. The simplifying assumptions used here are detailed in  [JJK23].

While  (DIV) implies that the velocity is divergence-free, this does not mean that the density is constant along particle trajectories. In fact, the density is governed by  (c-CON), a transport equation with a complicated source term. This inhomogeneity on the RHS is the principal limiting factor of the analysis, since it forces us to ensure that the normal fluid density does not become zero (vacuum) or negative (unphysical). By integrating  (c-CON) over 𝕋3\mathbb{T}^{3}, the advective term vanishes and using the positivity of the operator BB  [JJK23, Lemma 2.7], we have

dd​t​∫𝕋3ρ​𝑑x=2​λ​Re​∫𝕋3ψ¯​B​ψ≥0.\frac{d}{dt}\int_{\mathbb{T}^{3}}\rho\ dx=2\lambda\real\int_{\mathbb{T}^{3}}\bar{\psi}B\psi\geq 0. (1.1)

This implies that the net mass of the normal fluid does not decrease with time. Due to conservation of total mass (of both fluids), we have

dd​t​∫𝕋3(ρ+|ψ|2)​𝑑x=0.\frac{d}{dt}\int_{\mathbb{T}^{3}}\left(\rho+\absolutevalue{\psi}^{2}\right)\ dx=0. (1.2)

In other words, the coupling results in a conversion of superfluid into normal fluid, on average. However, the RHS of  (c-CON) is not necessarily non-negative for all x∈𝕋3x\in\mathbb{T}^{3}, and we have to control the local mass transfer between the two phases. Indeed, this means bounding the L∞L^{\infty} norm (in space) of the source term, Re(ψ¯​B​ψ)\real(\overline{\psi}B\psi), as described in Section  2.2. Since the coupling is a second-order differential operator, this essentially translates into finding an estimate of ‖ψ‖Lx∞​‖Δ​ψ‖Lx∞\norm{\psi}_{L^{\infty}_{x}}\norm{\Delta\psi}_{L^{\infty}_{x}}, which dictates the required regularity of the wavefunction. In the case of no source term for the continuity equation, there are several results pertaining to the standard version of the incompressible, inhomogeneous NSE — see  [Kaz74, LS78, Kim87, Sim90, Lio96, Dan03, CK03, Dan06] among many other references.

The NLS, an archetypal dispersive PDE, is often used to study quantum systems with low-energy wave interactions, like dipolar gases  [CMS08, Soh11] and quantum ferrofluids  [BSG+18]. There is a rich literature on the mathematical analysis of the NLS (see  [Tao06] for a collection of results). One particular aspect of NLS that has attracted much attention is its quantum hydrodynamic (QHD) reformulation  [JMR02, Jün10, AS17, AMZ21, WG21, AMZ23]. Another connection with the compressible fluid dynamics community has been the study of Korteweg models  [HL94, HL96, BD04, CDS12], of which QHD and even capillary flows [AS22] are special cases. The linear drag included in  (c-NSE) is not unheard of: similar terms have been used previously to either prove the global existence of solutions  [Cha22], or to show relaxation to a steady state  [BGLVV22, SYZ22]. For QHD with a combination of linear drag and electrostatic forces, see [JL04, AM09, AM12].

Given the vast mathematical literature that exists on the NSE and NLS independently, it is surprising that there are hardly any on a combination of the two, like the Pitaevskii model or others  [Kha69, Car96]. The first attempt to study a combined model was by Antonelli and Marcati  [AM15], in which a fractional time-step method pioneered by the same authors  [AM09, AMS12] was utilized. In this approach, the standard, uncoupled NLS is solved over a small time interval at the end of which the wavefunction is “updated” to account for the interactions with the normal fluid. However, this was still a uni-directionally coupled system, i.e., the wavefunction ψ\psi was dependent on the density ρ\rho and velocity uu, but not the other way around. This yields the standard form of the continuity equation: without a source term.

The small-data global existence of solutions for the Pitaevskii model on 𝕋2\mathbb{T}^{2} was established in  [JJK23] for the nonlinearity exponent p∈[1,4)p\in[1,4). Moreover, for p≥4p\geq 4, an almost-global existence was shown, wherein the existence time grows with decreasing data size, exponentially for p=4p=4 and polynomially for p>4p>4. The path to the required a priori bounds of ψ\psi is by energy-type estimates, up to Lt∞​Hx52L^{\infty}_{t}H^{\frac{5}{2}}_{x}.

In the current work, we obtain the global existence of weak solutions in 𝕋3\mathbb{T}^{3}, for the entire range of power nonlinearities (characterized by 1≤p<∞1\leq p<\infty), thus proving that the inter-phase mass transfer may be controlled for small initial data. We emphasize that we deal with density that is simply in Lx∞L^{\infty}_{x}, and thus not necessarily differentiable. This limits the highest regularity that one can subject the momentum equation (and the velocity) to. In turn, through the coupling operator BB, this also restricts the regularity we can achieve for the wavefunction. The primary objective is to derive time-independent a priori bounds on ψ\psi that ensure that the RHS of  (c-CON) will not lead to non-positive densities.

The approach used for 2D is not applicable in 3D, owing to the insufficiency of the Lt∞​Hx1∩Lt2​Hx2L^{\infty}_{t}H^{1}_{x}\cap L^{2}_{t}H^{2}_{x} bound for the velocity. This regularity of uu is not enough to derive an energy estimate of the required order for ψ\psi in 3D. Any higher regularity of uu would mean taking the derivatives of ρ\rho, which is off limits. Another roadblock is the extremely slow mass decay for high values of pp, i.e., when the superfluid interacts much more strongly with itself rather than the normal fluid. In such a scenario, a purely energy-based method [JJK23] led to a bifurcation in the existence time of solutions (global or almost-global) for different ranges of the nonlinear index pp. The mass decay rate (independent of dimension) pervades all levels of energy estimates, dominating the decay of higher norms as well. We overcome these challenges by a hybrid approach: combining the decay of superfluid mass with a maximal regularity estimate for parabolic equations (Section  2.3). The time-control of the mass conversion and of the higher order energy norm are presented in Lemmas 3.1 and 3.3, and state that the Lx2L^{2}_{x} and H˙x2\dot{H}^{2}_{x} norms of the wavefunction decrease as (1+t)−1p(1+t)^{-\frac{1}{p}} and (1+t)−12−1p(1+t)^{-\frac{1}{2}-\frac{1}{p}}, respectively. These allow us to evaluate the integral in  (2.13), independent of the final time TT, leading to global control of the solution. In order to apply maximal regularity, the initial wavefunction ψ0\psi_{0} must belong to an interpolation (Besov) space, which is carefully chosen to be marginally larger than H2H^{2}, so that the assumption ψ0∈H2\psi_{0}\in H^{2} is sufficient for our purposes. The resulting solution ψ\psi is shown to belong to C​([0,∞);H2)C([0,\infty);H^{2}), among other spaces. To summarize, we demonstrate that wielding the power of parabolic regularity allows us to guarantee global solutions, even when the mass decay is exceedingly small.

We now briefly outline the notation used in this article. Following this, we state and discuss the main result in Section  2. Several a priori estimates are derived in Section  3, which ends with an argument on ensuring a positive lower bound for the density. In this paper, we only present the required a priori estimates, as the general construction of solutions (and the density renormalization) follows as in  [JJK23, Section 4].

1.1. Notation

We denote by Hs​(𝕋3)H^{s}(\mathbb{T}^{3}) the completion of C∞​(𝕋3)C^{\infty}(\mathbb{T}^{3}) under the Sobolev norm HsH^{s}. When referring to the homogeneous Sobolev spaces, we use H˙s​(𝕋3)\dot{H}^{s}(\mathbb{T}^{3}). Consider a 3D vector-valued function u≡(u1,u2,u3)∈C∞​(𝕋3)u\equiv(u_{1},u_{2},u_{3})\in C^{\infty}(\mathbb{T}^{3}). The set of all divergence-free, smooth 3D functions uu defines Cd∞​(𝕋3)C^{\infty}_{d}(\mathbb{T}^{3}). Then, Hds​(𝕋3)H^{s}_{d}(\mathbb{T}^{3}) is the completion of Cd∞​(𝕋3)C^{\infty}_{d}(\mathbb{T}^{3}) under the HsH^{s} norm. There are many equivalent ways of defining Besov spaces, and the most appropriate choice for our purposes is through the method of real interpolation between Sobolev spaces [AF78, AF03]. For 1≤q≤∞,0<θ<11\leq q\leq\infty,0<\theta<1, and s=(1−θ)​s1+θ​s2s=(1-\theta)s_{1}+\theta s_{2}, we define

B2,qs:=(Hs1,Hs2)θ,q.B^{s}_{2,q}:=(H^{s_{1}},H^{s_{2}})_{\theta,q}.

The L2L^{2} inner product, denoted by ⟨⋅,⋅⟩\langle\cdot,\cdot\rangle, is sesquilinear (the first argument is complex conjugated, indicated by an overbar) to accommodate the complex nature of the Schrödinger equation. Explicitly, ⟨ϕ,ψ⟩=∫𝕋3ϕ¯​ψ​𝑑x\langle\phi,\psi\rangle=\int_{\mathbb{T}^{3}}\bar{\phi}\psi\ dx.

We use the subscript xx on a Banach space to denote that the Banach space is defined over 𝕋3\mathbb{T}^{3}. For instance, Lxr:=Lr​(𝕋3)L^{r}_{x}:=L^{r}(\mathbb{T}^{3}) and Hd,xs:=Hds​(𝕋3)H^{s}_{d,x}:=H^{s}_{d}(\mathbb{T}^{3}). For spaces/norms over time, the subscript tt is used, such as LtrL^{r}_{t}.

We also use the notation X≲YX\lesssim Y and X≳YX\gtrsim Y to imply that there exists a positive constant CC such that X≤C​YX\leq CY and C​X≥YCX\geq Y, respectively. When appropriate, the dependence of the constant on various parameters shall be denoted using a subscript as X≲k1,k2YX\lesssim_{k_{1},k_{2}}Y or X≤Ck1,k2​YX\leq C_{k_{1},k_{2}}Y. Throughout the article, CC is used to denote a (possibly large) constant that depends on the system parameters listed in  (2.4), while κ\kappa is used to represent a (small) positive number. The values of CC and κ\kappa can vary across the different steps of calculations.

2. Main result and discussion

2.1. Weak solutions and the existence theorem

First, we define the notion of a weak solution used here.

Definition 2.1 (Weak solutions111See Remark  2.5.).

For a given time T>0T>0, a triplet (ψ,u,ρ)(\psi,u,\rho) is a weak solution to the Pitaevskii model if

  1. (i)

    ψ∈L2​(0,T;H3​(𝕋3)),u∈L2​(0,T;Hd2​(𝕋3)),ρ∈L∞​([0,T]×𝕋3)\psi\in L^{2}(0,T;H^{3}(\mathbb{T}^{3})),u\in L^{2}(0,T;H^{2}_{d}(\mathbb{T}^{3})),\rho\in L^{\infty}([0,T]\times\mathbb{T}^{3}), and

  2. (ii)

    ψ\psi, uu, and ρ\rho satisfy the governing equations in the sense of distributions for all test functions, i.e.,

    −∫0T∫𝕋3(ψ​∂tφ¯+12​i​∇ψ⋅∇φ¯−λ​φ¯​B​ψ−i​μ​φ¯​|ψ|p​ψ)​𝑑x​𝑑t\displaystyle-\int_{0}^{T}\int_{\mathbb{T}^{3}}\left(\psi\partial_{t}\bar{\varphi}+\frac{1}{2i}\nabla\psi\cdot\nabla\bar{\varphi}-\lambda\bar{\varphi}B\psi-i\mu\bar{\varphi}\lvert\psi\rvert^{p}\psi\right)dx\ dt (2.1)
    =∫𝕋3(ψ0​φ¯​(0)−ψ​(T)​φ¯​(T))​𝑑x\displaystyle\quad=\int_{\mathbb{T}^{3}}\Big{(}\psi_{0}\bar{\varphi}(0)-\psi(T)\bar{\varphi}(T)\Big{)}dx

    and

    −∫0T∫𝕋3(ρu⋅∂tΦ+ρu⊗u:∇Φ−ν∇u:∇Φ−2λΦ⋅Im(∇ψ¯Bψ)+αρu⋅Φ)dxdt\displaystyle-\int_{0}^{T}\int_{\mathbb{T}^{3}}\Big{(}\rho u\cdot\partial_{t}\Phi+\rho u\otimes u:\nabla\Phi-\nu\nabla u:\nabla\Phi-2\lambda\Phi\cdot\imaginary(\nabla\bar{\psi}B\psi)+\alpha\rho u\cdot\Phi\Big{)}dx\ dt (2.2)
    =∫𝕋3(ρ0​u0​Φ​(0)−ρ​(T)​u​(T)​Φ​(T))​𝑑x\displaystyle\quad=\int_{\mathbb{T}^{3}}\Big{(}\rho_{0}u_{0}\Phi(0)-\rho(T)u(T)\Phi(T)\Big{)}dx

    and

    −∫0T∫𝕋3(ρ​∂tσ+ρ​u⋅∇σ+2​λ​σ​Re(ψ¯​B​ψ))​𝑑x​𝑑t=∫𝕋3(ρ0​σ​(0)−ρ​(T)​σ​(T))​𝑑x-\int_{0}^{T}\int_{\mathbb{T}^{3}}\Big{(}\rho\partial_{t}\sigma+\rho u\cdot\nabla\sigma+2\lambda\sigma\real(\bar{\psi}B\psi)\Big{)}dx\ dt=\int_{\mathbb{T}^{3}}\Big{(}\rho_{0}\sigma(0)-\rho(T)\sigma(T)\Big{)}dx (2.3)

    where ψ0∈H2​(𝕋3)\psi_{0}\in H^{2}(\mathbb{T}^{3}), u0∈Hd1​(𝕋3)u_{0}\in H^{1}_{d}(\mathbb{T}^{3}) and ρ0∈L∞​(𝕋3)\rho_{0}\in L^{\infty}(\mathbb{T}^{3}) are the initial data. The test functions are:

    1. (a)

      a complex-valued scalar field φ∈H1​(0,T;L2​(𝕋3))∩L2​(0,T;H1​(𝕋3))\varphi\in H^{1}(0,T;L^{2}(\mathbb{T}^{3}))\cap L^{2}(0,T;H^{1}(\mathbb{T}^{3})),

    2. (b)

      a real-valued, divergence-free (3D) vector field Φ∈H1​(0,T;Ld2​(𝕋3))∩L2​(0,T;Hd1​(𝕋3))\Phi\in H^{1}(0,T;L^{2}_{d}(\mathbb{T}^{3}))\cap L^{2}(0,T;H^{1}_{d}(\mathbb{T}^{3})), and

    3. (c)

      a real-valued scalar field σ∈H1​(0,T;L2​(𝕋3))∩L2​(0,T;H1​(𝕋3))\sigma\in H^{1}(0,T;L^{2}(\mathbb{T}^{3}))\cap L^{2}(0,T;H^{1}(\mathbb{T}^{3})).

Remark 2.2.

The last two terms in  (c-NSE) are pure gradients, and thus we can absorb them into the pressure, relabeling the latter as qq. Due to the use of the divergence-free test functions, all gradient terms in the definition of the weak solution disappear.

Now, we state the main result.

Theorem 2.3 (Global existence).

Fix p∈[1,∞)p\in[1,\infty), and choose 0<δ<min⁡{13,1p−1}0<\delta<\min\{\frac{1}{3},\frac{1}{p-1}\}. Let ψ0∈H2​(𝕋3)\psi_{0}\in~{}H^{2}(\mathbb{T}^{3}), and let u0∈Hd1​(𝕋3)u_{0}\in H^{1}_{d}(\mathbb{T}^{3}). Suppose 0<mi≤ρ0≤Mi<∞0<m_{i}\leq\rho_{0}\leq M_{i}<\infty a.e. in  𝕋3\mathbb{T}^{3}. Then, there exists a global weak solution (ψ,u,ρ)(\psi,u,\rho) to the Pitaevskii model such that the density is always bounded between mf∈(0,mi)m_{f}\in(0,m_{i}) and Mf:=Mi+mi−mfM_{f}:=M_{i}+m_{i}-m_{f}, provided the initial data satisfy the smallness condition

‖ψ0‖Hx22+‖u0‖Hx12+‖ψ0‖Lxp+2p+2≤ε0​(λ,μ,ν,mi,Mi,mf,α,p).\norm{\psi_{0}}_{H^{2}_{x}}^{2}+\norm{u_{0}}_{H^{1}_{x}}^{2}+\norm{\psi_{0}}_{L^{p+2}_{x}}^{p+2}\leq\varepsilon_{0}(\lambda,\mu,\nu,m_{i},M_{i},m_{f},\alpha,p). (2.4)

The solution has the regularity

ψ∈C​([0,∞);H2​(𝕋3))∩L2​(0,∞;H3​(𝕋3))∩L1+δ​(0,∞;H˙72+δ1​(𝕋3))\displaystyle\psi\in C([0,\infty);H^{2}(\mathbb{T}^{3}))\cap L^{2}(0,\infty;H^{3}(\mathbb{T}^{3}))\cap L^{1+\delta}(0,\infty;\dot{H}^{\frac{7}{2}+\delta_{1}}(\mathbb{T}^{3})) (2.5)
u∈C​([0,∞);Hd1​(𝕋3))∩L2​(0,∞;H2​(𝕋3))\displaystyle u\in C([0,\infty);H^{1}_{d}(\mathbb{T}^{3}))\cap L^{2}(0,\infty;H^{2}(\mathbb{T}^{3})) (2.6)
ρ∈L∞​([0,∞)×𝕋3)∩C​([0,∞);Ls​(𝕋3)),\displaystyle\rho\in L^{\infty}([0,\infty)\times\mathbb{T}^{3})\cap C([0,\infty);L^{s}(\mathbb{T}^{3})), (2.7)

for a sufficiently small δ1>0\delta_{1}>0, and 1≤s≤61\leq s\leq 6. Additionally, the solution satisfies the energy equality

12​‖ρ​(t)​u​(t)‖Lx22+12​‖∇ψ​(t)‖Lx22+2​μp+2​‖ψ​(t)‖Lxp+2p+2\displaystyle\frac{1}{2}\norm{\sqrt{\rho(t)}u(t)}_{L^{2}_{x}}^{2}+\frac{1}{2}\norm{\nabla\psi(t)}_{L^{2}_{x}}^{2}+\frac{2\mu}{p+2}\norm{\psi(t)}_{L^{p+2}_{x}}^{p+2} (2.8)
+ν​‖∇u‖L[0,t]2​Lx22+α​‖ρ​u‖L[0,t]2​Lx22+2​λ​‖B​ψ‖L[0,t]2​Lx22\displaystyle\quad+\nu\norm{\nabla u}_{L^{2}_{[0,t]}L^{2}_{x}}^{2}+\alpha\norm{\sqrt{\rho}u}_{L^{2}_{[0,t]}L^{2}_{x}}^{2}+2\lambda\norm{B\psi}_{L^{2}_{[0,t]}L^{2}_{x}}^{2}
=12​‖ρ0​u0‖Lx22+12​‖∇ψ0‖Lx22+2​μp+2​‖ψ0‖Lxp+2p+2a.e.t∈[0,∞).\displaystyle=\frac{1}{2}\norm{\sqrt{\rho_{0}}u_{0}}_{L^{2}_{x}}^{2}+\frac{1}{2}\norm{\nabla\psi_{0}}_{L^{2}_{x}}^{2}+\frac{2\mu}{p+2}\norm{\psi_{0}}_{L^{p+2}_{x}}^{p+2}\quad a.e.\ t\in[0,\infty).

The proof of Theorem  2.3 is based on a priori estimates, a semi-Galerkin scheme to construct solutions, and an adaptation of the classical renormalization procedure for the density  [Lio96, Theorem 2.4]. Since the coupling operator BB contains the velocity uu by itself (and not in combination with ρ\rho), we limit the calculations to when the density has a positive lower bound. This gives us an indication to the level of regularity expected of the RHS of  (c-CON), which in turn defines the spaces in which ψ\psi and uu belong to. In order to achieve this, we derive the required a priori control for the wavefunction and velocity, while ensuring that the density is neither differentiated nor does it become zero anywhere in the domain.

Before a more specific discussion on the method of proof, a few remarks about the result are warranted.

Remark 2.4.

Once we have ρ∈Lt,x∞\rho\in L^{\infty}_{t,x}, the renormalization procedure from  [DL89] can be used to show that ρ\rho indeed belongs to Ct​LxsC_{t}L^{s}_{x}. In a finite 3D domain, the Sobolev embedding H1⊂LsH^{1}\subset L^{s} for 1≤s≤61\leq s\leq 6 accounts for the integrability in Theorem  2.3. It is worth mentioning that in the analogous result in 2D, we get 1≤s<∞1\leq s<\infty due to the more favorable embedding.

Remark 2.5.

The regularity of the solutions seem to suggest that the wavefunction and velocity are strong solutions. Indeed this is true, as they are strongly continuous in their topologies. On the other hand, the density is truly a weak solution and is the reason for referring to the triplet as a weak solution. This low regularity of the density influences the nature of the calculations that follow, and in fact also prevent us from concluding uniqueness of the weak solutions. See [JT22b] for results akin to weak-strong uniqueness for the Pitaevskii model.

Remark 2.6.

In Lemma  3.1, we establish that the mass of superfluid decreases with time (algebraically) and goes to 0 as t→∞t\rightarrow\infty. Due to overall mass conservation, this means an increase in normal fluid mass, and an eventual conversion of all the superfluid into normal fluid. This inter-phase mass transfer is one of the underlying physical phenomena that the Pitaevskii model was designed to explain. In the macro-scale models of superfluidity (like the HVBK equations [Hol01, JT21], the coupling between the two fluids is suggestively called mutual friction, as it dissipates the overall energy of the system. In the micro-scale model that we are concerned with, such an energy sink can be interpreted as “heating up” the Bose-Einstein condensate (superfluid particles) into excited states (normal fluid particles).

Remark 2.7.

We point out that Theorem  2.3 is also valid for the 2D Pitaevskii model (with the same regularity, except for the density renormalization argument holding for all 1≤s<∞1\leq s<\infty).

Remark 2.8.

It would be interesting to consider the Pitaevskii model in ℝ3\mathbb{R}^{3}. It may help to localize the dynamics by using an external confining potential (in the NLS) that rises in strength with increasing distance from the origin. This would be akin to trapped-ion quantum systems in condensed matter physics. In such a scenario, it is plausible to expect that the current results from 𝕋3\mathbb{T}^{3} would continue to be valid in ℝ3\mathbb{R}^{3} as well. We thank one of the anonymous referees for posing this question.

2.2. The strategy

As indicated above, the main difficulty is to guarantee that if we begin from ρ0\rho_{0} that is bounded below, then the time evolution does not result in a degeneration where ρ\rho vanishes at certain points in the domain. Hence, we define our existence time so that ρ\rho does not go below a fixed lower bound until t=T∗t=T_{*}. So we aim to show that the chosen lower bound can be maintained for an arbitrarily long time.

Definition 2.9 (Local existence time).

Start with an initial density field 0<mi≤ρ0​(x)≤Mi<∞0<m_{i}\leq\rho_{0}(x)\leq M_{i}<\infty. Given 0<mf<mi0<m_{f}<m_{i}, we define the existence time for the solution as

T∗:=inf{t>0|inf𝕋3ρ(t,x)=mf}.T_{*}:=\inf\{t>0\ \lvert\ \inf_{\mathbb{T}^{3}}\rho(t,x)=m_{f}\}. (2.9)

Consider the Lagrangian path of a particle starting at y∈𝕋3y\in\mathbb{T}^{3}, as it is advected by the local velocity. These characteristic curves are denoted by Xy​(t)X_{y}(t) and solve the ODE given by

dd​t​Xy​(t)\displaystyle\frac{d}{dt}X_{y}(t) =u​(t,Xy​(t))\displaystyle=u(t,X_{y}(t)) (2.10)
Xy​(0)\displaystyle X_{y}(0) =y∈𝕋3,\displaystyle=y\in\mathbb{T}^{3},

where uu is the velocity of the normal fluid. Traveling along such a curve, we observe that

ρ​(t,Xy​(t))=ρ0​(y)+2​λ​Re​∫0tψ¯​B​ψ​(τ,Xy​(τ))​𝑑τ\rho(t,X_{y}(t))=\rho_{0}(y)+2\lambda\real\int_{0}^{t}\bar{\psi}B\psi(\tau,X_{y}(\tau))\ d\tau (2.11)

is a (formal) solution to the continuity equation. From  (2.9) and  (2.11), it is clear that a sufficient condition to ensure the density is bounded from below by mf∈(0,mi)m_{f}\in(0,m_{i}) is

2​λ​∫0T|ψ¯​B​ψ|​(τ,Xy​(τ))​𝑑τ<mi−mf.2\lambda\int_{0}^{T}\lvert\bar{\psi}B\psi\rvert(\tau,X_{y}(\tau))\ d\tau<m_{i}-m_{f}. (2.12)

This is, in turn, guaranteed by the sufficiency

2​λ​∫0T‖ψ‖Lx∞​‖B​ψ‖Lx∞<mi−mf.2\lambda\int_{0}^{T}\norm{\psi}_{L^{\infty}_{x}}\norm{B\psi}_{L^{\infty}_{x}}<m_{i}-m_{f}. (2.13)

By selecting small enough data so that all the arguments may be bootstrapped, it is possible to achieve  (2.13) independently of T>0T>0. Since B​ψB\psi involves a second-order derivative, its Lx∞L^{\infty}_{x} norm translates into high-regularity Sobolev spaces. For uu, this means showing that it belongs to Lt2​Hx2∩Ht1​Lx2L^{2}_{t}H^{2}_{x}\cap H^{1}_{t}L^{2}_{x}, which also proves useful in establishing strong continuity in time for the solution. As for the wavefunction, we make use of the parabolic nature of  (c-NLS) to derive the necessary regularity (see Lemma  2.11 below). It is important to remember that throughout these calculations, we handle the density only in Lx∞L^{\infty}_{x}, and not in any derivative spaces.

2.3. Elliptic operators and maximal parabolic regularity

We now define uniform ellipticity in the context of complex-valued Banach spaces, before stating the maximal parabolic regularity result that will be utilized in Lemma  3.4.

Definition 2.10 (Uniform (K,ζ)(K,\zeta)-ellipticity [PS01]).

For a complex-valued Banach space XX, consider the differential operator

A​(t,x)=∑|α|=2​maα​(t,x)​∂α,A(t,x)=\sum_{\absolutevalue{\alpha}=2m}a_{\alpha}(t,x)\partial^{\alpha}, (2.14)

with domain D​(A​(t,x))⊂XD(A(t,x))\subset X, where α\alpha is a multi-index and ∂α\partial^{\alpha} denotes spatial derivatives. The coefficients aαa_{\alpha} are bounded and uniformly continuous functions from [0,T]×ℝn[0,T]\times\mathbb{R}^{n} to ℂN×N\mathbb{C}^{N\times N} for some n,N∈ℕn,N\in\mathbb{N}. The principal symbol associated with this operator is

A~​(t,x,ξ)=(−1)m​∑|α|=2​maα​(t,x)​ξα.\tilde{A}(t,x,\xi)=(-1)^{m}\sum_{\absolutevalue{\alpha}=2m}a_{\alpha}(t,x)\xi^{\alpha}. (2.15)

The operator A​(t,x)A(t,x) is said to be uniformly (K,ζ)(K,\zeta)-elliptic if there exist K≥1K\geq 1 and ζ∈[0,π2)\zeta\in\left[0,\frac{\pi}{2}\right) such that

  1. (i)

    ∑|α|=2​m‖aα‖Lt,x∞≤K\sum_{\absolutevalue{\alpha}=2m}\norm{a_{\alpha}}_{L^{\infty}_{t,x}}\leq K,

  2. (ii)

    |A~​(t,x,ξ)−1|≤K\absolutevalue{\tilde{A}(t,x,\xi)^{-1}}\leq K, and

  3. (iii)

    σ​(A~​(t,x,ξ))⊂Σζ∖{0}\sigma\left(\tilde{A}(t,x,\xi)\right)\subset\Sigma_{\zeta}\setminus\{0\},

for (t,x)∈[0,T]×ℝn(t,x)\in[0,T]\times\mathbb{R}^{n}, and ξ∈ℝn\xi\in\mathbb{R}^{n} with |ξ|=1\absolutevalue{\xi}=1. Here, σ​(B)\sigma(B) refers to the spectrum of the operator BB, and Σζ:={z∈ℂ:|arg⁡z|≤ζ}\Sigma_{\zeta}:=\{z\in\mathbb{C}:\absolutevalue{\arg{z}}\leq\zeta\} is a sector in the right half of the complex plane.

It is possible to show that a uniformly (K,ζ)(K,\zeta)-elliptic operator generates an analytic semigroup of negative type, leading to the maximal regularity below.

Lemma 2.11 (Maximal parabolic regularity).

Let XX be a (complex-valued) reflexive Banach space and A:X1→XA\colon X_{1}\rightarrow X be a (K,ζ)(K,\zeta)-elliptic operator defined on D​(A)=X1⊂XD(A)=X_{1}\subset X. For T>0T>0, consider the initial value problem

∂tu​(t)+A​u​(t)=f​(t)u​(0)=u0,\begin{gathered}\partial_{t}u(t)+Au(t)=f(t)\\ u(0)=u_{0},\end{gathered} (2.16)

where f∈Lr​([0,T];X)f\in L^{r}([0,T];X) with 1<r<∞1<r<\infty and u0∈Xu_{0}\in X. If it is known that u0u_{0} belongs to the real interpolation space Y:=(X,X1)1−1r,rY:=(X,X_{1})_{1-\frac{1}{r},r}, then there exists a unique solution u∈W1,r​([0,T];X)∩Lr​([0,T];X1)u\in W^{1,r}([0,T];X)\cap L^{r}([0,T];X_{1}) to  (2.16) satisfying the maximal parabolic regularity estimate

‖u‖Lr​([0,T];X)+‖u‖Lr​([0,T];X1)+‖∂tu‖Lr​([0,T];X)≤Cr​(‖u0‖Y+‖f‖Lr​([0,T];X)).\norm{u}_{L^{r}([0,T];X)}+\norm{u}_{L^{r}([0,T];X_{1})}+\norm{\partial_{t}u}_{L^{r}([0,T];X)}\leq C_{r}\left(\norm{u_{0}}_{Y}+\norm{f}_{L^{r}([0,T];X)}\right). (2.17)

For the proof of this lemma, see  [PS01, Section 4].

3. A priori estimates

We now derive the required a priori estimates, using formal calculations. We assume the wavefunction and velocity are smooth functions and that the density is bounded from below by mf>0m_{f}>0 in [0,T][0,T]. Here, TT is any time less than the local existence time T∗T_{*}, and is extended to global existence in Section  3.5. The derivations of some estimates are identical to the 2D case  [JJK23], and are not repeated here.

3.1. Superfluid mass estimate

Lemma 3.1 (Algebraic decay rate of superfluid mass).

The mass S​(t)S(t) of the superfluid decays algebraically in time as (1+t)−2p(1+t)^{-\frac{2}{p}}. Specifically,

S​(t):=‖ψ‖Lx22​(t)≲S0(1+S0p2​t)2p,t∈[0,T],S(t):=\norm{\psi}_{L^{2}_{x}}^{2}(t)\lesssim\frac{S_{0}}{\left(1+S_{0}^{\frac{p}{2}}t\right)^{\frac{2}{p}}},\quad t\in[0,T], (3.1)

where S0:=‖ψ0‖Lx22S_{0}:=\norm{\psi_{0}}_{L^{2}_{x}}^{2} is the initial mass of the superfluid.

Proof.

See Lemma 3.1 of  [JJK23]. ∎

3.2. Energy estimate

The energy of the system is defined as

E​(t):=12​‖ρ​(t)​u​(t)‖Lx22+12​‖∇ψ​(t)‖Lx22+2​μp+2​‖ψ​(t)‖Lxp+2p+2.E(t):=\frac{1}{2}\norm{\sqrt{\rho(t)}u(t)}_{L^{2}_{x}}^{2}+\frac{1}{2}\norm{\nabla\psi(t)}_{L^{2}_{x}}^{2}+\frac{2\mu}{p+2}\norm{\psi(t)}_{L^{p+2}_{x}}^{p+2}. (3.2)
Lemma 3.2.

The energy balance of the system is given by

dd​t​E​(t)+ν​‖∇u‖Lx22+α​‖ρ​u‖Lx22+2​λ​‖B​ψ‖Lx22=0.\frac{d}{dt}E(t)+\nu\norm{\nabla u}_{L^{2}_{x}}^{2}+\alpha\norm{\sqrt{\rho}u}_{L^{2}_{x}}^{2}+2\lambda\norm{B\psi}_{L^{2}_{x}}^{2}=0. (3.3)
Proof.

See Section 3.2 of  [JJK23]. ∎

Integrating  (3.3) in time, we observe that the energy is bounded from above as

E​(t)+ν​‖∇u‖L[0,T]2​Lx22+α​‖ρ​u‖L[0,T]2​Lx22+2​λ​‖B​ψ‖L[0,T]2​Lx22=E0,t∈[0,T],E(t)+\nu\norm{\nabla u}_{L^{2}_{[0,T]}L^{2}_{x}}^{2}+\alpha\norm{\sqrt{\rho}u}_{L^{2}_{[0,T]}L^{2}_{x}}^{2}+2\lambda\norm{B\psi}_{L^{2}_{[0,T]}L^{2}_{x}}^{2}\\ =E_{0},\quad t\in[0,T], (3.4)

where

E0:=12​‖ρ0​u0‖Lx22+12​‖∇ψ0‖Lx22+2​μp+2​‖ψ0‖Lxp+2p+2E_{0}:=\frac{1}{2}\norm{\sqrt{\rho_{0}}u_{0}}_{L^{2}_{x}}^{2}+\frac{1}{2}\norm{\nabla\psi_{0}}_{L^{2}_{x}}^{2}+\frac{2\mu}{p+2}\norm{\psi_{0}}_{L^{p+2}_{x}}^{p+2} (3.5)

denotes the initial energy of the system. Next, we show that the energy is not just bounded, but also decays (algebraically) with time. In order to achieve this, we observe that ‖B​ψ‖Lx2\norm{B\psi}_{L^{2}_{x}} can be rewritten as ‖D2​ψ‖Lx2\norm{D^{2}\psi}_{L^{2}_{x}}, at the expense of some nonlinear terms on the RHS. More precisely (see equation (3.14) in  [JJK23] for the exact derivation),

‖B​ψ‖Lx22≥18​‖D2​ψ‖Lx22−C​‖|u|2​ψ‖Lx22−C​‖u⋅∇ψ‖Lx22+1C​‖ψ‖Lx2​p+22​p+2+1C​‖∇|ψ|p2+1‖Lx22.\displaystyle\norm{B\psi}_{L^{2}_{x}}^{2}\geq\frac{1}{8}\norm{D^{2}\psi}_{L^{2}_{x}}^{2}-C\norm{\absolutevalue{u}^{2}\psi}_{L^{2}_{x}}^{2}-C\norm{u\cdot\nabla\psi}_{L^{2}_{x}}^{2}+\frac{1}{C}\norm{\psi}_{L^{2p+2}_{x}}^{2p+2}+\frac{1}{C}\norm{\nabla\absolutevalue{\psi}^{\frac{p}{2}+1}}_{L^{2}_{x}}^{2}.

Thus,  (3.3) now becomes

d​Ed​t+ν​‖∇u‖Lx22+α​‖ρ​u‖Lx22+λ4​‖D2​ψ‖Lx22+1C​‖ψ‖Lx2​p+22​p+2+1C​‖∇|ψ|p2+1‖Lx22\displaystyle\frac{dE}{dt}+\nu\norm{\nabla u}_{L^{2}_{x}}^{2}+\alpha\norm{\sqrt{\rho}u}_{L^{2}_{x}}^{2}+\frac{\lambda}{4}\norm{D^{2}\psi}_{L^{2}_{x}}^{2}+\frac{1}{C}\norm{\psi}_{L^{2p+2}_{x}}^{2p+2}+\frac{1}{C}\norm{\nabla\absolutevalue{\psi}^{\frac{p}{2}+1}}_{L^{2}_{x}}^{2} (3.6)
≲‖|u|2​ψ‖Lx22+‖u⋅∇ψ‖Lx22\displaystyle\lesssim\norm{\absolutevalue{u}^{2}\psi}_{L^{2}_{x}}^{2}+\norm{u\cdot\nabla\psi}_{L^{2}_{x}}^{2}
=:I1+I2.\displaystyle=:I_{1}+I_{2}.

The first term on the RHS is estimated as

I1≲‖u‖Lx64​‖ψ‖Lx62≲‖u‖Hx14​‖ψ‖Hx12I_{1}\lesssim\norm{u}_{L^{6}_{x}}^{4}\norm{\psi}_{L^{6}_{x}}^{2}\lesssim\norm{u}_{H^{1}_{x}}^{4}\norm{\psi}_{H^{1}_{x}}^{2}

using Hölder’s inequality and Sobolev embedding. For the second term in  (3.6), we interpolate the Lx3L^{3}_{x} norm, and apply the Poincaré, Hölder’s, and Young’s inequalities, as well as Sobolev embedding to get

I2≲‖u‖Lx62​‖∇ψ‖Lx32≲‖u‖Hx12​‖∇ψ‖Lx2​‖D2​ψ‖Lx2≤Cκ​‖u‖Hx14​‖∇ψ‖Lx22+κ​‖D2​ψ‖Lx22.\displaystyle I_{2}\lesssim\norm{u}_{L^{6}_{x}}^{2}\norm{\nabla\psi}_{L^{3}_{x}}^{2}\lesssim\norm{u}_{H^{1}_{x}}^{2}\norm{\nabla\psi}_{L^{2}_{x}}\norm{D^{2}\psi}_{L^{2}_{x}}\leq C_{\kappa}\norm{u}_{H^{1}_{x}}^{4}\norm{\nabla\psi}_{L^{2}_{x}}^{2}+\kappa\norm{D^{2}\psi}_{L^{2}_{x}}^{2}.

We also use the Poincaré inequality to convert the last term on the LHS of  (3.6) into a coercive term for the internal energy term 2​μp+2​‖ψ‖Lxp+2p+2\frac{2\mu}{p+2}\norm{\psi}_{L^{p+2}_{x}}^{p+2} in E​(t)E(t). To this end, we observe that

‖ψ‖Lxp+2p+2\displaystyle\norm{\psi}_{L^{p+2}_{x}}^{p+2} ≤‖|ψ|p2+1−1|𝕋3|​∫𝕋3|ψ|p2+1‖Lx22+‖1|𝕋3|​∫𝕋3|ψ|p2+1‖Lx22≲‖∇|ψ|p2+1‖Lx22+‖ψ‖Lp2+1p+2\displaystyle\leq\norm{\absolutevalue{\psi}^{\frac{p}{2}+1}-\frac{1}{\absolutevalue{\mathbb{T}^{3}}}\int_{\mathbb{T}^{3}}\absolutevalue{\psi}^{\frac{p}{2}+1}}_{L^{2}_{x}}^{2}+\norm{\frac{1}{\absolutevalue{\mathbb{T}^{3}}}\int_{\mathbb{T}^{3}}\absolutevalue{\psi}^{\frac{p}{2}+1}}_{L^{2}_{x}}^{2}\lesssim\norm{\nabla\absolutevalue{\psi}^{\frac{p}{2}+1}}_{L^{2}_{x}}^{2}+\norm{\psi}_{L^{\frac{p}{2}+1}}^{p+2} (3.7)
≤C​‖∇|ψ|p2+1‖Lx22+κ​‖ψ‖Lxp+2p+2+Cκ​‖ψ‖Lx2p+2.\displaystyle\leq C\norm{\nabla\absolutevalue{\psi}^{\frac{p}{2}+1}}_{L^{2}_{x}}^{2}+\kappa\norm{\psi}_{L^{p+2}_{x}}^{p+2}+C_{\kappa}\norm{\psi}_{L^{2}_{x}}^{p+2}.

In the last inequality, we interpolated between the Lxp+2L^{p+2}_{x} and Lx2L^{2}_{x} norms, which is valid for p>2p>2. For a sufficiently small κ\kappa, the second term on the RHS is absorbed into the LHS. On the other hand, when 1≤p≤21\leq p\leq 2, the finite size of the domain implies Lx2⊆Lxp2+1L^{2}_{x}\subseteq L^{\frac{p}{2}+1}_{x}, which again leads to (3.7). Thus, for any p≥1p\geq 1,  (3.6) becomes

d​Ed​t+ν​‖∇u‖Lx22+α​‖ρ​u‖Lx22+1C​‖D2​ψ‖Lx22+1C​‖ψ‖Lxp+2p+2+1C​‖ψ‖Lx2​p+22​p+2\displaystyle\frac{dE}{dt}+\nu\norm{\nabla u}_{L^{2}_{x}}^{2}+\alpha\norm{\sqrt{\rho}u}_{L^{2}_{x}}^{2}+\frac{1}{C}\norm{D^{2}\psi}_{L^{2}_{x}}^{2}+\frac{1}{C}\norm{\psi}_{L^{p+2}_{x}}^{p+2}+\frac{1}{C}\norm{\psi}_{L^{2p+2}_{x}}^{2p+2} (3.8)
≤C​‖ψ‖Lx2p+2+C​‖u‖Hx14​‖ψ‖Hx12.\displaystyle\leq C\norm{\psi}_{L^{2}_{x}}^{p+2}+C\norm{u}_{H^{1}_{x}}^{4}\norm{\psi}_{H^{1}_{x}}^{2}.

In order to show a decaying norm, we need coercive terms on the LHS, which have been achieved. To control the RHS (particularly the second term), we derive a balance equation for a higher-order energy X​(t)X(t), defined in (3.19). Combining E​(t)E(t) with X​(t)X(t) allows us to close the estimates.

3.3. Higher-order energy estimate

We now present more a priori bounds for ψ\psi and uu, involving one more derivative than the energy EE.

3.3.1. The Schrödinger equation

Acting upon  (c-NLS) with the Laplacian −Δ-\Delta, multiplying by −Δ​ψ¯-\Delta\bar{\psi}, taking the real part, and integrating over the domain yields

12​dd​t​‖Δ​ψ‖Lx22\displaystyle\frac{1}{2}\frac{d}{dt}\norm{\Delta\psi}_{L^{2}_{x}}^{2} =−λ​Re​∫𝕋3(Δ2​ψ¯)​B​ψ+μ​Im​∫𝕋3(Δ2​ψ¯)​|ψ|p​ψ\displaystyle=-\lambda\real\int_{\mathbb{T}^{3}}(\Delta^{2}\bar{\psi})B\psi+\mu\imaginary\int_{\mathbb{T}^{3}}(\Delta^{2}\bar{\psi})\absolutevalue{\psi}^{p}\psi (3.9)
=:I3+I4.\displaystyle=:I_{3}+I_{4}.

The first term on the RHS of  (c-NLS) leads to a term which vanishes due to the periodic boundary conditions. We now estimate the RHS of  (3.9). For the first term, we have

I3\displaystyle I_{3} =λ​Re​∫𝕋3∇(Δ​ψ¯)⋅∇(−12​Δ​ψ+12​|u|2​ψ+i​u⋅∇ψ+μ​|ψ|p​ψ)\displaystyle=\lambda\real\int_{\mathbb{T}^{3}}\nabla(\Delta\bar{\psi})\cdot\nabla\left(-\frac{1}{2}\Delta\psi+\frac{1}{2}\absolutevalue{u}^{2}\psi+iu\cdot\nabla\psi+\mu\absolutevalue{\psi}^{p}\psi\right)
=−λ2​‖D3​ψ‖Lx22+λ​Re​∫𝕋3∇(Δ​ψ¯)⋅∇(12​|u|2​ψ+i​u⋅∇ψ+μ​|ψ|p​ψ)\displaystyle=-\frac{\lambda}{2}\norm{D^{3}\psi}_{L^{2}_{x}}^{2}+\lambda\real\int_{\mathbb{T}^{3}}\nabla(\Delta\bar{\psi})\cdot\nabla\left(\frac{1}{2}\absolutevalue{u}^{2}\psi+iu\cdot\nabla\psi+\mu\absolutevalue{\psi}^{p}\psi\right)
≤−λ4​‖D3​ψ‖Lx22+C​‖∇(|u|2​ψ)‖Lx22+C​‖∇(u⋅∇ψ)‖Lx22+C​‖∇(|ψ|p​ψ)‖Lx22.\displaystyle\leq-\frac{\lambda}{4}\norm{D^{3}\psi}_{L^{2}_{x}}^{2}+C\norm{\nabla(\absolutevalue{u}^{2}\psi)}_{L^{2}_{x}}^{2}+C\norm{\nabla(u\cdot\nabla\psi)}_{L^{2}_{x}}^{2}+C\norm{\nabla(\absolutevalue{\psi}^{p}\psi)}_{L^{2}_{x}}^{2}.

The first term on the RHS acts as the dissipative term for ψ\psi at this higher-order energy level. For I4I_{4}, we integrate by parts and use Hölder’s inequality to obtain

I4=−μ​Im​∫𝕋3∇(Δ​ψ¯)⋅∇(|ψ|p​ψ)≤λ8​‖D3​ψ‖Lx22+C​‖∇(|ψ|p​ψ)‖Lx22.I_{4}=-\mu\imaginary\int_{\mathbb{T}^{3}}\nabla(\Delta\bar{\psi})\cdot\nabla(\absolutevalue{\psi}^{p}\psi)\leq\frac{\lambda}{8}\norm{D^{3}\psi}_{L^{2}_{x}}^{2}+C\norm{\nabla(\absolutevalue{\psi}^{p}\psi)}_{L^{2}_{x}}^{2}.

Thus,  (3.9) becomes

dd​t​‖Δ​ψ‖Lx22+1C​‖D3​ψ‖Lx22\displaystyle\frac{d}{dt}\norm{\Delta\psi}_{L^{2}_{x}}^{2}+\frac{1}{C}\norm{D^{3}\psi}_{L^{2}_{x}}^{2} ≲‖∇(|u|2​ψ)‖Lx22+‖∇(u⋅∇ψ)‖Lx22+‖∇(|ψ|p​ψ)‖Lx22\displaystyle\lesssim\norm{\nabla\left(\absolutevalue{u}^{2}\psi\right)}_{L^{2}_{x}}^{2}+\norm{\nabla(u\cdot\nabla\psi)}_{L^{2}_{x}}^{2}+\norm{\nabla\left(\absolutevalue{\psi}^{p}\psi\right)}_{L^{2}_{x}}^{2} (3.10)
=:I5+I6+I7.\displaystyle=:I_{5}+I_{6}+I_{7}.

The first term is bounded using the Poincaré inequality, Sobolev embedding, and Lebesgue interpolation as

I5\displaystyle I_{5} ≲‖u‖Lx62​‖∇u‖Lx32​‖ψ‖Lx∞2+‖u‖Lx64​‖∇ψ‖Lx62\displaystyle\lesssim\norm{u}_{L^{6}_{x}}^{2}\norm{\nabla u}_{L^{3}_{x}}^{2}\norm{\psi}_{L^{\infty}_{x}}^{2}+\norm{u}_{L^{6}_{x}}^{4}\norm{\nabla\psi}_{L^{6}_{x}}^{2}
≲‖u‖Hx12​‖∇u‖Lx2​‖Δ​u‖Lx2​‖ψ‖Hx22+‖u‖Hx14​‖Δ​ψ‖Lx22\displaystyle\lesssim\norm{u}_{H^{1}_{x}}^{2}\norm{\nabla u}_{L^{2}_{x}}\norm{\Delta u}_{L^{2}_{x}}\norm{\psi}_{H^{2}_{x}}^{2}+\norm{u}_{H^{1}_{x}}^{4}\norm{\Delta\psi}_{L^{2}_{x}}^{2}
≤Cκ​‖u‖Hx14​‖∇u‖Lx22​‖ψ‖Hx24+κ​‖Δ​u‖Lx22+C​‖u‖Hx14​‖Δ​ψ‖Lx22.\displaystyle\leq C_{\kappa}\norm{u}_{H^{1}_{x}}^{4}\norm{\nabla u}_{L^{2}_{x}}^{2}\norm{\psi}_{H^{2}_{x}}^{4}+\kappa\norm{\Delta u}_{L^{2}_{x}}^{2}+C\norm{u}_{H^{1}_{x}}^{4}\norm{\Delta\psi}_{L^{2}_{x}}^{2}.

We applied Young’s inequality to extract out dissipative terms in the last step. Again, κ\kappa denotes a small number whose value shall be fixed later on, and CκC_{\kappa} is a constant whose value depends on κ\kappa and the system parameters. For the second term on the RHS of  (3.10), we have

I6\displaystyle I_{6} ≲‖∇u‖Lx32​‖∇ψ‖Lx62+‖u‖Lx62​‖D2​ψ‖Lx32\displaystyle\lesssim\norm{\nabla u}_{L^{3}_{x}}^{2}\norm{\nabla\psi}_{L^{6}_{x}}^{2}+\norm{u}_{L^{6}_{x}}^{2}\norm{D^{2}\psi}_{L^{3}_{x}}^{2}
≲‖∇u‖Lx2​‖Δ​u‖Lx2​‖Δ​ψ‖Lx22+‖u‖Hx12​‖Δ​ψ‖Lx2​‖D3​ψ‖Lx2\displaystyle\lesssim\norm{\nabla u}_{L^{2}_{x}}\norm{\Delta u}_{L^{2}_{x}}\norm{\Delta\psi}_{L^{2}_{x}}^{2}+\norm{u}_{H^{1}_{x}}^{2}\norm{\Delta\psi}_{L^{2}_{x}}\norm{D^{3}\psi}_{L^{2}_{x}}
≤Cκ​‖∇u‖Lx22​‖Δ​ψ‖Lx24+κ​‖Δ​u‖Lx22+Cκ​‖u‖Hx14​‖Δ​ψ‖Lx22+κ​‖D3​ψ‖Lx22\displaystyle\leq C_{\kappa}\norm{\nabla u}_{L^{2}_{x}}^{2}\norm{\Delta\psi}_{L^{2}_{x}}^{4}+\kappa\norm{\Delta u}_{L^{2}_{x}}^{2}+C_{\kappa}\norm{u}_{H^{1}_{x}}^{4}\norm{\Delta\psi}_{L^{2}_{x}}^{2}+\kappa\norm{D^{3}\psi}_{L^{2}_{x}}^{2}

Finally, we apply the Poincaré inequality and Sobolev embedding to bound I7I_{7}. This leads to

I7\displaystyle I_{7} ≲‖|ψ|p​|∇ψ|‖Lx22≲‖ψ‖Lx∞2​p​‖∇ψ‖Lx22≲(‖ψ‖Lx22​p+‖Δ​ψ‖Lx22​p)​‖∇ψ‖Lx22\displaystyle\lesssim\norm{\absolutevalue{\psi}^{p}\absolutevalue{\nabla\psi}}_{L^{2}_{x}}^{2}\lesssim\norm{\psi}_{L^{\infty}_{x}}^{2p}\norm{\nabla\psi}_{L^{2}_{x}}^{2}\lesssim\left(\norm{\psi}_{L^{2}_{x}}^{2p}+\norm{\Delta\psi}_{L^{2}_{x}}^{2p}\right)\norm{\nabla\psi}_{L^{2}_{x}}^{2} (3.11)
≲‖ψ‖Lx22​p​‖Δ​ψ‖Lx22+‖Δ​ψ‖Lx22​p+2.\displaystyle\lesssim\norm{\psi}_{L^{2}_{x}}^{2p}\norm{\Delta\psi}_{L^{2}_{x}}^{2}+\norm{\Delta\psi}_{L^{2}_{x}}^{2p+2}.

Combining all these inequalities into  (3.10), and absorbing κ​‖D3​ψ‖Lx22\kappa\norm{D^{3}\psi}_{L^{2}_{x}}^{2} into the LHS, we end up with

dd​t​‖Δ​ψ‖Lx22+1C​‖D3​ψ‖Lx22\displaystyle\frac{d}{dt}\norm{\Delta\psi}_{L^{2}_{x}}^{2}+\frac{1}{C}\norm{D^{3}\psi}_{L^{2}_{x}}^{2} (3.12)
≤Cκ​(‖∇u‖Lx22​‖Δ​ψ‖Lx24+‖u‖Hx14​‖∇u‖Lx22​‖ψ‖Hx24)+C​‖u‖Hx14​‖Δ​ψ‖Lx22+C​‖ψ‖Lx22​p​‖Δ​ψ‖Lx22\displaystyle\leq C_{\kappa}\left(\norm{\nabla u}_{L^{2}_{x}}^{2}\norm{\Delta\psi}_{L^{2}_{x}}^{4}+\norm{u}_{H^{1}_{x}}^{4}\norm{\nabla u}_{L^{2}_{x}}^{2}\norm{\psi}_{H^{2}_{x}}^{4}\right)+C\norm{u}_{H^{1}_{x}}^{4}\norm{\Delta\psi}_{L^{2}_{x}}^{2}+C\norm{\psi}_{L^{2}_{x}}^{2p}\norm{\Delta\psi}_{L^{2}_{x}}^{2}
+C​‖Δ​ψ‖Lx22​p+2+κ​‖Δ​u‖Lx22.\displaystyle\quad+C\norm{\Delta\psi}_{L^{2}_{x}}^{2p+2}+\kappa\norm{\Delta u}_{L^{2}_{x}}^{2}.

This constitutes the higher-order estimate for the wavefunction. Next, we combine this with corresponding estimates for the velocity.

3.3.2. The Navier-Stokes equation

We begin by rewriting  (c-NSE) in the non-conservative form, and applying the Leray projector (see Remark  2.2) to get

𝒫​(ρ​∂tu+ρ​u⋅∇u−ν​Δ​u+α​ρ​u)=𝒫​(−2​λ​Im(∇ψ¯​B​ψ)−2​λ​u​Re(ψ¯​B​ψ)).\mathcal{P}\left(\rho\partial_{t}u+\rho u\cdot\nabla u-\nu\Delta u+\alpha\rho u\right)=\mathcal{P}\left(-2\lambda\imaginary(\nabla\bar{\psi}B\psi)-2\lambda u\real(\bar{\psi}B\psi)\right). (c-NSE’)

Here, 𝒫\mathcal{P} is the Leray projector, which projects a Hilbert space into its divergence-free subspace, thus removing any purely gradient terms. Next, we multiply  (c-NSE’) by ∂tu\partial_{t}u and integrate over the domain. This leads to

∫𝕋3ρ​|∂tu|2+ν2​dd​t​‖∇u‖Lx22\displaystyle\int_{\mathbb{T}^{3}}\rho\absolutevalue{\partial_{t}u}^{2}+\frac{\nu}{2}\frac{d}{dt}\norm{\nabla u}_{L^{2}_{x}}^{2} =−∫𝕋3ρ​u⋅∇u⋅∂tu−2​λ​∫𝕋3∂tu⋅Im(∇ψ¯​B​ψ)\displaystyle=-\int_{\mathbb{T}^{3}}\rho u\cdot\nabla u\cdot\partial_{t}u-2\lambda\int_{\mathbb{T}^{3}}\partial_{t}u\cdot\imaginary(\nabla\bar{\psi}B\psi) (3.13)
−2​λ​∫𝕋3∂tu⋅u​Re(ψ¯​B​ψ)−α​∫𝕋3ρ​u⋅∂tu\displaystyle\qquad-2\lambda\int_{\mathbb{T}^{3}}\partial_{t}u\cdot u\real(\bar{\psi}B\psi)-\alpha\int_{\mathbb{T}^{3}}\rho u\cdot\partial_{t}u
=:I8+I9+I10+I11.\displaystyle=:I_{8}+I_{9}+I_{10}+I_{11}.

Henceforth, we repeatedly use the fact that the density is bounded both above and below (mf≤ρ≤Mf=Mi+mi−mfm_{f}\leq\rho\leq M_{f}=M_{i}+m_{i}-m_{f}) to control the RHS. In particular, ‖u‖Lx2\norm{u}_{L^{2}_{x}} and ‖∂tu‖Lx2\norm{\partial_{t}u}_{L^{2}_{x}} are equivalent to ‖ρ​u‖Lx2\norm{\sqrt{\rho}u}_{L^{2}_{x}} and ‖ρ​∂tu‖Lx2\norm{\sqrt{\rho}\partial_{t}u}_{L^{2}_{x}}, respectively. Thus, for the first term,

I8\displaystyle I_{8} ≤18​‖ρ​∂tu‖Lx22+C​∫𝕋3|u|2​|∇u|2≤18​‖ρ​∂tu‖Lx22+C​‖u‖Lx62​‖∇u‖Lx32\displaystyle\leq\frac{1}{8}\norm{\sqrt{\rho}\partial_{t}u}_{L^{2}_{x}}^{2}+C\int_{\mathbb{T}^{3}}\absolutevalue{u}^{2}\absolutevalue{\nabla u}^{2}\leq\frac{1}{8}\norm{\sqrt{\rho}\partial_{t}u}_{L^{2}_{x}}^{2}+C\norm{u}_{L^{6}_{x}}^{2}\norm{\nabla u}_{L^{3}_{x}}^{2}
≤18​‖ρ​∂tu‖Lx22+Cκ​‖u‖Hx14​‖∇u‖Lx22+κ​‖Δ​u‖Lx22.\displaystyle\leq\frac{1}{8}\norm{\sqrt{\rho}\partial_{t}u}_{L^{2}_{x}}^{2}+C_{\kappa}\norm{u}_{H^{1}_{x}}^{4}\norm{\nabla u}_{L^{2}_{x}}^{2}+\kappa\norm{\Delta u}_{L^{2}_{x}}^{2}.

In going from the second line to the third, we use the Sobolev embeddings and Lebesgue interpolation. Finally, Young’s inequality lets us extract the required dissipative term. For the second integral of  (3.13), we have

I9\displaystyle I_{9} ≤18​‖ρ​∂tu‖Lx22+C​‖∇ψ‖Lx62​‖B​ψ‖Lx32\displaystyle\leq\frac{1}{8}\norm{\sqrt{\rho}\partial_{t}u}_{L^{2}_{x}}^{2}+C\norm{\nabla\psi}_{L^{6}_{x}}^{2}\norm{B\psi}_{L^{3}_{x}}^{2}
≤18​‖ρ​∂tu‖Lx22+C​‖Δ​ψ‖Lx22​‖B​ψ‖Lx2​‖B​ψ‖Hx1\displaystyle\leq\frac{1}{8}\norm{\sqrt{\rho}\partial_{t}u}_{L^{2}_{x}}^{2}+C\norm{\Delta\psi}_{L^{2}_{x}}^{2}\norm{B\psi}_{L^{2}_{x}}\norm{B\psi}_{H^{1}_{x}}
≤18​‖ρ​∂tu‖Lx22+Cκ​‖B​ψ‖Lx22​(‖Δ​ψ‖Lx24+‖Δ​ψ‖Lx22)+κ​‖∇(B​ψ)‖Lx22\displaystyle\leq\frac{1}{8}\norm{\sqrt{\rho}\partial_{t}u}_{L^{2}_{x}}^{2}+C_{\kappa}\norm{B\psi}_{L^{2}_{x}}^{2}\left(\norm{\Delta\psi}_{L^{2}_{x}}^{4}+\norm{\Delta\psi}_{L^{2}_{x}}^{2}\right)+\kappa\norm{\nabla(B\psi)}_{L^{2}_{x}}^{2}

where the B​ψB\psi term is handled via interpolation and Young’s inequality, while the term ∇ψ\nabla\psi is bounded using Sobolev embedding. For the third integral,

I10\displaystyle I_{10} ≤18​‖ρ​∂tu‖Lx22+C​‖u‖Lx62​‖ψ‖Lx∞2​‖B​ψ‖Lx32\displaystyle\leq\frac{1}{8}\norm{\sqrt{\rho}\partial_{t}u}_{L^{2}_{x}}^{2}+C\norm{u}_{L^{6}_{x}}^{2}\norm{\psi}_{L^{\infty}_{x}}^{2}\norm{B\psi}_{L^{3}_{x}}^{2}
≤18​‖ρ​∂tu‖Lx22+Cκ​‖B​ψ‖Lx22​(‖u‖Hx12​‖ψ‖Hx22+‖u‖Hx14​‖ψ‖Hx24)+κ​‖∇(B​ψ)‖Lx22\displaystyle\leq\frac{1}{8}\norm{\sqrt{\rho}\partial_{t}u}_{L^{2}_{x}}^{2}+C_{\kappa}\norm{B\psi}_{L^{2}_{x}}^{2}\left(\norm{u}_{H^{1}_{x}}^{2}\norm{\psi}_{H^{2}_{x}}^{2}+\norm{u}_{H^{1}_{x}}^{4}\norm{\psi}_{H^{2}_{x}}^{4}\right)+\kappa\norm{\nabla(B\psi)}_{L^{2}_{x}}^{2}

where the B​ψB\psi term is handled just like in I9I_{9}. Finally, for the last term, we integrate by parts and use  (c-CON), which results in

I11=−α2​dd​t​‖ρ​u‖Lx22+α2​∫𝕋3ρ​u⋅∇|u|2+α​λ​∫𝕋3Re(ψ¯​B​ψ)⁡|u|2.I_{11}=-\frac{\alpha}{2}\frac{d}{dt}\norm{\sqrt{\rho}u}_{L^{2}_{x}}^{2}+\frac{\alpha}{2}\int_{\mathbb{T}^{3}}\rho u\cdot\nabla\absolutevalue{u}^{2}+\alpha\lambda\int_{\mathbb{T}^{3}}\real(\overline{\psi}B\psi)\absolutevalue{u}^{2}. (3.14)

We estimate the second term in  (3.14) via interpolation and a Sobolev embedding. This gives

α2​∫𝕋3ρ​u⋅∇|u|2\displaystyle\frac{\alpha}{2}\int_{\mathbb{T}^{3}}\rho u\cdot\nabla\absolutevalue{u}^{2} ≲‖u‖Lx2​‖u‖Lx3​‖∇u‖Lx6\displaystyle\lesssim\norm{u}_{L^{2}_{x}}\norm{u}_{L^{3}_{x}}\norm{\nabla u}_{L^{6}_{x}}
≤Cκ​‖u‖Lx23​‖u‖Lx6+κ​‖Δ​u‖Lx22≤Cκ​‖u‖Hx14+κ​‖Δ​u‖Lx22.\displaystyle\leq C_{\kappa}\norm{u}_{L^{2}_{x}}^{3}\norm{u}_{L^{6}_{x}}+\kappa\norm{\Delta u}_{L^{2}_{x}}^{2}\leq C_{\kappa}\norm{u}_{H^{1}_{x}}^{4}+\kappa\norm{\Delta u}_{L^{2}_{x}}^{2}.

Similarly, for the third term in  (3.14),

α​λ​∫𝕋3Re(ψ¯​B​ψ)⁡|u|2\displaystyle\alpha\lambda\int_{\mathbb{T}^{3}}\real(\overline{\psi}B\psi)\absolutevalue{u}^{2} ≲‖ψ‖Lx6​‖u‖Lx62​‖B​ψ‖Lx2≤Cκ​‖ψ‖Hx12​‖u‖Hx14+κ​‖B​ψ‖Lx22.\displaystyle\lesssim\norm{\psi}_{L^{6}_{x}}\norm{u}_{L^{6}_{x}}^{2}\norm{B\psi}_{L^{2}_{x}}\leq C_{\kappa}\norm{\psi}_{H^{1}_{x}}^{2}\norm{u}_{H^{1}_{x}}^{4}+\kappa\norm{B\psi}_{L^{2}_{x}}^{2}.

Putting together the above estimates into  (3.13), we end up with

ν​dd​t​‖∇u‖Lx22+‖ρ​∂tu‖Lx22+α​dd​t​‖ρ​u‖Lx22\displaystyle\nu\frac{d}{dt}\norm{\nabla u}_{L^{2}_{x}}^{2}+\norm{\sqrt{\rho}\partial_{t}u}_{L^{2}_{x}}^{2}+\alpha\frac{d}{dt}\norm{\sqrt{\rho}u}_{L^{2}_{x}}^{2} (3.15)
≤Cκ​(‖u‖Hx14​‖∇u‖Lx22+‖u‖Hx14+‖ψ‖Hx12​‖u‖Hx14)\displaystyle\leq C_{\kappa}\left(\norm{u}_{H^{1}_{x}}^{4}\norm{\nabla u}_{L^{2}_{x}}^{2}+\norm{u}_{H^{1}_{x}}^{4}+\norm{\psi}_{H^{1}_{x}}^{2}\norm{u}_{H^{1}_{x}}^{4}\right)
+Cκ​‖B​ψ‖Lx22​(‖Δ​ψ‖Lx22+‖u‖Hx12​‖ψ‖Hx22+‖Δ​ψ‖Lx24+‖u‖Hx14​‖ψ‖Hx24)\displaystyle\quad+C_{\kappa}\norm{B\psi}_{L^{2}_{x}}^{2}\left(\norm{\Delta\psi}_{L^{2}_{x}}^{2}+\norm{u}_{H^{1}_{x}}^{2}\norm{\psi}_{H^{2}_{x}}^{2}+\norm{\Delta\psi}_{L^{2}_{x}}^{4}+\norm{u}_{H^{1}_{x}}^{4}\norm{\psi}_{H^{2}_{x}}^{4}\right)
+κ​‖B​ψ‖Lx22+κ​‖∇(B​ψ)‖Lx22+κ​‖Δ​u‖Lx22\displaystyle\quad+\kappa\norm{B\psi}_{L^{2}_{x}}^{2}+\kappa\norm{\nabla(B\psi)}_{L^{2}_{x}}^{2}+\kappa\norm{\Delta u}_{L^{2}_{x}}^{2}

where CκC_{\kappa} depends on κ\kappa as well as the system parameters.

In order to obtain the higher-order velocity dissipation ‖Δ​u‖Lx22\norm{\Delta u}_{L^{2}_{x}}^{2}, we multiply  (c-NSE’) by −θ​Δ​u-\theta\Delta u, with θ>0\theta>0 to be fixed shortly, and integrate over the domain. This gives

θ​ν​‖Δ​u‖Lx22\displaystyle\theta\nu\norm{\Delta u}_{L^{2}_{x}}^{2} =θ​∫𝕋3ρ​∂tu⋅Δ​u+θ​∫𝕋3(ρ​u⋅∇u)⋅Δ​u+2​λ​θ​∫𝕋3Im(∇ψ¯​B​ψ)⋅Δ​u\displaystyle=\theta\int_{\mathbb{T}^{3}}\rho\partial_{t}u\cdot\Delta u+\theta\int_{\mathbb{T}^{3}}\left(\rho u\cdot\nabla u\right)\cdot\Delta u+2\lambda\theta\int_{\mathbb{T}^{3}}\imaginary(\nabla\bar{\psi}B\psi)\cdot\Delta u (3.16)
+2​λ​θ​∫𝕋3u​Re(ψ¯​B​ψ)⋅Δ​u+α​θ​∫𝕋3ρ​u⋅Δ​u\displaystyle\qquad+2\lambda\theta\int_{\mathbb{T}^{3}}u\real(\bar{\psi}B\psi)\cdot\Delta u+\alpha\theta\int_{\mathbb{T}^{3}}\rho u\cdot\Delta u
=:I12+I13+I14+I15+I16.\displaystyle=:I_{12}+I_{13}+I_{14}+I_{15}+I_{16}.

For the first term, we have

I12≤θ​ν10​‖Δ​u‖Lx22+C​θ​‖ρ​∂tu‖Lx22.I_{12}\leq\frac{\theta\nu}{10}\norm{\Delta u}_{L^{2}_{x}}^{2}+C\theta\norm{\sqrt{\rho}\partial_{t}u}_{L^{2}_{x}}^{2}.

The second integral is manipulated just as I8I_{8}, namely,

I13≤θ​ν20​‖Δ​u‖Lx22+Cθ​∫𝕋3|u|2​|∇u|2≤θ​ν10​‖Δ​u‖Lx22+Cθ​‖u‖Hx14​‖∇u‖Lx22.\displaystyle I_{13}\leq\frac{\theta\nu}{20}\norm{\Delta u}_{L^{2}_{x}}^{2}+C_{\theta}\int_{\mathbb{T}^{3}}\absolutevalue{u}^{2}\absolutevalue{\nabla u}^{2}\leq\frac{\theta\nu}{10}\norm{\Delta u}_{L^{2}_{x}}^{2}+C_{\theta}\norm{u}_{H^{1}_{x}}^{4}\norm{\nabla u}_{L^{2}_{x}}^{2}.

The integral I14I_{14} requires Sobolev embedding, the Poincaré inequality, and Lebesgue norm interpolation, and reads

I14\displaystyle I_{14} ≤θ​ν10​‖Δ​u‖Lx22+Cθ​‖∇ψ‖Lx62​‖B​ψ‖Lx32\displaystyle\leq\frac{\theta\nu}{10}\norm{\Delta u}_{L^{2}_{x}}^{2}+C_{\theta}\norm{\nabla\psi}_{L^{6}_{x}}^{2}\norm{B\psi}_{L^{3}_{x}}^{2}
≤θ​ν10​‖Δ​u‖Lx22+Cθ​‖Δ​ψ‖Lx22​‖B​ψ‖Lx2​‖B​ψ‖Hx1\displaystyle\leq\frac{\theta\nu}{10}\norm{\Delta u}_{L^{2}_{x}}^{2}+C_{\theta}\norm{\Delta\psi}_{L^{2}_{x}}^{2}\norm{B\psi}_{L^{2}_{x}}\norm{B\psi}_{H^{1}_{x}}
≤θ​ν10​‖Δ​u‖Lx22+Cκ,θ​‖B​ψ‖Lx22​(‖Δ​ψ‖Lx22+‖Δ​ψ‖Lx24)+κ​‖∇(B​ψ)‖Lx22.\displaystyle\leq\frac{\theta\nu}{10}\norm{\Delta u}_{L^{2}_{x}}^{2}+C_{\kappa,\theta}\norm{B\psi}_{L^{2}_{x}}^{2}\left(\norm{\Delta\psi}_{L^{2}_{x}}^{2}+\norm{\Delta\psi}_{L^{2}_{x}}^{4}\right)+\kappa\norm{\nabla(B\psi)}_{L^{2}_{x}}^{2}.

In a similar manner, we have

I15\displaystyle I_{15} ≤θ​ν10​‖Δ​u‖Lx22+Cθ​‖u‖Lx62​‖ψ‖Lx∞2​‖B​ψ‖Lx32\displaystyle\leq\frac{\theta\nu}{10}\norm{\Delta u}_{L^{2}_{x}}^{2}+C_{\theta}\norm{u}_{L^{6}_{x}}^{2}\norm{\psi}_{L^{\infty}_{x}}^{2}\norm{B\psi}_{L^{3}_{x}}^{2}
≤θ​ν10​‖Δ​u‖Lx22+Cθ​‖u‖Hx12​‖ψ‖Hx22​‖B​ψ‖Lx2​‖B​ψ‖Hx1\displaystyle\leq\frac{\theta\nu}{10}\norm{\Delta u}_{L^{2}_{x}}^{2}+C_{\theta}\norm{u}_{H^{1}_{x}}^{2}\norm{\psi}_{H^{2}_{x}}^{2}\norm{B\psi}_{L^{2}_{x}}\norm{B\psi}_{H^{1}_{x}}
≤θ​ν10​‖Δ​u‖Lx22+Cκ,θ​‖B​ψ‖Lx22​(‖u‖Hx12​‖ψ‖Hx22+‖u‖Hx14​‖ψ‖Hx24)+κ​‖∇(B​ψ)‖Lx22.\displaystyle\leq\frac{\theta\nu}{10}\norm{\Delta u}_{L^{2}_{x}}^{2}+C_{\kappa,\theta}\norm{B\psi}_{L^{2}_{x}}^{2}\left(\norm{u}_{H^{1}_{x}}^{2}\norm{\psi}_{H^{2}_{x}}^{2}+\norm{u}_{H^{1}_{x}}^{4}\norm{\psi}_{H^{2}_{x}}^{4}\right)+\kappa\norm{\nabla(B\psi)}_{L^{2}_{x}}^{2}.

The last integral in  (3.16) requires, like I12I_{12}, only Hölder’s and Young’s inequalities. This provides

I16≤θ​ν10​‖Δ​u‖Lx22+C​θ​‖ρ​u‖Lx22.I_{16}\leq\frac{\theta\nu}{10}\norm{\Delta u}_{L^{2}_{x}}^{2}+C\theta\norm{\sqrt{\rho}u}_{L^{2}_{x}}^{2}.

In the end,  (3.16) becomes

θ​ν2​‖Δ​u‖Lx22\displaystyle\frac{\theta\nu}{2}\norm{\Delta u}_{L^{2}_{x}}^{2} ≤C​θ​‖ρ​∂tu‖Lx22+C​θ​‖ρ​u‖Lx22+Cθ​‖u‖Hx14​‖∇u‖Lx22\displaystyle\leq C\theta\norm{\sqrt{\rho}\partial_{t}u}_{L^{2}_{x}}^{2}+C\theta\norm{\sqrt{\rho}u}_{L^{2}_{x}}^{2}+C_{\theta}\norm{u}_{H^{1}_{x}}^{4}\norm{\nabla u}_{L^{2}_{x}}^{2} (3.17)
+Cκ,θ​‖B​ψ‖Lx22​(‖Δ​ψ‖Lx22+‖u‖Hx12​‖ψ‖Hx22+‖Δ​ψ‖Lx24+‖u‖Hx14​‖ψ‖Hx24)\displaystyle\quad+C_{\kappa,\theta}\norm{B\psi}_{L^{2}_{x}}^{2}\left(\norm{\Delta\psi}_{L^{2}_{x}}^{2}+\norm{u}_{H^{1}_{x}}^{2}\norm{\psi}_{H^{2}_{x}}^{2}+\norm{\Delta\psi}_{L^{2}_{x}}^{4}+\norm{u}_{H^{1}_{x}}^{4}\norm{\psi}_{H^{2}_{x}}^{4}\right)
+κ​‖∇(B​ψ)‖Lx22.\displaystyle\quad+\kappa\norm{\nabla(B\psi)}_{L^{2}_{x}}^{2}.

We now add  (3.12),  (3.15) and  (3.17). Then, we note from the definition of B​ψB\psi that

‖∇(B​ψ)‖Lx22≲‖D3​ψ‖Lx22+‖∇(|u|2​ψ)‖Lx22+‖∇(u⋅∇ψ)‖Lx22+‖∇(|ψ|p​ψ)‖Lx22,\norm{\nabla(B\psi)}_{L^{2}_{x}}^{2}\lesssim\norm{D^{3}\psi}_{L^{2}_{x}}^{2}+\norm{\nabla(\absolutevalue{u}^{2}\psi)}_{L^{2}_{x}}^{2}+\norm{\nabla(u\cdot\nabla\psi)}_{L^{2}_{x}}^{2}+\norm{\nabla(\absolutevalue{\psi}^{p}\psi)}_{L^{2}_{x}}^{2},

where the last three terms on the RHS are exactly I5I_{5}, I6I_{6}, and I7I_{7}. Using sufficiently small values for θ\theta and κ\kappa, we also absorb ‖ρ​∂tu‖Lx22\norm{\sqrt{\rho}\partial_{t}u}_{L^{2}_{x}}^{2} and ‖Δ​u‖Lx22\norm{\Delta u}_{L^{2}_{x}}^{2} on the RHS into the LHS. Finally, we are left with

dd​t​[‖Δ​ψ‖Lx22+ν​‖∇u‖Lx22+α​‖ρ​u‖Lx22]+1C​‖D3​ψ‖Lx22+1C​‖ρ​∂tu‖Lx22+1C​‖Δ​u‖Lx22\displaystyle\frac{d}{dt}\left[\norm{\Delta\psi}_{L^{2}_{x}}^{2}+\nu\norm{\nabla u}_{L^{2}_{x}}^{2}+\alpha\norm{\sqrt{\rho}u}_{L^{2}_{x}}^{2}\right]+\frac{1}{C}\norm{D^{3}\psi}_{L^{2}_{x}}^{2}+\frac{1}{C}\norm{\sqrt{\rho}\partial_{t}u}_{L^{2}_{x}}^{2}+\frac{1}{C}\norm{\Delta u}_{L^{2}_{x}}^{2} (3.18)
≤C​((1+‖u‖Hx14)​‖∇u‖Lx22​‖ψ‖Hx24+‖u‖Hx14​‖Δ​ψ‖Lx22+(1+‖ψ‖Hx12)​‖u‖Hx14)\displaystyle\leq C\Bigg{(}\left(1+\norm{u}_{H^{1}_{x}}^{4}\right)\norm{\nabla u}_{L^{2}_{x}}^{2}\norm{\psi}_{H^{2}_{x}}^{4}+\norm{u}_{H^{1}_{x}}^{4}\norm{\Delta\psi}_{L^{2}_{x}}^{2}+\left(1+\norm{\psi}_{H^{1}_{x}}^{2}\right)\norm{u}_{H^{1}_{x}}^{4}\Bigg{)}
+C​(‖ψ‖Lx22​p​‖Δ​ψ‖Lx22+‖Δ​ψ‖Lx22​p+2+‖u‖Hx14​‖∇u‖Lx22)\displaystyle\quad+C\Big{(}\norm{\psi}_{L^{2}_{x}}^{2p}\norm{\Delta\psi}_{L^{2}_{x}}^{2}+\norm{\Delta\psi}_{L^{2}_{x}}^{2p+2}+\norm{u}_{H^{1}_{x}}^{4}\norm{\nabla u}_{L^{2}_{x}}^{2}\Big{)}
+C​‖B​ψ‖Lx22​(‖Δ​ψ‖Lx22+‖u‖Hx12​‖ψ‖Hx22+‖Δ​ψ‖Lx24+‖u‖Hx14​‖ψ‖Hx24)\displaystyle\quad+C\norm{B\psi}_{L^{2}_{x}}^{2}\left(\norm{\Delta\psi}_{L^{2}_{x}}^{2}+\norm{u}_{H^{1}_{x}}^{2}\norm{\psi}_{H^{2}_{x}}^{2}+\norm{\Delta\psi}_{L^{2}_{x}}^{4}+\norm{u}_{H^{1}_{x}}^{4}\norm{\psi}_{H^{2}_{x}}^{4}\right)
+C​θ​‖ρ​u‖Lx22+κ​‖B​ψ‖Lx22.\displaystyle\quad+C\theta\norm{\sqrt{\rho}u}_{L^{2}_{x}}^{2}+\kappa\norm{B\psi}_{L^{2}_{x}}^{2}.

This is the higher-order energy estimate. Using similar arguments to Section 3.3.3 of  [JJK23], we can use the Grönwall inequality to control the higher-order energy and dissipation. The results are summarized in the following lemma.

Lemma 3.3 (Algebraic decay rate for energies).

We label the higher-order energy as

X:=‖Δ​ψ​(t)‖Lx22+ν​‖∇u​(t)‖Lx22.X:=\norm{\Delta\psi(t)}_{L^{2}_{x}}^{2}+\nu\norm{\nabla u(t)}_{L^{2}_{x}}^{2}. (3.19)

Then, the sum Z:=X+EZ:=X+E decays as

Z​(t)≤Z0​e−tC+C​S0p2+1(1+S0p2​t)1+2p≲Z0+S0p2+1,Z(t)\leq Z_{0}e^{-\frac{t}{C}}+\frac{CS_{0}^{\frac{p}{2}+1}}{\left(1+S_{0}^{\frac{p}{2}}t\right)^{1+\frac{2}{p}}}\lesssim Z_{0}+S_{0}^{\frac{p}{2}+1}, (3.20)

where Z0:=Z​(0)Z_{0}:=Z(0). Moreover, the time-integral of the corresponding dissipation terms is also bounded. Specifically,

‖D3​ψ‖L[0,T]2​Lx22+‖ρ​∂tu‖L[0,T]2​Lx22+‖Δ​u‖L[0,T]2​Lx22+‖∇u‖L[0,T]2​Lx22+‖ρ​u‖L[0,T]2​Lx22+‖B​ψ‖L[0,T]2​Lx22\displaystyle\norm{D^{3}\psi}_{L^{2}_{[0,T]}L^{2}_{x}}^{2}+\norm{\sqrt{\rho}\partial_{t}u}_{L^{2}_{[0,T]}L^{2}_{x}}^{2}+\norm{\Delta u}_{L^{2}_{[0,T]}L^{2}_{x}}^{2}+\norm{\nabla u}_{L^{2}_{[0,T]}L^{2}_{x}}^{2}+\norm{\sqrt{\rho}u}_{L^{2}_{[0,T]}L^{2}_{x}}^{2}+\norm{B\psi}_{L^{2}_{[0,T]}L^{2}_{x}}^{2} (3.21)
≲Z0+S0p​(Z0+S0)≲Z0+S0p+1.\displaystyle\lesssim Z_{0}+S_{0}^{p}(Z_{0}+S_{0})\lesssim Z_{0}+S_{0}^{p+1}.

In addition, by integrating the higher-order energy estimate over [t,2​t][t,2t] (for t≥1t\geq 1), we end up with a time-decaying estimate for the dissipation, given by

‖D3​ψ‖L[t,2​t]2​Lx22+‖ρ​∂tu‖L[t,2​t]2​Lx22+‖Δ​u‖L[t,2​t]2​Lx22+‖∇u‖L[t,2​t]2​Lx22+‖ρ​u‖L[t,2​t]2​Lx22+‖B​ψ‖L[t,2​t]2​Lx22\displaystyle\norm{D^{3}\psi}_{L^{2}_{[t,2t]}L^{2}_{x}}^{2}+\norm{\sqrt{\rho}\partial_{t}u}_{L^{2}_{[t,2t]}L^{2}_{x}}^{2}+\norm{\Delta u}_{L^{2}_{[t,2t]}L^{2}_{x}}^{2}+\norm{\nabla u}_{L^{2}_{[t,2t]}L^{2}_{x}}^{2}+\norm{\sqrt{\rho}u}_{L^{2}_{[t,2t]}L^{2}_{x}}^{2}+\norm{B\psi}_{L^{2}_{[t,2t]}L^{2}_{x}}^{2} (3.22)
≲Z0​e−tC+S0p2+1(1+S0p2​t)2p.\displaystyle\lesssim Z_{0}e^{-\frac{t}{C}}+\frac{S_{0}^{\frac{p}{2}+1}}{\left(1+S_{0}^{\frac{p}{2}}t\right)^{\frac{2}{p}}}.

The last inequality in  (3.21) is valid because Z0,S0≤1Z_{0},S_{0}\leq 1.

3.4. Maximal parabolic regularity for ψ\psi

From the previous analysis, we have obtained ψ∈L[0,T]2​Hx3\psi\in L^{2}_{[0,T]}H^{3}_{x}. However, as pointed out in the discussion following Definition  2.9, we seek B​ψ∈L[0,T]2​Lx∞B\psi\in L^{2}_{[0,T]}L^{\infty}_{x}, which follows from ψ∈L[0,T]2​Hx72+\psi\in L^{2}_{[0,T]}H^{\frac{7}{2}+}_{x}. In the 2D case [JJK23], this was achieved by taking advantage of the Sobolev embedding Hx1+δ⊂Lx∞H^{1+\delta}_{x}\subset L^{\infty}_{x} and deriving a “highest-order energy estimate” for ψ\psi. This approach does not work here due to the embedding Hx32+δ⊂Lx∞H^{\frac{3}{2}+\delta}_{x}\subset L^{\infty}_{x}, which would require higher-order estimates on uu and ρ\rho. Instead, we exploit the parabolic nature of  (c-NLS) and apply the method of maximal regularity to gain the necessary control of ψ\psi.

Lemma 3.4 (Maximal regularity for ψ\psi).

For δ\delta as defined in Theorem  2.3, and a sufficiently small δ1>0\delta_{1}>0, we have the maximal regularity bound

‖∂tψ‖L[0,T]1+δ​Hx32+δ1+‖Δ​ψ‖L[0,T]1+δ​Hx32+δ1≤f​(Z0,S0),\norm{\partial_{t}\psi}_{L^{1+\delta}_{[0,T]}H^{\frac{3}{2}+\delta_{1}}_{x}}+\norm{\Delta\psi}_{L^{1+\delta}_{[0,T]}H^{\frac{3}{2}+\delta_{1}}_{x}}\leq f\left(Z_{0},S_{0}\right), (3.23)

uniformly in time TT, where f:(ℝ+)2→ℝ+f\colon(\mathbb{R}_{+})^{2}\rightarrow\mathbb{R}_{+} is a continuous polynomial, with f​(0,0)=0f(0,0)=0.

Proof.

We begin by rewriting  (c-NLS) in a parabolic form as

∂tψ−λ+i2​Δ​ψ=−λ2​|u|2​ψ−i​λ​u⋅∇ψ−μ​(λ+i)​|ψ|p​ψ.\partial_{t}\psi-\frac{\lambda+i}{2}\Delta\psi=-\frac{\lambda}{2}\absolutevalue{u}^{2}\psi-i\lambda u\cdot\nabla\psi-\mu(\lambda+i)\absolutevalue{\psi}^{p}\psi. (3.24)

The differential operator in this case is A=−λ+i2​ΔA=-\frac{\lambda+i}{2}\Delta. Comparing this to  (2.14), we see that m=1m=1, and aα=−λ+i2a_{\alpha}=-\frac{\lambda+i}{2} when α∈{(2,0,0),(0,2,0),(0,0,2)}\alpha\in\{(2,0,0),(0,2,0),(0,0,2)\} and aα=0a_{\alpha}=0 otherwise. Thus, the first condition in Definition  2.10 is satisfied. The principal symbol of the operator is

A~​(ξ)=λ+i2​|ξ|2,\tilde{A}(\xi)=\frac{\lambda+i}{2}\absolutevalue{\xi}^{2},

from which it is clear that A~​(ξ)−1\tilde{A}(\xi)^{-1} is also bounded for |ξ|=1\absolutevalue{\xi}=1. Finally, the spectrum σ​(λ+i2​|ξ|2)\sigma\left(\frac{\lambda+i}{2}\absolutevalue{\xi}^{2}\right) belongs to the sector Σζ0\Sigma_{\zeta_{0}} for tan⁡(ζ0)>λ−1>0\tan{\zeta_{0}}>\lambda^{-1}>0. Thus, the operator AA is uniformly (K,ζ0)(K,\zeta_{0})-elliptic for K=max⁡{1+λ22,21+λ2}K=\max\{\frac{\sqrt{1+\lambda^{2}}}{2},\frac{2}{\sqrt{1+\lambda^{2}}}\}, and the maximal parabolic regularity estimate is applicable.

We now act upon  (3.24) by (−Δ)34+δ12(-\Delta)^{\frac{3}{4}+\frac{\delta_{1}}{2}} for some δ1>0\delta_{1}>0 that will be determined shortly. We then apply Lemma  2.11 with X=L2​(𝕋3)X=L^{2}(\mathbb{T}^{3}) and X1=H2​(𝕋3)X_{1}=H^{2}(\mathbb{T}^{3}), which results in

‖∂t(−Δ)34+δ12​ψ‖Ltr​Lx2+‖(−Δ)34+δ12​ψ‖Ltr​Hx2\displaystyle\norm{\partial_{t}(-\Delta)^{\frac{3}{4}+\frac{\delta_{1}}{2}}\psi}_{L^{r}_{t}L^{2}_{x}}+\norm{(-\Delta)^{\frac{3}{4}+\frac{\delta_{1}}{2}}\psi}_{L^{r}_{t}H^{2}_{x}} (3.25)
≲‖(−Δ)34+δ12​ψ0‖(Lx2,Hx2)1−1r,r+‖(−Δ)34+δ12​(|u|2​ψ)‖Ltr​Lx2+‖(−Δ)34+δ12​(u⋅∇ψ)‖Ltr​Lx2\displaystyle\lesssim\norm{(-\Delta)^{\frac{3}{4}+\frac{\delta_{1}}{2}}\psi_{0}}_{(L^{2}_{x},H^{2}_{x})_{1-\frac{1}{r},r}}+\norm{(-\Delta)^{\frac{3}{4}+\frac{\delta_{1}}{2}}(\absolutevalue{u}^{2}\psi)}_{L^{r}_{t}L^{2}_{x}}+\norm{(-\Delta)^{\frac{3}{4}+\frac{\delta_{1}}{2}}(u\cdot\nabla\psi)}_{L^{r}_{t}L^{2}_{x}}
+‖(−Δ)34+δ12​(|ψ|p​ψ)‖Ltr​Lx2\displaystyle\quad+\norm{(-\Delta)^{\frac{3}{4}+\frac{\delta_{1}}{2}}(\absolutevalue{\psi}^{p}\psi)}_{L^{r}_{t}L^{2}_{x}}
:=I17+I18+I19+I20,\displaystyle:=I_{17}+I_{18}+I_{19}+I_{20},

for r=1+δr=1+\delta with δ∈(0,13)\delta\in(0,\frac{1}{3}), and this restriction will become clear when estimating I18I_{18} in (3.27). It is important to note that LtrL^{r}_{t} is calculated on [0,T][0,T], where T>0T>0 is the local existence time defined in (2.9). We now estimate each of the terms on the RHS. The norm of the initial condition is found to belong to a Besov space as a result of the interpolation (see  [AF03, Chapter 7]). Indeed, we have

I17\displaystyle I_{17} =‖(−Δ)34+δ12​ψ0‖(Lx2,Hx2)1−1r,r=‖(−Δ)34+δ12​ψ0‖(Lx2,Hx2)δ1+δ,1+δ=‖(−Δ)34+δ12​ψ0‖B2,1+δ2​δ1+δ\displaystyle=\norm{(-\Delta)^{\frac{3}{4}+\frac{\delta_{1}}{2}}\psi_{0}}_{(L^{2}_{x},H^{2}_{x})_{1-\frac{1}{r},r}}=\norm{(-\Delta)^{\frac{3}{4}+\frac{\delta_{1}}{2}}\psi_{0}}_{(L^{2}_{x},H^{2}_{x})_{\frac{\delta}{1+\delta},1+\delta}}=\norm{(-\Delta)^{\frac{3}{4}+\frac{\delta_{1}}{2}}\psi_{0}}_{B^{\frac{2\delta}{1+\delta}}_{2,1+\delta}} (3.26)
≲‖(−Δ)34+δ12​ψ0‖H2​δ1+δ+δ2≤‖ψ0‖H32+2​δ1+δ+δ1+δ2≤‖ψ0‖H2,\displaystyle\lesssim\norm{(-\Delta)^{\frac{3}{4}+\frac{\delta_{1}}{2}}\psi_{0}}_{H^{\frac{2\delta}{1+\delta}+\delta_{2}}}\leq\norm{\psi_{0}}_{H^{\frac{3}{2}+\frac{2\delta}{1+\delta}+\delta_{1}+\delta_{2}}}\leq\norm{\psi_{0}}_{H^{2}},

for sufficiently small values for δ,δ1\delta,\delta_{1}, and δ2\delta_{2}. The first inequality is due to the embedding Hs+δ2⊂B2,qsH^{s+\delta_{2}}\subset B^{s}_{2,q} for any δ2>0\delta_{2}>0 (see  [Lu21, Lemma 2.2]). Due to the restriction δ<13\delta<\frac{1}{3}, we have 2​δ1+δ<12\frac{2\delta}{1+\delta}<\frac{1}{2}, implying that we may bound I17I_{17} with the initial data, as in the last inequality of (3.26). Next, we deal with the second term on the RHS of (3.25) as

I18\displaystyle I_{18} ≲‖|u|2​ψ‖Lt1+δ​Hx2≲‖‖u‖Lx∞​‖u‖Hx2​‖ψ‖Lx∞‖Lt1+δ+‖‖u‖Lx∞2​‖ψ‖Hx2‖Lt1+δ\displaystyle\lesssim\norm{\absolutevalue{u}^{2}\psi}_{L^{1+\delta}_{t}H^{2}_{x}}\lesssim\norm{\norm{u}_{L^{\infty}_{x}}\norm{u}_{H^{2}_{x}}\norm{\psi}_{L^{\infty}_{x}}}_{L^{1+\delta}_{t}}+\norm{\norm{u}_{L^{\infty}_{x}}^{2}\norm{\psi}_{H^{2}_{x}}}_{L^{1+\delta}_{t}} (3.27)
≲‖‖u‖Hx112​‖u‖Hx232​‖ψ‖Hx2‖Lt1+δ+‖‖u‖Hx1​‖u‖Hx2​‖ψ‖Hx2‖Lt1+δ\displaystyle\lesssim\norm{\norm{u}_{H^{1}_{x}}^{\frac{1}{2}}\norm{u}_{H^{2}_{x}}^{\frac{3}{2}}\norm{\psi}_{H^{2}_{x}}}_{L^{1+\delta}_{t}}+\norm{\norm{u}_{H^{1}_{x}}\norm{u}_{H^{2}_{x}}\norm{\psi}_{H^{2}_{x}}}_{L^{1+\delta}_{t}}
≲‖u‖Lt2​(1+δ)1−3​δ​Hx112​‖u‖Lt2​Hx232​‖ψ‖Lt∞​Hx2+‖u‖Lt2​(1+δ)1−δ​Hx1​‖u‖Lt2​Hx2​‖ψ‖Lt∞​Hx2\displaystyle\lesssim\norm{u}_{L^{\frac{2(1+\delta)}{1-3\delta}}_{t}H^{1}_{x}}^{\frac{1}{2}}\norm{u}_{L^{2}_{t}H^{2}_{x}}^{\frac{3}{2}}\norm{\psi}_{L^{\infty}_{t}H^{2}_{x}}+\norm{u}_{L^{\frac{2(1+\delta)}{1-\delta}}_{t}H^{1}_{x}}\norm{u}_{L^{2}_{t}H^{2}_{x}}\norm{\psi}_{L^{\infty}_{t}H^{2}_{x}}
≲(Z0+S01+2​p​δ1+δ)14​(Z0+S0p+1)34​(Z0+S0p2+1)12\displaystyle\lesssim\left(Z_{0}+S_{0}^{1+\frac{2p\delta}{1+\delta}}\right)^{\frac{1}{4}}\left(Z_{0}+S_{0}^{p+1}\right)^{\frac{3}{4}}\left(Z_{0}+S_{0}^{\frac{p}{2}+1}\right)^{\frac{1}{2}}
+(Z0+S01+p​δ1+δ)12​(Z0+S0p+1)12​(Z0+S0p2+1)12.\displaystyle\quad+\left(Z_{0}+S_{0}^{1+\frac{p\delta}{1+\delta}}\right)^{\frac{1}{2}}\left(Z_{0}+S_{0}^{p+1}\right)^{\frac{1}{2}}\left(Z_{0}+S_{0}^{\frac{p}{2}+1}\right)^{\frac{1}{2}}.

The second inequality follows from the product rule for Sobolev norms [MB02, Lemma 3.4], and the third inequality from Agmon’s inequality and Sobolev embedding. The fourth inequality is due to Hölder’s inequality, while the final step follows from  (3.20) and  (3.21). In a similar way, we can analyze the third term on the RHS of  (3.25), yielding

I19\displaystyle I_{19} ≲‖u⋅∇ψ‖Lt1+δ​Hx2\displaystyle\lesssim\norm{u\cdot\nabla\psi}_{L^{1+\delta}_{t}H^{2}_{x}} (3.28)
≲‖‖u‖Lx∞​‖∇ψ‖Hx2‖Lt1+δ+‖‖u‖Hx2​‖∇ψ‖Lx∞‖Lt1+δ\displaystyle\lesssim\norm{\norm{u}_{L^{\infty}_{x}}\norm{\nabla\psi}_{H^{2}_{x}}}_{L^{1+\delta}_{t}}+\norm{\norm{u}_{H^{2}_{x}}\norm{\nabla\psi}_{L^{\infty}_{x}}}_{L^{1+\delta}_{t}}
≲‖u‖Lt2​(1+δ)1−3​δ​Hx112​‖u‖Lt2​Hx212​‖D3​ψ‖Lt2​Lx2+‖u‖Lt2​Hx2​‖D2​ψ‖Lt2​(1+δ)1−3​δ​Lx212​‖D3​ψ‖Lt2​Lx212\displaystyle\lesssim\norm{u}_{L^{\frac{2(1+\delta)}{1-3\delta}}_{t}H^{1}_{x}}^{\frac{1}{2}}\norm{u}_{L^{2}_{t}H^{2}_{x}}^{\frac{1}{2}}\norm{D^{3}\psi}_{L^{2}_{t}L^{2}_{x}}+\norm{u}_{L^{2}_{t}H^{2}_{x}}\norm{D^{2}\psi}_{L^{\frac{2(1+\delta)}{1-3\delta}}_{t}L^{2}_{x}}^{\frac{1}{2}}\norm{D^{3}\psi}_{L^{2}_{t}L^{2}_{x}}^{\frac{1}{2}}
≲(Z0+S01+2​p​δ1+δ)14​(Z0+S0p+1)34.\displaystyle\lesssim\left(Z_{0}+S_{0}^{1+\frac{2p\delta}{1+\delta}}\right)^{\frac{1}{4}}\left(Z_{0}+S_{0}^{p+1}\right)^{\frac{3}{4}}.

Since we have p≥1p\geq 1, the last term on the RHS of  (3.25) is bounded using  (3.1) and  (3.20) as

I20\displaystyle I_{20} ≲‖|ψ|p​ψ‖Lt1+δ​Hx2≲‖‖ψ‖Hx2p+1‖Lt1+δ\displaystyle\lesssim\norm{\absolutevalue{\psi}^{p}\psi}_{L^{1+\delta}_{t}H^{2}_{x}}\lesssim\norm{\norm{\psi}_{H^{2}_{x}}^{p+1}}_{L^{1+\delta}_{t}} (3.29)
≲‖(‖ψ‖Lx2+‖Δ​ψ‖Lx2)p+1‖Lt1+δ≲S0δ​p2​(1+δ)+12+Z0p+12+S0p24+p​(1+3​δ)4​(1+δ)+12.\displaystyle\lesssim\norm{\left(\norm{\psi}_{L^{2}_{x}}+\norm{\Delta\psi}_{L^{2}_{x}}\right)^{p+1}}_{L^{1+\delta}_{t}}\lesssim S_{0}^{\frac{\delta p}{2(1+\delta)}+\frac{1}{2}}+Z_{0}^{\frac{p+1}{2}}+S_{0}^{\frac{p^{2}}{4}+\frac{p(1+3\delta)}{4(1+\delta)}+\frac{1}{2}}.

Putting together the estimates in  (3.26)–(3.29) gives us the desired result. ∎

We conclude that ∂tψ,Δ​ψ∈L[0,T]1+δ​Hx32+δ1\partial_{t}\psi,\Delta\psi\in L^{1+\delta}_{[0,T]}H^{\frac{3}{2}+\delta_{1}}_{x}, uniformly in TT, and that their norms can be made small by an appropriate choice of S0S_{0} and Z0Z_{0}.

3.5. Ensuring global-in-time positive density

We have now obtained all the a priori estimates needed to return to  (2.13).

Proof of Theorem  2.3: Using  (CPL), we have

‖B​ψ‖Lx∞\displaystyle\norm{B\psi}_{L^{\infty}_{x}} ≲‖Δ​ψ‖Lx∞+‖|u|2​ψ‖Lx∞+‖u⋅∇ψ‖Lx∞+‖|ψ|p​ψ‖Lx∞\displaystyle\lesssim\norm{\Delta\psi}_{L^{\infty}_{x}}+\norm{\absolutevalue{u}^{2}\psi}_{L^{\infty}_{x}}+\norm{u\cdot\nabla\psi}_{L^{\infty}_{x}}+\norm{\absolutevalue{\psi}^{p}\psi}_{L^{\infty}_{x}} (3.30)
≲‖D2​ψ‖Hx32+δ1+‖u‖Hx1​‖u‖Hx2​‖ψ‖Hx2+‖u‖Hx112​‖u‖Hx212​‖D3​ψ‖Lx2+‖ψ‖Hx2p+1,\displaystyle\lesssim\norm{D^{2}\psi}_{H^{\frac{3}{2}+\delta_{1}}_{x}}+\norm{u}_{H^{1}_{x}}\norm{u}_{H^{2}_{x}}\norm{\psi}_{H^{2}_{x}}+\norm{u}_{H^{1}_{x}}^{\frac{1}{2}}\norm{u}_{H^{2}_{x}}^{\frac{1}{2}}\norm{D^{3}\psi}_{L^{2}_{x}}+\norm{\psi}_{H^{2}_{x}}^{p+1},

where the second step is a consequence of Agmon’s inequality and Sobolev embedding. We now substitute  (3.30) into the LHS of  (2.13) and also use the Sobolev embedding ‖ψ‖Lx∞≲‖ψ‖Hx2\norm{\psi}_{L^{\infty}_{x}}\lesssim\norm{\psi}_{H^{2}_{x}}. This leads to

∫0T‖ψ‖Lx∞​‖B​ψ‖Lx∞\displaystyle\int_{0}^{T}\norm{\psi}_{L^{\infty}_{x}}\norm{B\psi}_{L^{\infty}_{x}} ≲∫0T‖ψ‖Hx2​‖D2​ψ‖Hx32+δ1+∫0T‖u‖Hx1​‖u‖Hx2​‖ψ‖Hx22\displaystyle\lesssim\int_{0}^{T}\norm{\psi}_{H^{2}_{x}}\norm{D^{2}\psi}_{H^{\frac{3}{2}+\delta_{1}}_{x}}+\int_{0}^{T}\norm{u}_{H^{1}_{x}}\norm{u}_{H^{2}_{x}}\norm{\psi}_{H^{2}_{x}}^{2} (3.31)
+∫0T‖ψ‖Hx2​‖u‖Hx112​‖u‖Hx212​‖D3​ψ‖Lx2+∫0T‖ψ‖Hx2p+2\displaystyle\quad+\int_{0}^{T}\norm{\psi}_{H^{2}_{x}}\norm{u}_{H^{1}_{x}}^{\frac{1}{2}}\norm{u}_{H^{2}_{x}}^{\frac{1}{2}}\norm{D^{3}\psi}_{L^{2}_{x}}+\int_{0}^{T}\norm{\psi}_{H^{2}_{x}}^{p+2}
:=I21+I22+I23+I24.\displaystyle:=I_{21}+I_{22}+I_{23}+I_{24}.

We now show that each of these four terms can be made as small as required, by choosing sufficiently small values of the data, i.e., S0S_{0} and Z0Z_{0}. For the first term (I21)(I_{21}), we use Hölder’s inequality, (3.1), and  (3.20) to write

I21\displaystyle I_{21} =∫0T‖ψ‖Hx2​‖D2​ψ‖Hx32+δ1≤‖ψ‖L[0,T]1+1δ​Hx2​‖D2​ψ‖L[0,T]1+δ​Hx32+δ1\displaystyle=\int_{0}^{T}\norm{\psi}_{H^{2}_{x}}\norm{D^{2}\psi}_{H^{\frac{3}{2}+\delta_{1}}_{x}}\leq\norm{\psi}_{L^{1+\frac{1}{\delta}}_{[0,T]}H^{2}_{x}}\norm{D^{2}\psi}_{L^{1+\delta}_{[0,T]}H^{\frac{3}{2}+\delta_{1}}_{x}} (3.32)
≲(S012​(1−p​δ1+δ)+Z012)​‖D2​ψ‖L[0,T]1+δ​Hx32+δ1.\displaystyle\lesssim\left(S_{0}^{\frac{1}{2}\left(1-\frac{p\delta}{1+\delta}\right)}+Z_{0}^{\frac{1}{2}}\right)\norm{D^{2}\psi}_{L^{1+\delta}_{[0,T]}H^{\frac{3}{2}+\delta_{1}}_{x}}.

The above calculation assumes that p<1+1δp<1+\frac{1}{\delta} (which is consistent with the definition of δ\delta in Theorem  2.3), so that a uniform-in-TT bound can be obtained. From Lemma  3.4, we conclude the smallness (in terms of the initial data) of the last factor in the RHS, i.e., ‖D2​ψ‖L[0,T]1+δ​Hx32+δ1\norm{D^{2}\psi}_{L^{1+\delta}_{[0,T]}H^{\frac{3}{2}+\delta_{1}}_{x}}. Thus, I21I_{21} is independent of TT, and can be made sufficiently small by appropriate initial data.

Moving on to the remaining terms in  (3.31), we have

I22≲‖u‖Lt2​Hx1​‖u‖Lt2​Hx2​‖ψ‖Lt∞​Hx22.I_{22}\lesssim\norm{u}_{L^{2}_{t}H^{1}_{x}}\norm{u}_{L^{2}_{t}H^{2}_{x}}\norm{\psi}_{L^{\infty}_{t}H^{2}_{x}}^{2}. (3.33)

All the terms are, once again, bounded in terms of the data according to  (3.20) and  (3.21). In the same way, we have

I23≲‖ψ‖Lt∞​Hx2​‖u‖Lt2​Hx112​‖u‖Lt2​Hx212​‖D3​ψ‖Lt2​Lx2,I_{23}\lesssim\norm{\psi}_{L^{\infty}_{t}H^{2}_{x}}\norm{u}_{L^{2}_{t}H^{1}_{x}}^{\frac{1}{2}}\norm{u}_{L^{2}_{t}H^{2}_{x}}^{\frac{1}{2}}\norm{D^{3}\psi}_{L^{2}_{t}L^{2}_{x}}, (3.34)

where all the terms are controlled by the data. Finally, using  (3.1) and  (3.20), we arrive at

I24\displaystyle I_{24} ≲∫0T(S0p2+1(1+S0p2​t)1+2p+Z0p2+1​e−tC+S0(p+2)24(1+S0p2​t)(p+2)22​p)​𝑑t\displaystyle\lesssim\int_{0}^{T}\left(\frac{S_{0}^{\frac{p}{2}+1}}{\left(1+S_{0}^{\frac{p}{2}}t\right)^{1+\frac{2}{p}}}+Z_{0}^{\frac{p}{2}+1}e^{-\frac{t}{C}}+\frac{S_{0}^{\frac{(p+2)^{2}}{4}}}{\left(1+S_{0}^{\frac{p}{2}}t\right)^{\frac{(p+2)^{2}}{2p}}}\right)dt (3.35)
≲S0+Z0p2+1+S0(p+1)2+34,\displaystyle\lesssim S_{0}+Z_{0}^{\frac{p}{2}+1}+S_{0}^{\frac{(p+1)^{2}+3}{4}},

and this calculation holds for all values of p≥1p\geq 1. From the analysis in  (3.32)–(3.35), we conclude that the LHS of  (3.31) can be made sufficiently small to satisfy the constraint given by  (2.13), for all values of time T>0T>0. This implies that the density always remains bounded below, and thus solutions are global in time for p<1+1δp<1+\frac{1}{\delta}. No matter how large (but finite) pp is, it is possible to choose δ\delta small enough so that the constraint in (2.13) is met. ∎

Acknowledgments

J.J. was supported by the NSF grants DMS-2009458 and DMS-2306910, while I.K. was supported by the NSF grant DMS-2205493. The authors appreciate the comments of the anonymous referees, which helped to improve the manuscript.

Conflict of interest declaration

On behalf of all the authors, the corresponding author states that there is no conflict of interest.

Data availability statement

Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

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