A multidimensional nonlinear sixth-order quantum diffusion equation

Ann. I. H. Poincaré – AN 30 (2013) 337–365

www.elsevier.com/locate/anihpc

A multidimensional nonlinear sixth-order quantum diffusion equation

Mario Bukal

, Ansgar Jüngel

, Daniel Matthes

aInstitute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstraße 8-10, 1040 Wien, Austria bZentrum für Mathematik, Technische Universität München, 85747 Garching, Germany

Received 6 May 2011; received in revised form 19 June 2012; accepted 20 August 2012 Available online 3 September 2012

Abstract

This paper is concerned with the analysis of a sixth-order nonlinear parabolic equation whose solutions describe the evolution of the particle density in a quantum fluid. We prove the global-in-time existence of weak nonnegative solutions in two and three space dimensions under periodic boundary conditions. Moreover, we show that these solutions are smooth and classical whenever the particle density is strictly positive, and we prove the long-time convergence to the spatial homogeneous equilibrium at a universal exponential rate. Our analysis strongly uses the Lyapunov property of the entropy functional.

©2012 Elsevier Masson SAS. All rights reserved.

MSC:35B45; 35G25; 35K55

Keywords:Higher-order diffusion equations; Quantum diffusion model; Entropy-dissipation estimate; Gradient flow

1. Introduction

Degond et al. derived in[5] a nonlocal quantum diffusion model for charged particles in, for instance, semicon- ductors or cold plasmas by applying a moment method to a Wigner–BGK model. An asymptotic expansion of the nonlocal model in terms of the scaled Planck constanth¯² leads to a family of parabolic equations for the particle densities n(t;x). The first member of this family is the classical heat equation ∂_tn=n. The second one is the fourth-order Derrida–Lebowitz–Speer–Spohn (DLSS) equation, see (3) below, which is analyzed in[9,13]. This pa- per is concerned with the third family member, obtained from an expansion to orderh¯⁴(see[3, Appendix]), which reads as

∂_tn=div

n∇ 1

2

∂_ij²logn2

+1 n∂_ij²

n∂_ij²logn

. (1)

✩ The authors acknowledge partial support from the Austrian Science Fund (FWF), grants P20214, P22108, and I395; the Austria–Croatia Project HR 01/2010; the Austria–France Project FR 07/2010; and the Austria–Spain Project ES 08/2010 of the Austrian Exchange Service (ÖAD).

* Corresponding author.

E-mail addresses:mbukal@asc.tuwien.ac.at(M. Bukal),juengel@tuwien.ac.at(A. Jüngel),matthes@ma.tum.de(D. Matthes).

0294-1449/$ – see front matter ©2012 Elsevier Masson SAS. All rights reserved.

http://dx.doi.org/10.1016/j.anihpc.2012.08.003

Here and in the following, we employ the notations∂_i=∂/∂x_i,∂_ij² =∂²/∂x_i∂x_j, etc. and the summation convention over repeated indices from 1 tod. We study the initial-value problem for (1) in thed-dimensional torusT^d∼= [0,1]^d (imposing periodic boundary conditions) in dimensionsd=2 andd=3. The one-dimensional problem has recently been studied in[14].

Specifically, we establish and compare two solution concepts for (1). The first concept is concerned with weak nonnegative solutions; in this framework we generalize the global existence result from[14]to themultidimensional situation. The second concept is that of positive classical solutions; in analogy to the results obtained by Bleher et al.

for the fourth-order DLSS equation[2], we are able to establish the existence of such regular solutions for (1) locally in time. Naturally, a classical solution is also a weak solution on the time interval of its existence. Vice versa, from a given weak solution, one obtains classical solutions on all time subintervals on which the weak solution is strictly positive and has a uniformly bounded energy (see below for the definition). Since we are not able to rule out the loss of strict positivity due to the evolution, it thus might happen that the classical solution concept breaks down on certain, possibly even infinite time intervals along the globally well-defined weak solution.

We shall provide further motivations to study (1) in Section2below. At this point, we simply want to put Eq. (1) into the general context of higher-order parabolic equations. Mainly initiated by the research on pattern formation in Cahn–Hilliard and related models in the late 1980’s, the literature on the rich mathematical structure of nonlinear fourth-order and sixth-order equations has grown rapidly over the last two decades. Particular interest has been devoted to equations that arepositivity preserving: such equations allow for the introduction of a suitable solution concept such that a nonnegative initial datum leads to a nonnegative global solution. Clearly, this is a core feature for equations that model the evolution of particle densities, etc. On the other hand, positivity preservation is a rare property, since general parabolic equations of fourth- or higher-order do not obey comparison principles. For instance, even the linear equation∂_tn+(−)^mn=0 isnotpositivity preserving ifm >1.

Among the positivity preserving models, the probably most famous study object is the fourth-order thin-film equa- tion

∂tn+div

n^β∇n

=0. (2)

It describes the surface tension-dominated motion of thin viscous films of heightn(t;x)0 under free slip (β=2) or no-slip (β=3) boundary conditions. The available literature on the existence, (non-)uniqueness and qualitative properties of solutions is huge and steadily growing; see[1]for a collection of references.

Other models for thin viscous films lead tosixth-orderequations. One example is

∂_tn=div

n^β∇²n ,

which models the spreading of a thin viscous fluid under the driving force of an elastic plate[8]. The model was first introduced in[15, Formula (A8)]in space dimension d=1 withβ=3 together with a more general form of this equation arising in the isolation oxidation of silicon. Another application for such thin-film equations concerns the bonding of Silicon–Germanium films to silicon substrates[8]. Further examples of sixth-order equations can be found in[7,14,16].

The previously mentioned DLSS (orquantum diffusion) equation

∂_tn+∂_ij²

n∂_ij²logn

=0, (3)

provides another well-studied example of a fourth-order equation. Originally, the one-dimensional version of (3) arose in the context of spin systems. Derrida et al.[6]derived it in the course of studying fluctuations of the interface between the regions of predominantly positive and negative particle spins in the Toom model. There are numerous results concerning the existence of weak solutions and their long-time behavior. We refer to[13]for some references.

Eq. (1) can be seen as a sixth-order extension of (3).

In the existence analysis for equations like (2), one of the main difficulties is to establish nonnegativity of the solu- tions. Typically, sophisticated regularizations are constructed that lead to smooth andstrictly positiveapproximative solutions. The limit of vanishing regularizations then provides a nonnegative weak solution.

For our Eq. (1), the situation is more delicate since—like for the DLSS equation (3)—the nonlinearity in the equation is not well-defined whennvanishes. This is a problem: Although nonnegativity of the solution is expected on physical grounds, the possibility that a vacuum (localized in time and space) is created from an initially strictly

positive density cannot be ruled out. Thus, atop of constructing strictly positive approximations, we need to define a solution concept that works also for merely nonnegative densities with the property that the passage to the limit of vanishing regularizations is possible.

The key idea here is to rewrite the nonlinearity in (1) in a way that substitutes the logarithm by an expression that is still well-defined forn=0. It turns out that the following equivalent representation of Eq. (1),

∂tn=³n+∂_{ij k}³ F₁^{(ij k)}(n)+∂_ij²F₂^{(ij )}(n), (4)

with the nonlinear operators F₁^{(ij k)}(n)=4∂i

√n 4∂j ⁴

√n∂k ⁴

√n−3∂_{j k}²√ n

, F₂^{(ij )}(n)=8

d k=1

∂_ik²√

n−4∂i ⁴

√n∂k⁴

√n

∂_{j k}² √

n−4∂j ⁴

√n∂k ⁴

√n

(5) is appropriate to study both concepts of solutions: weak and classical.

The construction of strictly positive approximative solutions uses yet another transformation of the nonlinearity.

First, (1) is discretized in time with the implicit Euler scheme. The semi-discrete equation is regularized by an addi- tional term of the formε(³−1)logn. Each time step then requires the solution of astrictlyelliptic problem in terms ofy=logn. Classical elliptic theory providesL^∞-bounds onyand thus strict positivity ofn=exp(y).

The required compactness to perform the deregularization limitε↓0 and later the passage to the time-continuous limit is obtained from the dissipation of a distinguished Lyapunov functional: The physical entropy

H[u] =

T^d

u(logu−1)+1

dx (6)

is nonincreasing along the solutions. In fact, using the entropy construction method of[12], which is based on system- atic integration by parts, we are able to prove that the entropy dissipation−dH/dtcontrols certain spatial derivatives,

−dH[n]

dt κ

T^d

∇³√n²+ |∇√⁶ n|⁶

dx, (7)

where∇^k denotes the tensor of all partial derivatives of orderk. The resulting estimates are sufficient to pass to the limit.

Our main results about weak solutions are the following two theorems.

Theorem 1 (Global existence of weak solutions). Let n₀ ∈L¹(T^d) be a nonnegative function with finite en- tropy H[n₀] <∞. Then there exists a nonnegative function n ∈ W_loc^1,4/3(0,∞; H⁻³(T^d)), satisfying √

n ∈ L²_loc(0,∞;H³(T^d))andn(0)=n₀, that is a solution to(4)in the following weak sense:

∞ 0

∂_tn, ϕdt+ ∞ 0

T^d

∂_{ij k}³ ϕ∂_{ij k}³ n+∂_{ij k}³ ϕF₁^{(ij k)}(n)−∂_ij²ϕF₂^{(ij )}(n)

dxdt=0 (8)

for all test functionsϕ∈L⁴(0, T;H³(T^d)).

It is not trivial at all to see that all integrals on the right-hand side of (8) are well-defined for functionsnof the stated regularity. At this point, we just mention that under these hypotheses,√⁴

nis a well-defined Sobolev function;

see Lemma26in Appendix A as well as[17]and[9, Section 3]for a discussion about the regularity of square and fourth roots of nonnegative functions. The relevant estimates on the pairings inside the integrals are established in the course of the proof; see, e.g., Lemma10below. Since dimension-dependent Sobolev embeddings are involved, this particular concept of weak solution does not carry over to space dimensionsd4.

We recall that H^k(T^d)etc. are spaces of functions that are 1-periodic in each spatial coordinate direction. The derivation of the sixth-order equation (1) in[3]was performed onR^dand hence, it does not include the derivation of

physically relevant boundary conditions. In this work, we have chosen periodic boundary conditions to simplify the analysis. In particular, integration by parts plays a pivotal role in our derivation of a priori estimates, and the boundary integrals vanish for periodic functions. We note that in[3], radially symmetric solutions for (1) satisfying no-flux-type boundary conditions have been considered instead.

Theorem 2(Exponential time decay). Letn₀∈L¹(T^d)be a nonnegative function with finite entropyH[n₀]<∞and unit mass _Tdn₀dx=1. Letn be the weak solution to(4)constructed in Theorem1. Then there exists a constant λ >0, depending ond, such that for allt >0,

n(t; ·)−1

L¹(T^d)

2H[n₀]e^−λt.

Since Eq. (4) is semi-linear parabolic, it is accessible by methods from the theory of analytic semigroups. This approach leads to the following result on classical solutions.

Theorem 3 (Existence and uniqueness of a classical solution). Let n0∈H²(T^d) be strictly positive. Then there existT_∗>0and precisely one smooth and strictly positive classical solutionn∈C^∞((0, T_∗);C^∞(T^d))to(4)with n(t )→n₀inH²(T^d)ast↓0. Moreover, eitherT_∗= +∞, or there exists a limiting profilen_∗∈H²(T^d)such that n(t )→n_∗inH²(T^d)ast↑T_∗andmin_x∈Tdn_∗(x)=0.

In other words, the only possibility for a classical solution to break down is the loss of strict positivity. This result parallels the one of[2]for the fourth-order DLSS equation in space dimensiond=1. Since stronger Sobolev embeddings are available for the sixth-order equation (4), our result holds in dimensionsd=2 andd=3 as well. It is an open problem if loss of positivity can occur att >0 or not.

Naturally, we shall establish a connection between the concept of weak solutions, defined in (8), and classical solutions. To do so, we need to introduce theenergy: For a positive and smooth functionu∈C^∞(T^d), define

E[u] =1 2

T^d

u∇²logu²dx. (9)

This functional is equivalent to theL²-norm of∇²√

uin the sense that c∇²√

u²

L²E[u]C∇²√ u²

L² (10)

for some constants 0< cC[9,13]. For smooth and positive solutions to (4), one easily proves thatEis a Lyapunov functional, see Lemma6below. The functionalE[u]extends in a weakly lower semi-continuous manner to all non- negative functionsuwith√

u∈H²(T^d); see[9, Section 3]for details. Hence, ifnis a weak solution in the sense of Theorem1, thenE[n(t )]is well-defined for almost everyt >0. We expect thatE is a Lyapunov functional also for weak solutions, but currently we are not able to prove this conjecture, mainly becauseEis not a convex functional.

Theorem 4.Assume that the weak solutionnfrom Theorem1has the property thatE[n(t )]is uniformly bounded on some interval (T₁, T₂), and that it is strictly positive at some timet₀∈ [T₁, T₂); hereT₁=0 and/orT₂= +∞are admissible. Then there existsT_∗∈(T₁, T₂]such thatnequals to the classical solution from Theorem3 on(t₀, T_∗).

Moreover, eitherT_∗=T2orn(t )loses strict positivity ast↑T_∗in the sense of Theorem3.

It is well known for the fourth-order equation (3), that weak solutions may not be unique[13]. We expect the same phenomenon to occur for (1). On the other hand, Theorem4asserts that a new weak solutionn_∗can branch off from a given classical solutionnat some timeT >0onlyif eithernloses strict positivity, limt↑Tinf_x∈T^dn(t, x)=0, or if n_∗has locally unbounded energy, lim sup_t↓TE[n_∗(t)] = +∞. This shows consistency between the notions of weak and classical solutions. In view of Theorem2, it is reasonable to conjecture that all weak solutions become classical eventually ast→ ∞.

The paper is organized as follows. Section2provides some background information on the derivation and prop- erties of (1). In Section3, we derive the alternative formulation (4) of (1) and we prove the entropy inequality (7).

Sections4,5,6, and7are devoted to the proofs of Theorems1,2,3, and4, respectively. Finally, in Appendix A, we collect some technical lemmas and recall some known results which are used in the existence analysis.

Notations.All functionsudefined on the torusT^dare assumed to be one-periodic in each coordinate. Specifically, u:[0,1]^d→Ris said to belong to the function spaceL^p(T^d),W^m,p(T^d)orC^∞(T^d), respectively, if its periodic extension Eu:R^d→R, defined by Eu(x)=u(xmodT^d), belongs toL^p_loc(R^d),W_loc^m,p(R^d)orC^∞(R^d). Lebesgue and Sobolev norms are calculated by integrating the respective powers of Euand its weak derivatives (which are periodic functions onR^d) over the unit cube[0,1]^d.

2. Derivation, motivation, and open problems

In this section, we indicate several motivations to study Eq. (1) by reviewing its derivation from the nonlocal quan- tum model, putting it in the context of gradient flows, and establishing connections to the heat and DLSS equations.

2.1. On the derivation from the nonlocal quantum model

Degond et al. derived in[5]the nonlocal and nonlinear quantum diffusion model

∂tn=div(n∇A) inR^d, t >0, (11)

where the potentialAis defined implicitly as the unique solution to n(t;x)=

R^d

Exp

A(t;x)−|p|² 2

dp.

The so-called quantum exponential Exp is defined as the Wigner transformed operator exponential: Denoting byW the Wigner transformation and byW⁻¹the corresponding Weyl quantization, then Exp(f )=W⁻¹◦exp◦W (f ); see [5]for details.

In the semi-classical limith¯↓0, the expression Exp(A−|p|²/2)converges toe^A, so thatA=logn, and we recover from (11) the classical heat equation. Forh >¯ 0, however, the quantum exponential is a complicated, genuinely non- local operator. An asymptotic expansion in terms ofh¯ has been performed in[3, Appendix], leading to the following local approximation ofAin terms ofn:

A=A₀+ ¯h²

12A₁+ ¯h⁴

360A₂+O

¯ h⁶

(12) with the local expressions

A₀=logn, A₁= −2√

√ n

n , A₂=1

2∇²logn²+1 n∂_ij²

n∂_ij²logn .

ReplacingAin (11) byA₀,A₁, orA₂yields, respectively, the heat equation, the DLSS equation (3), or the sixth-order equation (1). In this sense, (3) and (1) constitute, respectively, the primary and secondary quantum corrections to the classical diffusion equation.

2.2. Gradient-flow structure

Eq. (1) possesses—at least on a formal level—a variational structure. The divergence form implies that solutions nformally conserve the total mass, i.e., the integralm= _Tdn(t;x)dxis independent oft. By homogeneity, we can assumem=1 without loss of generality. Thus, any solution to (1) defines a curvet→n(t )in the space of probability measures onT^d. Provided thatnis regular enough, this curve realizes a steepest descent in the energy landscape of the energy functionalEfrom (9) with respect to theL²-Wasserstein metric. Indeed, by a formal calculation, we obtain the gradient-flow representation

∂_tn=div

n∇δE[n] δn

from (11) withA≡A₂, whereA₂=δE[n]/δnis the variational derivative ofE.

This variational structure is a remarkable property by itself. Atop of that, it establishes yet another connection to the heat and DLSS equations. It is well known since the seminal paper[11]that the heat equation is the gradient flow

of the entropy functionalHfrom (6) with respect to theL²-Wasserstein distance. The dissipation ofHalong its own gradient flow amounts to theFisher information,

F[n] = −1 2

dH[n] dt =1

2

T^d

n|∇logn|²dx,

while the second-order time derivative produces the energy from (9), E[n] =1

4 d²H[n]

dt² =1 2

T^d

n∇²logn²dx.

The Fisher information, in turn, has been proven to generate the DLSS equation (3) as a gradient flow with respect to theL²-Wasserstein distance[9]. It is readily checked thatE also equals the first-order time derivative of the en- tropy along solutions of the DLSS equation. In this sense, the sixth-order equation (1) is related to the fourth-order equation (3) in the same way as (3) itself is related to the heat equation.

We mention this point because the intimate relation between the heat and the DLSS equations (and, more generally, between second-order porous medium and fourth-order diffusion equations) has been the key tool in obtaining optimal rates for the intermediate asymptotics of solutions to (3) in[18]. It would be interesting to derive estimates on the long- time behavior of solutions to (1) by similar means.

2.3. Open problems

Finally, we propose several questions about Eq. (1) that we consider of interest:

• With our methods, we are able to prove the dissipation property (7) only in dimensionsd3. IsHstill a Lyapunov functional in higher dimensionsd4?

• Is the Fisher informationF a Lyapunov functional? Our only result in this direction so far is a formal proof of dissipation ofFin dimensiond=1.

• Is the energyEmonotone along the weak solutions constructed here? If the answer is affirmative, then the addi- tional hypotheses on the uniform boundedness of the energy could be removed from Theorem4.

• Does (1) admit global weak solutions in dimensionsd4? Even if we assume that an inequality of the form (7) continues to hold, it is far from clear how to rewrite the weak formulation (8) in a form that does not take advantage of Sobolev embeddings in low dimensions.

• If (1) is posed on R^d instead ofT^d, one readily verifies that there exists a family of self-similar solutions us, namely

u_s(t;x)=λ(t )^−dU

λ(t )⁻¹x

withλ(t )=(1+6t )^1/6, with the Gaussian profile

U (z)=exp

−|z|² 2√³

2

.

Do these “spreading Gaussians” play the same role for (1) as they do for the heat equation and for the DLSS equation? In other words, isUan attracting stationary solution of (1) after the self-similar rescaling withx=λ(t )ξ andt=(e^6τ−1)/6, and do arbitrary solutions converge toUat a universal exponential rate? In dimensiond=1, there is numerical evidence for an affirmative answer.

3. Alternative formulations and functional inequalities

In this section, we derive two alternative formulations of the sixth-order equation (1) and prove an energy- dissipation formula and an entropy-dissipation estimate. First, we show that (1) can be written as the sum of a symmetric sixth-order term and a fourth-order remainder, and as the sum of a linear sixth-order part and a fifth-order remainder.

Lemma 5.Eq.(1)can be written for smooth positive solutions equivalently as

∂_tn=∂_{ij k}³

n∂_{ij k}³ logn +2∂_ij²

n∂_ik² logn∂_{j k}² logn

inT^d, t >0, (13)

and also equivalently as

∂_tn=³n+∂_{ij k}³ F₁^{(ij k)}(n)+∂_ij²F₂^{(ij )}(n) inT^d, t >0, (14)

where the nonlinear operatorsF₁^{(ij k)}andF₂^{(ij )}are defined in(5).

We recall that we have employed the summation convention in the above formulas.

Proof. For the following formal calculations, we introduce the shorthand notations y=logn,yi=∂ilogn,yij =

∂_ij²logn, etc. Observing that∂kn=nyk,n∂k(1/n)= −(∂kn)/n= −yk, we calculate 1

2n∂_k

∂_ij²logn2

=n∂_ij²y∂_ij²y_k, and

n∂_k 1

n∂_ij²

n∂_ij²logn

=∂_{ij k}³ (ny_ij)−y_k∂_ij²(ny_ij)

=∂_ij²

y_k(ny_ij)+ny_{ij k}

−y_k∂_ij²(ny_ij)

=∂_i

y_k∂_j(ny_ij)+y_{j k}(ny_ij)

−y_k∂_ij²(ny_ij)+∂_ij²(ny_{ij k})

=yik∂j(nyij)+yij k(nyij)+yj k∂i(nyij)+∂_ij² n∂_ij²yk

=2yik∂_j(ny_ij)+n∂_ij²y∂_ij²y_k+∂_ij² n∂_ij²y_k

=2∂j(ny_ijy_ik)−n∂_ij²y∂_ij²y_k+∂_ij² n∂_ij²y_k

. Summing these results, we obtain

1 2n∂k

∂_ij²logn2

+n∂k

1 n∂_ij²

n∂_ij²logn

=∂_ij² n∂_ij²yk

+2∂j(nyijyik).

Differentiation with respect toxkyields

∂_k 1

2n∂_k

∂_ij²logn2

+n∂_k 1

n∂_ij²

n∂_ij²logn

=∂_{ij k}³ n∂_{ij k}³ y

+2∂_{j k}²(ny_ijy_ik), which shows (13).

Similarly, introducingu=√⁴

n,u_i =∂_iu, u_ij =∂_ij²u, etc. and observing that∂_kn=4u³u_k,∂_ij²n=12u²u_iu_j + 4u³u_ij, anduu_ij=∂_ij²(u²)/2−u_iu_j, we calculate

n∂_{ij k}³ y=∂_{ij k}³ n−3

n∂_ij²n∂_kn+ 2

n²∂_in∂_jn∂_kn

=∂_{ij k}³ n−48u²uijuk−16uuiujuk

=∂_{ij k}³ n−12∂_ij² u²

∂_k u²

+16uiu_j∂_k u²

=∂_{ij k}³ n+4∂k

√n 4∂i ⁴

√n∂_j√⁴

n−3∂_ij²√ n

=∂_{ij k}³ n+F₁^{(ij k)}(n), 2nyiky_{j k}=32u⁴

u_ik u −u_iu_k

u² uj k

u −ujuk

u²

=8

∂_ik² u²

−4uiu_k

∂_{j k}² u²

−4uju_k

=8

∂_ik²√

n−4∂i ⁴

√n∂_k√⁴ n

∂_{j k}²√

n−4∂j ⁴

√n∂_k√⁴ n

=F₂^{(ij )}(n). (15)

Differentiating both equations and summing them leads to

∂_{ij k}³ n∂_{ij k}³ y

+2∂_ij²(ny_iky_{j k})=³n+∂_{ij k}³ F₁^{(ij k)}+∂_ij²F₂^{(ij )}, (16)

which gives (14). 2

In the next lemma, we make our claim about the Lyapunov property of the energyE, defined in (9), more precise.

Lemma 6.Ifn∈C^∞((t₁, t₂);C^∞(T^d))is a positive and classical solution to(1), then the energyE[n(t )]is a smooth and nonincreasing function on the interval(t₁, t₂). In fact, the energy is dissipated according to

d dtE

n(t )

= −

T^d

n(t ) ∇

1 2

∂_ij²logn(t )2

+ 1 n(t )∂_ij²

n(t )∂_ij²logn(t )

2

dx, t >0. (17)

Proof. The smoothness ofE[n(t )]follows since on the set of positive functionsu∈C^∞(T^d), the operationu→logu is a smooth map fromC^∞(T^d)to itself. Dissipation formula (17) follows by using formulation (1) and integration by parts:

d dtE[n] =

T^d

1 2∂_tn

∂_ij²logn2

+n∂_ij²(logn)∂_ij² ∂tn

n

dx

=

T^d

∂tn 1

2

∂_ij²logn2

+1 n∂_ij²

n∂_ij²logn dx

= −

T^d

n ∇

1 2

∂_ij²logn2

+1 n∂_ij²

n∂_ij²logn

2

dx,

which shows the claim. 2

Finally, we prove the entropy-dissipation inequality (7).

Lemma 7.Letd3 and letu∈H³(T^d)be strictly positive onT^d. Then there existsκ >0, depending only ond, such that

T^d

∂_{ij k}³ (logu)∂_{ij k}³ u+∂_{ij k}³ (logu)F₁^{(ij k)}(u)−∂_ij²(logu)F₂^{(ij )}(u) dx

κ

T^d

∇³√u²+ |∇√⁶ u|⁶

dx. (18)

Proof. The proof is based on an extension of the entropy construction method developed in[12]for one-dimensional equations. A proof for d =1 is given in [14]. Therefore, we restrict ourselves to the cases d =2 and d =3.

By (16), (18) is equivalent to, up to a factor,

T^d

u

∂_{ij k}³ logu2

−2∂_ij²logu

∂_ik² logu∂_{j k}² logu dx κ

12

T^d

2⁶∇³√

u²+6⁶|∇√⁶ u|⁶

dx. (19)

Settingy=logn,yi=∂ilogn,yij=∂_ij²logn, etc., a computation shows that (19) is equivalent to

T^d

u

12S[u] −κR[u]

dx0, (20)

whereS[u] =y_{ij k}² −2yijy_{j k}y_ki and

R[u] =2y_i²y_j²y_k²+12y_i²y_jy_{j k}y_k+8yiy_jy_ky_{ij k}+24yiy_ijy_{j k}y_k+12y_i²y_{j k}² +48yiy_{ij k}y_{j k}+16y_{ij k}² .

The idea of the entropy construction method is to find the “right” integrations by parts which are neces- sary to write the integrand of (20) as a sum of squares. To this end, we define the vector-valued function v = (v¹, . . . , v^d):T^d→R^dby

v^k=

2y_i²y_j²+y_iiy_j²+5yijy_iy_j+5yiijy_j y_k+

3y_i²y_j+11yiij+24yiy_ij

y_{j k}−(5yiy_j+11yij)y_{ij k}. A straightforward computation shows that the weighted divergence

T[u] =1

udiv(uv)=e^−y∂_k e^yv^k can be written as

T[u] =2y_i²y_j²y_k²+3y_i²y_j²y_kk+16y_i²y_jy_{j k}y_k+9y_i²y_jy_{j kk}+y_i²y_jjy_kk+7yiiy_jy_{j k}y_k+40yiy_ijy_{j k}y_k +3y_i²y_{j k}² +5yiy_ijjy_kk+40yiy_ijy_{j kk}+3yiy_{ij k}y_{j k}+11yijjy_ikk−11y_{ij k}² +24yijy_{j k}y_ki. By the divergence theorem, we have

T^d

uT[u]dx=0.

Hence, (20) is equivalent to

T^d

u

12S[u] −κR[u] +T[u]

dx0. (21)

We prove that there existsκ >0 such that the integrand is nonnegative. The expressionT[u]turns out to be the “right”

integration by parts formula allowing us to prove the nonnegativity of the above integral. At this point, we need to distinguish the space dimension.

First, considerd=2. Letx∈T^d be fixed. Without loss of generality, we may assume that∇u(x)points into the first coordinate direction, i.e.y₂=0 atx. Then we compute

12S[u] −εR[u] +T[u]

=(2−2ε)y₁⁶+3y₁⁴(y₁₁+y₂₂)+4(4−3ε)y₁⁴y₁₁+9y₁³(y₁₁₁+y₁₂₂)−8εy₁³y₁₁₁+y₁²(y₁₁+y₂₂)² +7y₁²y11(y11+y22)+8(5−3ε)y₁²

y₁₁² +y₁₂²

+3(1−4ε)y₁²

y₁₁² +2y²₁₂+y²₂₂ +5y1(y111+y122)(y11+y22)+40y1

y11(y111+y122)+y12(y122+y222) +3(1−16ε)y1(y11y111+2y12y112+y22y122)+11

(y111+y122)²+(y112+y222)² +(1−16ε)

y₁₁₁² +3y₁₁₂² +3y₁₂₂² +y₂₂₂²

=ξAεξ+ηBεη, whereξ andηare the vectors

ξ=

y₁³, y₁y₁₁, y₁y₂₂, y₁₁₁, y₁₂₂

, η=(y₁y₁₂, y₁₁₂, y₂₂₂), and the symmetric matricesA_εandB_εare defined by

A_ε=1 2

⎛

⎜⎜

⎜⎝

4−4ε 19−12ε 3 9−8ε 9

19−12ε 102−72ε 9 48−48ε 45

3 9 8−24ε 5 8−48ε

9−8ε 48−48ε 5 24−32ε 22

9 45 8−48ε 22 28−96ε

⎞

⎟⎟

⎟⎠,

B_ε=

46−48ε 23−48ε 20 23−48ε 14−48ε 11

20 11 12−16ε

.

Sylvester’s criterion shows that the unperturbed matricesA₀andB₀are positive definite. Indeed, the principal minors ofA₀are 2, 47/4, 20, 13, and 149/4, and the principal minors ofB₀are 46, 115, and 334. Since the set of (strictly) positive definite matrices is open in the set of all real symmetric matrices, there existsε₀>0 such that for all 0<

ε < ε₀, the matricesA_εandB_εare positive definite, too. This shows that 12S[u] −εR[u] +T[u]0 for 0< ε < ε₀, which implies (21).

Next, letd =3. This case is similar to the previous one, but technically more involved. Again, we fix somex∈ T^d and assume that∇u(x)is parallel to the first coordinate direction, i.e.y₂=y₃=0. For easier presentation, we introduce the abbreviations

p₊=y22+y33, p₋=y22−y33,

q_j+=y_j22+y_j33, q_j−=y_j22−y_j33, j=1,2,3.

Observe that 2

y₂₂² +y₃₃²

=p²₊+p²₋, 2

y_j22² +y²_j33

=q_j²₊+q_j²₋,

2(y22y_j22+y33yj33)=p₊q_j++p₋q_j−. With these notations, we find that

12S[u] −εR[u] +T[u]

=(2−2ε)y₁⁶+3y₁⁴(y11+p₊)+4(4−3ε)y₁⁴y11+9y₁³(y111+q1+)

−8εy₁³y111+y₁²(y11+p₊)²+7y₁²y11(y11+p₊)+8(5−3ε)y₁²

y₁₁² +y₁₂² +y₁₃² +3(1−4ε)y₁²

y₁₁² +1

2

p₋² +p₊² +2

y²₁₂+y²₁₃+y²₂₃

+5y1(y₁₁₁+q₁₊)(y₁₁+p₊) +40y1

y11(y111+q1+)+y12(y112+q2+)+y13(y113+q3+) +3(1−16ε)y1

y₁₁₁y₁₁+1

2(q₁₊p₊+q₁₋p₋)+2(y112y₁₂+y₁₁₃y₁₃+y₁₂₃y₂₃) +11

(y₁₁₁+q₁₊)²+(y₁₁₂+q₂₊)²+(y₁₁₃+q₃₊)² +(1−16ε)

y₁₁₁² +3

y₁₁₂² +y₁₁₃² +3

2

q₁²₊+q₁²₋+q₂²₊+q₂²₋+q₃²₊+q₋²₃ +6y₁₂₃²

=ξAεξ+ 3 j=2

η_jBεηj+ζCεζ +2νCεν+1

4(1−16ε)

q₂²₊+q₂²₋ , where

ξ=

y₁³, y₁y₁₁, y₁p₊, y₁₁₁, q₁₊

, η_j=(y₁y_1j, y_11j, q_j+), ζ =(y1p₋, q1−), ν=(y1y23, y123).

The matricesAεandBεare almost identical to those given above, with minor modifications in the third and fifth rows and columns:

A_ε=1 2

⎛

⎜⎜

⎜⎝

4−4ε 19−12ε 3 9−8ε 9

19−12ε 102−72ε 9 48−48ε 45

3 9 5−12ε 5 13/2−24ε

9−8ε 48−48ε 5 24−32ε 22

9 45 13/2−24ε 22 25−48ε

⎞

⎟⎟

⎟⎠,

B_ε=

46−48ε 23−48ε 20 23−48ε 14−48ε 11

20 11 47/4−12ε

.

Furthermore, the matrixC_ε is given by C_ε=

3−12ε 3/2−24ε 3/2−24ε 3−48ε

.

Again, the Sylvester criterion shows thatA₀,B₀, andC₀are positive definite. The principal minors ofA₀are 2, 47/4, 19/8, 5/8, and 453/64, while those ofB₀are 46, 115, and 1221/4, and those ofC₀are 3 and 27/4. Thus, there exists ε₀>0 such that for all 0< ε < ε₀, alsoA_ε,B_ε, andC_εare positive definite. 2

4. Existence of weak solutions

The proof of Theorem1is divided into several steps.

4.1. Solution of the semi-discretized problem

LetT >0 andτ >0 be given. We wish to solve, for a given initial datumn0∈L¹(T^d), the semi-discrete problem 1

τ(n−n₀)=³n+∂_{ij k}³ F₁^{(ij k)}(n)+∂_ij²F₂^{(ij )}(n) inT^d, whereF₁^{(ij k)}andF₂^{(ij )}are defined in (5).

Proposition 8.For a nonnegative functionn₀∈L¹(T^d)of unit mass,n₀L¹=1, and of finite entropy,H[n₀]<∞, there exists a sequence of solutionsn^τ₁, n^τ₂, . . .inH³(T^d)to the elliptic problems

1 τ

T^d

n^τ_k−n^τ_k−1 φdx+

T^d

∂_{ij k}³ φ∂_{ij k}³ n^τ_k+∂_{ij k}³ φF₁^{(ij k)} n^τ_k

−∂_ij²φF₂^{(ij )} n^τ_k

dx=0, (22)

holding for all test functionsφ∈H³(T^d), with the initial solutionn^τ₀=n₀. These solutions are of unit mass, and the entropy estimate

H n^τ_k

+κτ

T^d

∇³

n^τ_k²+∇⁶ n^τ_k⁶

dxH n^τ_k−1

, k1, (23)

holds withκ >0given in Lemma7.

Proof. For simplicity, we only give the argument for the construction ofn=n^τ₁ fromn₀. The passage from n^τ_k to n^τ_k+1works precisely in the same way since finiteness of the entropy is inherited from one step to the next.

Regularized problem. In a first step, we are going to construct strictly positive solutionsn_ε∈H³(T^d)to the regu- larized problem

1

τ(n−n0)=³n+∂_{ij k}³ F₁^{(ij k)}(n)+∂_ij²F₂^{(ij )}(n)+ε

³logn−logn

. (24)

Writingn=e^y, it follows from (15) that ³n=∂_{ij k}³

n∂_{ij k}³ y

−∂_{ij k}³ F₁^{(ij k)}(n).

Thus, assuming strict positivity andH³-regularity ofn, we can reformulate (24) as 1

τ(n−n₀)=∂_{ij k}³

(n+ε)∂_{ij k}³ y

−εy+∂_ij²F₂^{(ij )}(n), (25)

which is an equation inH⁻³(T^d).

Fixed point operator. We define the continuous mapSε:X× [0,1] →W^2,4(T^d)on the set X=

u∈W^2,4 T^d

: min

x∈T^du(x) >0