Rigidity in isothermal fluids

# Rigidity results in generalized isothermal fluids

Rémi Carles Kleber Carrapatoso  and  Matthieu Hillairet Institut Montpelliérain Alexander Grothendieck
CNRS
Univ. Montpellier
France
###### Abstract.

We investigate the long-time behavior of solutions to the isothermal Euler, Korteweg or quantum Navier Stokes equations, as well as generalizations of these equations where the convex pressure law is asymptotically linear near vacuum. By writing the system with a suitable time-dependent scaling we prove that the densities of global solutions display universal dispersion rate and asymptotic profile. This result applies to weak solutions defined in an appropriate way. In the exactly isothermal case, we establish the compactness of bounded sets of such weak solutions, by introducing modified entropies adapted to the new unknown functions.

This work was partially supported by the EFI project ANR-17-CE40-0030 and the Dyficolti project ANR-13-BS01-0003-01 of the French National Research Agency (ANR)

## 1. Introduction

In the isentropic case , the Euler equation on , ,

 (1.1) {∂tρ+div(ρu)=0,∂t(ρu)+div(ρu⊗u)+∇(ργ)=0,

enjoys the formal conservations of mass,

 M(t)=∫Rdρ(t,x)dx≡M(0),

and entropy (or energy),

 E(t)=12∫Rdρ(t,x)|u(t,x)|2dx+1γ−1∫Rdρ(t,x)γdx≡E(0).

In general, smooth solutions are defined only locally in time (see [MUK86, JYC90, Xin98]). However, for some range of , if the initial velocity has a special structure and the initial density is sufficiently small, the classical solution is defined globally in time. In addition the large time behavior of the solution can be described rather precisely, as established in [Serre97]. We restate some results from [Serre97] in the following theorem:

###### Theorem 1.1 (From [Serre97]).

Let and . There exists such that the following holds.
If are such that , then the system (1.1) with initial data and admits a unique global solution, in the sense that , where . In addition, there exists such that

 (1.2) ∥∥∥(ρ(t,x)−1tdR∞(xt),u(t,x)−x1+t−11+tU∞(xt))∥∥∥L∞(Rd)⟶t→∞0.

Conversely, if are such that , then there exists such that the solution to (1.1) with and is global in time in the same sense as above, and (1.2) holds.

In particular, in the frame of small data (in the sense described above), the dispersion

 ∥ρ(t)∥L∞(Rd)∼t→∞∥R∞∥L∞(Rd)td

is universal but the asymptotic profile can be arbitrary. Typically, given any function , will be allowed provided that is sufficiently small. For completeness we provide a brief proof of the above theorem in appendix.

We emphasize that the structure of the velocity is crucial: the initial velocity is a small (decaying) perturbation of a linear velocity. In a way, the above result is the Euler generalization of the global existence results for the Burgers equation with expanding data. Refinements of this result can be found in [GrSe97, Gra98, Serre2017].

The isothermal Euler equation corresponds to the value in (1.1),

 (1.3) {∂tρ+div(ρu)=0,∂t(ρu)+div(ρu⊗u)+κ∇ρ=0,κ>0,

The mass is still formally conserved, and the energy now reads

 E(t)=12∫Rdρ(t,x)|u(t,x)|2dx+∫Rdρ(t,x)lnρ(t,x)dx≡E(0).

Unlike in the isentropic case, the energy has an indefinite sign, a property which causes many technical problems. In this paper, we show that the isothermal Euler equation on , , with asymptotically vanishing density, , displays a specific large time behavior, in the sense that if the solution is global in time, then the density disperses with a rate different from the above one, and possesses a universal asymptotic Gaussian profile. This property remains when the convex pressure law satisfies , as well as for the Korteweg and quantum Navier-Stokes equations:

 (1.4) ⎧⎪ ⎪⎨⎪ ⎪⎩∂tρ+div(ρu)=0,∂t(ρu)+div(ρu⊗u)+∇P(ρ)=ε22ρ∇(Δ√ρ√ρ)+νdiv(ρDu),

with , where denotes the symmetric part of the gradient,

 Du:=12(∇u+t∇u).

For this system, we still have conservation of mass and the energy

 (1.5) E(t)=12∫ρ(t,x)|u(t,x)|2dx+ε22∫|∇√ρ(t,x)|2dx+∫F(ρ(t,x))dx,

where

 F(ρ)=ρ∫ρ1P(r)r2dr,

satisfies

 ˙E(t)=−ν∫ρ|Du|2.

In the case and equation (1.4) is the precise system derived in [Bru-Me10], as a correction to the isothermal quantum Euler equation. We emphasize that, because of the lack of positivity of the term in the energy functional, only the barotropic variant – where with – is studied in references. Classically, a Bohm potential (corresponding to the term multiplied by in (1.4)) is also added, see [AS-comp, Gis-VV15, Jungel, VasseurYu] for instance. In the case where the dissipation is absent (), but with capillarity (), we refer to [BDD07, AnMa09, AnMa12, AuHa17].

A loose statement of our main result reads (a more precise version is provided in the next section, see Theorem 2.9):

###### Theorem 1.2.

Let be a global weak solution to (1.4) with initial density/velocity satisfying

 (1+|x|2+|u0|2)1/2√ρ0∈L2(Rd).

Then there exists a mapping such that

 τ(t)∼t→∞2t√P′(0)ln(t), ρ(t,x)∼t→∞∥ρ0∥L1πd/2exp(−|x|2/τ(t)2)τ(t)d.

This theorem entails that, in contrast with the isentropic case, the density of solutions to (1.4) disperses as follows :

 ∥ρ(t)∥L∞(Rd)∼t→∞∥ρ0∥L1(Rd)(2P′(0)√π)d×1(t√ln(t))d,

with a universal profile. This result applies to a notion of “weak solution” that is based on standard a priori estimates satisfied by smooth solutions to (1.4). We make precise the definition of such solutions in the next section, see Definition 2.1.

The main ingredient of the proof is to translate in terms of our isothermal equations a change of unknown functions introduced for the dispersive logarithmic Schrödinger equation in [CaGa-p]. This enables to transform (1.4) into a system with unknowns for which the associated energy is positive-definite. A second feature of the new system is that, asymptotically in time, it reads (keeping only the dominating terms):

 (1.6)

where is the time-dependent scaling mentioned in Theorem 1.2. By taking the divergence of the second equation and replace with the first one, we obtain then (keeping again only the dominating terms):

 {∂tR=0,∂tR−P′(0)LR=0,

where is the Fokker-Planck operator In this last system, the first equation implies that converges to a stationary solution to the second equation. The analysis of the long-time behavior of solutions to this Fokker-Planck equation, as provided in [AMTU01], entails the expected result.

The outline of the paper is as follows. In the next section, we provide rigorous definitions of weak solutions and precise statements for our main result. Section 3 is then devoted to the long-time behavior of solutions to (1.4). In this section, we compute at first explicit solutions to (1.4) with Gaussian densities. These explicit computations motivate the introduction of the change of variable that we use afterwards. In what remains of this section we give an exhaustive proof of the precise version for Theorem 1.2. The long-time analysis mentioned here is based on the a priori existence of solutions. However, in the compressible setting, global existence of solutions is questionable. So, in the last section of the paper, we focus on the notion of weak solutions that we consider. At first, we present the a priori estimates which motivate their definition. We end the paper by proving a sequential compactness result. This sequential compactness property is a cornerstone for the proof of existence of weak solutions, see e.g. [Lio98, Fei04]. As for the large time behavior, we simply state a loose version of our result here (see Theorem 4.10 for the precise statement):

###### Theorem 1.3.

Assume , , with , and let . Let be a sequence of weak solutions to (1.4) on , enjoying uniformly the conservation of mass, as well as a suitable notion of energy dissipation, BD-entropy dissipation, and Mellet-Vasseur type inequality. Then up to the extraction of a subsequence, converges to a weak solution of (1.4) on .

It is for the system (2.8) in terms of , as mentioned above, that fairly natural a priori estimates are required in the above statement. Even though the notions of solution for (1.4) and (2.8) are equivalent (Lemma 2.6 below), we did not find a direct approach to express the pseudo energy, pseudo BD-entropy and Mellet-Vasseur type inequality mentioned above in a direct way in terms of , that is, without resorting to .

## 2. Weak solutions and large time behavior

We now state a precise definition regarding the notion of solution that we consider in this paper. Even though, in (1.4), the fluid genuine unknowns are and , the mathematical theory that we develop in Section 4 suits better to the unknowns and . Therefore we state our definition of weak solution in terms of these latter unknowns. Nevertheless, we shall keep these notations, even though no fluid velocity field underlies the computation of

###### Definition 2.1.

Let and Given , we call weak solution to (1.4) on any pair such that there is a collection satisfying

• The following regularities:

 (⟨x⟩+|u|)√ρ∈L∞(0,T;L2(Rd)),where ⟨x⟩=√1+|x|2, (ε+ν)∇√ρ∈L∞(0,T;L2(Rd)), ε√ν∇2√ρ∈L2(0,T;L2(Rd)), √νTN∈L2(0,T;L2(Rd)),

with the compatibility conditions

 √ρ⩾0 a.e. on (0,T)×Rd,√ρu=0 a.e. on {√ρ=0}.
• Euler case : The following equations in

 (2.1) {∂tρ+div(ρu)=0,∂t(ρu)+div(√ρu⊗√ρu)+∇P(ρ)=0.
• Korteweg and Navier-Stokes cases : The following equations in

 (2.2) ⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩∂t√ρ+div(√ρu)=12Trace(TN),∂t(ρu)+div(√ρu⊗√ρu)+∇P(ρ)=div(ν√ρSN+ε22SK),

with the symmetric part of , and the compatibility conditions:

 (2.3) √ρTN=∇(√ρ√ρu)−2√ρu⊗∇√ρ, (2.4) SK=√ρ∇2√ρ−∇√ρ⊗∇√ρ.

We emphasize that the above definition is essentially the “standard” one, up to the fact that we require . The reason for this assumption will become clear in the Subsection 4.1 where we will recall the a priori estimates motivating this definition (see Lemma 2.6, as well as the definition of the pseudo-energy in (2.13)).

Several remarks are in order. When the symbol alone appears, it must be understood as , while when the symbol appears alone, it is defined by Under the compatibility condition of item this yields a well-defined vector-field. As for the stress-tensors involved in the momentum equation (2.2), we emphasize that (2.3) reads formally

An originality of the previous definition is that in the case , we do not ask for the continuity equation in terms of but in terms of However, we prove here that the usual continuity equation as written in (2.2) is a consequence to this definition thanks to the regularity of and This is the content of the following lemma:

###### Lemma 2.2.

Let . Assume that is a weak solution to (2.2) on in the sense of Definition 2.1. Then it satisfies

 ∂tρ+div(ρu)=0 in D′((0,T)×Rd).
###### Proof.

By definition, we have

 ∂t√ρ+div(√ρu)=12Trace(TN)

Here we note that (so that ). We can then multiply this equation by We obtain:

 ∂tρ=−2√ρdiv(√ρu)+√ρTrace(TN).

At this point we remark that, by definition of

 div(ρu)=√ρTrace(TN)+2√ρu⋅∇√ρ

and, since , the products of the identity below are well-defined:

 div(ρu)=√ρdiv(√ρu)+√ρu⋅∇√ρ.

Combining these equation entails

 div(ρu)=2√ρdiv(√ρu)−√ρTraceTN.

We conclude thus that:

 ∂tρ=−div(ρu).

### 2.1. Rewriting of (1.4) with a suitable time-dependent scaling

In the case where the density is defined for all time and is dispersive (in the sense that it goes to zero pointwise), it is natural to examine the behavior of near , since it gives an “asymptotic pressure law” as time goes to infinity. A consequence of our result is that the large time behavior in (1.4) is very different according to or . Herein, we assume that with and . Typically, when , we recover the isothermal case, , and we can also consider

 P(ρ)=κρ+N∑j=1κjργj,N⩾1, κj>0, γj>1,

with no other restriction on (in any dimension), or even the exotic case . The most general class of pressure laws that we shall consider is fixed by the following assumptions:

###### Assumption 2.3 (Pressure law).

The pressure is convex ( for all ), and satisfies

 κ:=P′(0)>0.

Resuming the approach from [CaGa-p] (the link between Schrödinger equation and Euler-Korteweg equation is formally given by the Madelung transform), we change the unknown functions as follows. Introduce solution of the ordinary differential equation

 (2.5) ¨τ=2κτ,τ(0)=1,˙τ(0)=0.

The reason for considering this equation will become clear in Subsection 3.1. We find in [CaGa-p], for slightly more general initial data:

###### Lemma 2.4.

Let , . Consider the ordinary differential equation

 (2.6) ¨τ=2κτ,τ(0)=α,˙τ(0)=β.

It has a unique solution , and it satisfies, as ,

 τ(t)=2t√κlnt(1+O(ℓ(t))),˙τ(t)=2√κlnt(1+O(ℓ(t))),

where

 ℓ(t):=lnlntlnt⋅

We sketch the proof of this lemma in Appendix B, without paying attention to the quantitative estimate of the remainder term. We now introduce the Gaussian , and we set

 (2.7) ρ(t,x)=1τ(t)dR(t,xτ(t))∥ρ0∥L1∥Γ∥L1,u(t,x)=1τ(t)U(t,xτ(t))+˙τ(t)τ(t)x,

where we denote by the spatial variable for and . Denoting , (1.4) becomes, in terms of these new unknowns,

 (2.8) ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩∂tR+1τ2div(RU)=0,∂t(RU)+1τ2div(RU⊗U)+2κyR+P′(θRτd)∇R∂t(Rw)+1τ2div=ε22τ2R∇(Δ√R√R)+ντ2div(RDU)+ν˙ττ∇R.

The analogue of Definition 2.1 is the following:

###### Definition 2.5.

Let and Given , we call weak solution to (2.8) on any pair such that there exists a collection satisfying

• The following regularities:

 (⟨y⟩+|U|)√R∈L∞(0,T;L2(Rd)), (ε+ν)∇√R∈L∞(0,T;L2(Rd)), ε√ν∇2√R∈L2(0,T;L2(Rd)), √νTN∈L2(0,T;L2(Rd)),

with the compatibility conditions

 √R⩾0 a.e. on (0,T)×Rd,√RU=0 a.e. on {√R=0}.
• Euler case : The following equations in

 (2.9) ⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩∂tR+1τ2div(RU)=0,∂t(RU)+1τ2div(√RU⊗√RU)+2κyR+P′(θRτd)∇R=0.
• Korteweg and Navier-Stokes cases : The following equations in

 (2.10) ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩∂t√R+1τ2div(√RU)=12τ2Trace(TN),∂t(RU)+1τ2div(√RU⊗√RU)+2κyR+P′(θRτd)∇R∂t(RU)+1τ2div=div(ντ2√RSN+ε22τ2SK)+ν˙ττ∇R,

with the symmetric part of and the compatibility conditions:

 (2.11) √RTN=∇(√R√RU)−2√RU⊗∇√R, (2.12) SK=√R∇2√R−∇√R⊗∇√R,

Mimicking the proof of Lemma 2.2, we see that in the case , if is a weak solution to (2.10) on in the sense of Definition 2.5, then it satisfies

 ∂tR+1τ2div(RU)=0 in D′((0,T)×Rd).

In view of (2.7), we check directly:

###### Lemma 2.6 (Equivalence of the notions of solution).

Let . Then is a weak solution of (1.4) on if and only if is a weak solution of (2.8) on , where and are related through (2.7).

###### Remark 2.7.

If in Definition 2.1, we had required only , then the above equivalence would not hold. In the same spirit, the change of unknown (2.7) would make the notion of solution rather delicate in the case of the Newtonian Navier-Stokes equation, a case where typically . More generally, we do not consider velocities enjoying integrability properties, unless the density appear as a weight in the integral.

We define the pseudo-energy of the system (2.8) by

 (2.13) E(t) :=12τ2∫R|U|2+ε22τ2∫|∇√R|2+P′(0)∫(R|y|2+RlnR) +τdθ∫G(θRτd),

where

 G(u)=∫u0∫v0P′(σ)−P′(0)σdσdv,

which formally satisfies

 (2.14) ˙E(t)=−D(t)−ν˙τ(t)τ(t)3∫R(t,y)divU(t,y)dy,

where the dissipation is defined by

 (2.15) D(t) :=˙ττ3∫R|U|2+d˙τττd(∫[P(σ)−σP′(0)]|σ=θRτd) +ε2˙ττ3∫|∇√R|2+ντ4∫|SN|2.

By convexity we have , and for , so . Note also the identities

 (2.16) F′′(σ)=P′(0)σ+G′′(σ),F(ρ)=P′(0)ρlnρ+G(ρ).

Recall the Csiszár-Kullback inequality (see e.g. [LogSob, Th. 8.2.7]): for with ,

 ∥f−g∥2L1(Rd)⩽2∥f∥L1(Rd)∫f(x)ln(f(x)g(x))dx.

Writing

 ∫(R|y|2+RlnR)=∫RlnRΓ,

the (formal) conservation of the mass for and the definition (2.7) imply that the pseudo-energy is non-negative, .

As for global solutions, we have the following natural definition:

###### Definition 2.8.

Let and We call global weak solution to (2.8) any pair which, by restriction, yields a weak solution to (2.8) on for arbitrary

### 2.2. Main result: large-time behavior of weak solutions to (2.10)

With the previous definitions and remarks, a quantitative and precise statement of Theorem 1.2 reads as follows:

###### Theorem 2.9.

Let . Assume that satisfies Assumption 2.3, and let be a global weak solution of (2.8), in the sense of Definition 2.8, with constant mass

 ∫RdR(t,y)dy=∫RdR(0,y)dy,∀t⩾0.
1. If then

 ∫RdyR(t,y)dy⟶t→∞0and∣∣∣∫Rd(RU)(t,y)dy∣∣∣⟶t→∞∞,

unless , a case where

 ∫RdyR(t,y)dy=∫Rd(RU)(t,y)dy≡0.
2. If then weakly in as .

3. If and the energy defined by (1.5) satisfies as , then

 ∫Rd|y|2R(t,y)dy⟶t→∞∫Rd|y|2Γ(y)dy.
###### Remark 2.10.

Unlike in Theorem 1.1, no smallness assumption is made on at ( may even be linear in space), so there is no such geometrical structure on the initial velocity as in [Serre97, Gra98].

###### Remark 2.11.

In view of (2.14)–(2.15) and the property , the assumptions of point (b) are fairly natural, after noticing that

 ˙τ(t)τ(t)3∫R|divU|dy ⩽˙ττ(∫Rdy)1/2(1τ4∫R|divU|2dy)1/2 ≲˙ττ(∫Rdy)1/2√D(t).

Similarly, at least in the case , the formal conservation of the energy defined by (1.5), encompasses the assumption of point (c).

###### Remark 2.12 (Wasserstein distance).

The points (b) and (c) of Theorem 2.9 imply the large time convergence of to in the Wasserstein distance , defined, for and probability measures, by

 Wp(ν1,ν2)=inf{(∫Rd×Rd|x−y|pdμ(x,y))1/p;(πj)♯μ=νj},

where varies among all probability measures on , and denotes the canonical projection onto the -th factor. This implies, for instance, the convergence of fractional momenta (see e.g. [Vi03, Theorem 7.12])

 (2.17) ∫|y|2sR(t,y)dy⟶t→∞∫|y|2sΓ(y)dy,0⩽s⩽1.

Back to the initial unknowns , Theorem 2.9 and (2.7) yield

 ρ(t,x)∼t→∞∥ρ0∥L1(Rd)πd/21τ(t)de−|x|2/τ(t)2,

as announced in Theorem 1.2, where the symbol means that only a weak limit is considered. However, in the special case of Gaussian initial data considered in Section 3.1, it is easy to check that all the assumptions of Theorem 2.9 are satisfied, and moreover that strongly in . Finally, another consequence of Lemma 2.4, the (proof of the) last point in Theorem 2.9, and (2.7) is

 12∫Rdρ(t,x)|u(t,x)|2dx∼t→∞P′(0)d∥ρ0∥L1(Rd)lnt∼t→∞−P′(0)∫Rdρ(t,x)lnρ(t,x)dx.

This shows that indeed, no a priori information can be directly extracted from the energy defined in (1.5).

## 3. From Gaussians to Theorem 2.9

This part of the paper is devoted to the large time behavior of solutions to (1.3) and its variants. We first compute explicit Gaussian solutions and then proceed to the proof of Theorem 2.9.

### 3.1. Explicit solution

In this section, we resume and generalize some results established in [Yuen12, CFY17]. The generalizations concern two aspects: we allow densities and velocities which are not centered at the same point (hence and below), and we consider the quantum Navier-Stokes equation.

#### 3.1.1. Euler and Newtonian Navier-Stokes equations

We recall the compressible Euler equation for isothermal fluids on

 (3.1) {∂tρ+div(ρu)=0,∂t(ρu)+div(ρu⊗u)+κ∇ρ=0,

where . As noticed in [Yuen12], (3.1) has a family of explicit solutions with Gaussian densities and affine velocities centered at the same point. Allowing different initial centers for these quantities leads to considering

 (3.2)

with , . Seeking a solution of the form

 ρ(t,x)=b(t)e−∑dj=1αj(t)(xj−¯¯¯xj)2,u(t,x)=⎛⎜ ⎜⎝β1(t)x1⋮βd(t)xd⎞⎟ ⎟⎠+⎛⎜ ⎜⎝c1(t)⋮cd(t)⎞⎟ ⎟⎠,

and plugging this ansatz into (3.1), we obtain a set of ordinary differential equations:

 (3.3) ˙αj+2αjβj=0,˙βj+β2j−2καj=0, (3.4) ˙¯¯¯xj=βj¯¯¯xj+cj,˙b=bd∑j=1(˙αj¯¯¯x2j+2αj¯¯¯xj˙¯¯¯xj−2αjcj¯¯¯xj−βj), (3.5) ˙cj+βjcj+2καj¯¯¯xj=0.

Mimicking [LiWa06], seeking and of the form

 αj(t)=α0jτj(t)2,βj(t)=˙τj(t)τj(t),

we check that the two equations in (3.3) are satisfied if and only if

 (3.6) ¨τj=2κα0jτj,τj(0)=1,˙τj(0)=β0j,

and we find

 b(t)=b0∏dj=1τj(t),¯¯¯xj(t)=c0jt,cj(t)=c0j(1−˙τj(t)τj(t)t).
###### Remark 3.1.

Since the velocity is affine in , this computation also yields explicit solutions for the isothermal (Newtonian) Navier-Stokes equations, but not for its quantum counterpart, as we will see below.

#### 3.1.2. Korteweg and quantum Navier-Stokes equations

As in [CFY17], we generalize (3.1) by allowing the presence of a Korteweg term (), and we extend this contribution by allowing a quantum dissipation (quantum Navier-Stokes equation, when ). We recall the isothermal Korteweg and quantum Navier-Stokes equations

 (3.7)

with , and where the Korteweg term is also equal to

 ε24∇Δρ−ε2div(∇√ρ⊗∇√ρ),

which is called the Bohm’s identity. Proceeding as in the previous subsection, (3.3)–(3.5) become

 (3.8) ˙αj+2αjβj=0,˙βj+β2j−2καj=ε2α2j−ναjβj, ˙¯¯¯xj=βj¯¯¯xj+cj,˙b=bd∑j=1(˙αj¯¯¯x2j+2αj¯¯¯xj˙¯¯¯xj−2αjcj¯¯¯xj−βj), ˙cj+βjcj+2καj¯¯¯xj=−ε2α2j¯¯¯xj+ναjβj¯¯¯xj.

Again, we seek and of the form

 αj(