Lq-functional inequalities and weighted porous media equations

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Lq-functional inequalities and weighted porous media equations

Jean Dolbeault, Ivan Gentil, Arnaud Guillin, Feng-Yu Wang

To cite this version:

Jean Dolbeault, Ivan Gentil, Arnaud Guillin, Feng-Yu Wang. Lq-functional inequalities and weighted porous media equations. Potential Analysis, Springer Verlag, 2008, 28, pp.35-59. �hal-00122415�

hal-00122415, version 1 - 1 Jan 2007

L

-functional inequalities and weighted porous media equations

Jean Dolbeault, Ivan Gentil, Arnaud Guillin and Feng-Yu Wang 1st January 2007

Abstract

Using measure-capacity inequalities we study new functional inequalities, namely L^q- Poincar´e inequalities

Varµ(f^q)^1/q≤CP

Z

|∇f|²dµ andL^q-logarithmic Sobolev inequalities

Entµ f^2q1/q

≤CLS

Z

|∇f|²dµ .

As a consequence, we establish the asymptotic behavior of the solutions to the so-called weighted porous media equation

∂u

∂t = ∆u^m− ∇ψ· ∇u^m form≥1, in terms ofL²-norms and entropies.

Mathematics Subject Classification 2000: 26D10; 35K55, 39B62, 46E27, 46E35, 60E15 Keywords: Logarithmic Sobolev Inequality; Poincar´e inequality; Porous media equation;

Asymptotic behaviour; Nonlinear parabolic equations; Variance; Entropy; Weighted Sobolev spaces; Large time asymptotics; Rate of convergence.

1 Introduction

In this paper we analyze decay rates of the entropies associated to nonlinear diffusion equations using inequalities relating entropy and entropy production functionals. Consider for instance the Ornstein-Uhlenbeck semi-group onR^d, which is governed by

∂u

∂t = ∆u−x· ∇u , t>0, x∈R^d, with an initial conditionu₀ ∈ C² R^d

∩L¹₊(R^d, dγ). Here dγ =γ dx is the Gaussian measure on R^d,γ(x) := (2π)^−d/2 exp(−|x|²/2). Two entropies are widely used, namely

Var_γ(u) :=

Z u−

Z u dγ

2

dγ and Ent_γ(u) :=

Z ulog

u R u dγ

dγ .

Ifuis a smooth solution of the Ornstein-Uhlenbeck equation, integrations by parts show that d

dtVar_γ(u) =−2 Z

| ∇u|²dγ and d

dtEnt_γ(u) =−4 Z

| ∇√

u|²dγ .

By the Poincar´e inequality in the first case,

∀f ∈ C¹(R^d), Varγ(f)6 Z

|∇f|²dγ ,

and Gross’ logarithmic Sobolev inequality, see [Gro75], in the second case,

∀f ∈ C¹(R^d), Ent_γ f² 6 2

Z

|∇f|²dγ , it follows from Gronwall’s lemma that for anyt>0,

Var_γ(u)6e^−2tVar_γ(u₀) and Ent_γ(u)6e^−2tEnt_γ(u₀)

for all smooth initial conditionsu₀ inL²_γ(R^d) in the first case, or such thatEnt_γ(u₀) is finite in the second case. With minor changes, the method can be extended to the semi-group generated by

∂u

∂t = ∆u− ∇ψ· ∇u , t>0, x∈R^d, if ψ is a smooth function such that Z := R

e^−ψdx is finite and the probability measure dµ_ψ =Z_ψ⁻¹e^−ψdxsatisfies a Poincar´e inequality or a logarithmic Sobolev inequality. See for example [ABC⁺00] for a review on this inequalities and there applications.

A natural question is how to extend the variance or the entropy convergence to nonlinear semi-groups. Letm >1, and consider the semi-group generated by theweighted porous media equation

∂u

∂t = ∆u^m− ∇ψ· ∇u^m, t>0, x∈R^d,

with a non-negative initial condition u(0, x) = u₀(x). This equation, in short (WPME) is a simple extension of the standard porous media equation, which corresponds toψ= 0. We shall refer to [V´az92] for an introduction on this topic. A major difference is that under appropriate conditions onψthe solution of (WPME) converges to its mean. In other words, the nonlinear semi-group converges to the limit measureµ_ψ. The variance of a solution solution of (WPME) now obeys to

d

dtVar_µψ(u) =− 8 (m+ 1)²

Z

| ∇u^m+12 |²dµ_ψ .

Classical Poincar´e and logarithmic Sobolev inequalities are of no more use and have to be replaced by adapted functional inequalities, which are the purpose of this paper. This paper extends some earlier results on solutions to the porous media equation on the torusS¹≡[0,1) and related functional inequalities, see [CDGJ06]. As for the functional inequalities, we will work in a more general framework involving two Borel probability measures µ and ν on a Riemannian manifold (M, g), which are not necessarily absolutely continuous with respect to the volume measure. To consider quantities like R

f^qdµ and R

|∇f|²dν, it is therefore natural to work in the space of functionsf ∈ C¹(M), although slightly more general function spaces can be introduced by density with respect to appropriate norms. If the measures were absolutely continuous with respect to the volume measure, we could take functions which are only locally Lipschitz continuous as in [BCR05].

In Section 2 we will define functional inequalities that we shall call L^q-Poincar´e and L^q- logarithmic Sobolev inequalities. Equivalence of these inequalities with capacity-measure cri- teria will be established, based on Maz’ja’s theory. Links with more classical inequalities such as weak Poincar´e or weak logarithmic Sobolev inequalities are then studied in Section 3. Ex- plicit criteria can be deduced from earlier works, mainly [BR03, BCR05, CGG05]. In Section 4

we will give applications to the weighted porous media equation. Using theL^q-Poincar´e and L^q-logarithmic Sobolev inequalities we describe the asymptotic behavior of the solutions in terms of variance or entropy. The proof of two variants of results of [BCR05] is given in an appendix, see Section 5.

Throughout this paper, we intend to work under minimal assumptions and do not require that the measures showing up on both sides of the inequalities are the same or that they are absolutely continuous with respect to the volume measure. However, when only one measure is specified, one has to understand that the measures µand ν are the same on both sides of the inequalities.

2 Two L^q-functional inequalities

2.1 L^q-Poincar´e inequalities

Definition 2.1 Let µ and ν be respectively a probability measure and a positive measure on M. Assume that q ∈(0,1]. We shall say that (µ, ν) satisfies a L^q-Poincar´e inequalitywith constant C_P if for all non-negative functions f ∈ C¹(M) one has

Var_µ(f^q)1/q

:=

"

Z

f^2qdµ− Z

f^qdµ 2#1/q

6C_P Z

|∇f|²dν . (1) Note that if q > 1, Inequality (1) is not true if µ is not a Dirac measure. Consider indeed f = 1 +ǫ g withǫ→0 and g bounded. By applying Inequality (1) we get

h

q²ǫ² Varµ g²

+o(1)i1/q

6ǫ²CP

Z

|∇g|²dν+o(ǫ²). Ifgis such thatVar_µ g²

andR

|∇g|²dνare both positive and finite, we obtain a contradiction by lettingǫ→0 if q >1.

Proposition 2.2 For any bounded positive function f, the function q 7→ Var_µ(f^q)^1/q is in- creasing with respect toq∈(0,1]. As a consequence, if theL^q1-Poincar´e inequality holds, then the L^q2-Poincar´e inequality also holds for any0< q₂6q₁ 61.

We shall say that L^q-Poincar´e inequalities form a hierarchy of inequalities. The classical Poincar´e inequality corresponding toq= 1 implies allL^q-Poincar´e inequalities forq∈(0,1).

Proof ⊳ Without loss of generality, we may assume that f is positive. For any q ∈(0,1), letF(q) :=Var_µ(f^q)^1/q. We have

F^′(q) F(q) = d

dqlogF(q) = 1 q

−logF(q) +µ(f^2qlogf²)−µ(f^q)µ(f^qlogf²) F(q)^q

.

Leth(t) :=µ(f^tqlogf²)/µ(f^tq),t∈(1,2) and observe that by the Cauchy-Schwarz inequality,

2

q[µ(f^tq)]²h^′(t) =µ(f^tq(logf²)²)µ(f^tq)−[µ(f^tq logf²)]² >0. This proves that h(2) = µ(f^2qlogf²)

µ(f^2q) > µ(f^qlogf²)

µ(f^q) =h(1). Hence we arrive at

qF^′(q)

F(q) ≥ −logF(q) +

1−^[µ(f_µ(f^q2q)]₎²

µ(f^2qlogf²)

F(q)^q =−logF(q) +µ(f^2qlogf^2q) q µ(f^2q)

≥ −logF(q) + log µ(f^2q)1/q

≥0

where the last two inequalities hold by Jensen’s inequality and by monotonicity of the loga- rithm.⊲

We will now give a characterization of the L^q-Poincar´e inequality in terms of the capacity measure criterion. Such a criterion has recently been applied in [BCR06, Che05, CGG05] to give necessary and sufficient conditions for the usual, weak or super Poincar´e inequality, and the usual or weak logarithmic Sobolev inequality or theF-Sobolev inequality. The capacity measure criterion allows to compare all these inequalities and can be characterized in terms of Hardy’s inequality, in the one-dimensional case.

Let µ and ν be respectively a probability measure and a positive measure on M. Given measurable setsA and Ω such that A⊂Ω⊂M,the capacity Cap_ν(A,Ω) is defined as

Cap_ν(A,Ω) := inf Z

|∇f|²dν : f ∈ C¹(M), I_A6f 6I_Ω

. If the set

f ∈ C¹(M) : I_A6f 6I_Ω is empty then, by convention, we set Cap_ν(A,Ω) :=

+∞. This the case of Cap_ν(A, A) = +∞ for any bounded measurable setA and any ν with a locally positive density.

Letq ∈(0,1) and define

β_P:= sup (

X

k∈Z

µ(Ω_k)1/(1−q)

Cap_ν(Ω_k,Ω_k+1)q/(1−q)

)(1−q)/q

where the supremum is taken over all Ω⊂M withµ(Ω)≤1/2 and all sequences (Ω_k)_k∈Z such that for allk∈Z, Ω_k⊂Ω_k+1 ⊂Ω.

Theorem 2.3 Letµandνbe respectively a probability measure and a positive measure onM. (i) If q ∈ [1/2,1) and (µ, ν) satisfies a L^q-Poincar´e inequality with a constant C_P, then

β_P62^1/qC_P.

(ii) If q ∈(0,1) and β_P<+∞, then (µ, ν) satisfies a L^q-Poincar´e inequality with constant C_P6κ_Pβ_P, for some constant κ_P which depends only on q.

Proof ⊳ The proof follows the main lines of Theorem 2.3.5 of [Maz85].

Proof of(i). Consider Ω⊂M such that µ(Ω)≤1/2 and let (Ω_k)_k∈Z be a sequence such that for all k∈Z, Ω_k⊂Ω_k+1⊂Ω. Fix N ∈N^∗ and fork∈ {−N, . . . N}, let f_k ∈ C¹(M) be such thatI_Ω

k 6f_k6I_Ω

k+1. If no suchf_k exists, that is if Cap_ν(Ω_k,Ω_k+1) = +∞, then we discard Ω_k+1 from the sequence and reindex it. Finally, let (τ_k)k∈{−N,...N} be a non-increasing family of non-negative reals numbers to be defined later. A functionf on M is defined as follows:

(1) f =τ_−N on Ω_−N,

(2) f = (τ_k−τ_k+1)f_k+τ_k+1 on Ω_k+1\Ω_k for allk∈ {−N, N −1}, (3) f =τ_Nf_N+1 on Ω_N+1\Ω_N and f = 0 on Ω\Ω_N+1.

Using the fact that f = 0 on Ω^c, it follows from the Cauchy-Schwarz inequality that Z

f^qdµ 2

6µ(Ω) Z

f^2qdµ ,

from which we get

Var_µ(f^q)> 1 2

Z

f^2qdµ . By the co-area formula, we obtain

Z

f^2qdµ= Z ∞

0

µ({f >t})d(t^2q)>

N−1

X

k=−N

Z τ_k τk+1

µ({f >τ_k})d(t^2q) =

N−1

X

k=−N

µ({f >τ_k})

τ_k^2q−τ_k+1^2q .

From 2q>1, we get

τ_k^2q−τ_k+1^2q

>(τ_k−τ_k+1)^2q, and

Var_µ(f^q)> 1 2

N−1

X

k=−N

µ(Ω_k) (τ_k−τ_k+1)^2q .

Using theL^q-Poincar´e inequality we get 1

2

N−1

X

k=−N

µ(Ω_k) (τ_k−τ_k+1)^2q

!1/q

6C_P Z

|∇f|²dν .

On the other hand, with the convention τ_N+1 = 0, we have Z

|∇f|²dν =

N

X

k=−N

(τ_k−τ_k+1)² Z

Ωk+1\Ωk

|∇f_k|²dν .

We may now take the infimum over all functions f_k and obtain 1

2

N−1

X

k=−N

µ(Ω_k) (τ_k−τ_k+1)^2q

!1/q

6C_P

N

X

k=−N

(τ_k−τ_k+1)² Cap_ν(Ω_k,Ω_k+1).

Next consider an appropriate choice of (τ_k)^N_k=−N : fork∈ {−N, . . . N}, let

τ_k=

N

X

j=k

µ(Ωj) Cap_ν(Ω_j,Ω_j+1)

_{2 (1−q)}¹ .

We observe that τ_k−τ_k+1=

µ(Ωk) Cap_ν(Ω_k,Ω_k+1)

1/(2(1−q))

and ( _N−1

X

k=−N

µ(Ω_k)^1/(1−q) Cap_ν(Ω_k,Ω_k+1)^q/(1−q)

)(1−q)/q

62^1/qC_P(1 +RN)

with RN := ^{µ(ΩN)1/(1−q)}

Cap_ν(ΩN,ΩN+1)^q/(1−q)

nPN−1

k=−N µ(Ωk)^1/(1−q) Cap_ν(Ωk,Ωk+1)^q/(1−q)

o−1

. By taking the limit as N goes to infinity, we obtainβ_P62^1/qC_P.

Proof of(ii). Letf be a smooth non-negative function onM and takeq∈(0,1]. For alla>0, Var_µ(f^q)6

Z

(f^q−a^q)²dµ6 Z

|f −a|^2qdµ .

Witha:=m(f), a median off with respect toµ, defineF+= (f−a)+and F− = (f−a)−= F₊−(f −a), so that

Varµ(f^q)6 Z

(f^q−a^q)²dµ6 Z

F₊^2qdµ+ Z

F₋^2qdµ .

We recall that m = m(f) is a median of f with respect to the measure µ if and only if µ({f >m})>1/2 and µ({f 6m})>1/2. The computation of the termR

F₋^2qdµ is exactly the same as the one of R

F₊^2qdµ, so we shall only detail one of them. Let us fix ρ ∈ (0,1), note Ω_k:=

F+>ρ^k for any k∈Z, and use again the co-area formula:

Z

F₊^2qdµ= Z +∞

0

µ({F+ >t})d(t^2q) =X

k∈Z

Z ρ^k ρ^k+1

µ({F+ >t})d(t^2q)6 1−ρ^2q ρ^2q

X

k∈Z

µ(Ω_k)ρ^2kq.

By H¨older’s inequality with parameters (1/(1−q),1/q) one gets Z

F₊^2qdµ 6 1−ρ^2q ρ^2q

X

k∈Z

µ(Ω_k)^1/(1−q) Cap_ν(Ω_k,Ω_k+1)^q/(1−q)

!1−q

X

k∈Z

ρ^2kCap_ν(Ω_k,Ω_k+1)

!q

6 1−ρ^2q

ρ^2q β^q_P X

k∈Z

ρ^2kCap_ν(Ω_k,Ω_k+1)

!q

.

Fork∈Z, defineg_k:= minn

1, ^F_ρ⁺_k−ρ^−ρ_k+1^k+1

+

o. Then we haveI_Ω

k 6g_k6I_Ω

k+1, Cap_ν(Ω_k,Ω_k+1)6

Z

Ωk+1\Ωk

|∇g_k|²dν = 1 ρ^2k(1−ρ)²

Z

Ωk+1\Ωk

|∇F+|²dν . Hence

Z

F₊^2qdµ6 1−ρ^2q ρ^2q(1−ρ)^2qβ^q_P

Z

|∇F₊|²dν q

. The same inequality holds forF₋:

Z

F₋^2qdµ6 1−ρ^2q ρ^2q(1−ρ)^2qβ^q_P

Z

|∇F₋|²dν q

.

Using the inequality a^q+b^q 62^1−q(a+b)^q for any a,b>0, ones gets (Var_µ(f^q))^1/q 6κ_Pβ_P

Z

|∇f|²dν

withκ_P := 2^(1−q)/q min_ρ∈(0,1) (^1−ρ2q)^1/q

ρ²(1−ρ)² . ⊲ 2.2 L^q-logarithmic Sobolev inequalities

Definition 2.4 Let µ and ν be respectively a probability measure and a positive measure on M and assume that q ∈(0,1]. We shall say that (µ, ν) satisfies a L^q-logarithmic Sobolev inequality with constant C_LS if and only if, for any non-negative functionf ∈ C¹(M),

Entµ f^2q1/q

:=

Z

f^2q logf^2q R f^2qdµdµ

1/q

6CLS

Z

|∇f|²dν .

It is well known thatEntµ f²

>Varµ(f) for any non-negative functionf, for any probability measure µ. Hence, for any q ∈ (0,1], any L^q-logarithmic Sobolev inequality results in a L^q- Poincar´e inequality with corresponding measures.

Letq ∈(0,1) and define

β_LS = sup





 X

k∈Z

h

µ(Ω_k) log

1 + _µ(Ω^e2

k)

i1/(1−q)

[Cap_ν(Ω_k,Ω_k+1)]^q/(1−q)







(1−q)/q

where the supremum is taken over all Ω⊂M withµ(Ω)≤1/2 and all sequence (Ω_k)_k∈Z such that, for allk∈Z, Ω_k⊂Ω_k+1⊂Ω.

Theorem 2.5 Letµandνbe respectively a probability measure and a positive measure onM. If q ∈ (0,1) and β_LS < +∞, then (µ, ν) satisfies a L^q-logarithmic Sobolev inequality with constant CLS6κLSβLS, where κLS depends only on q.

This theorem is the counterpart for theL^q-logarithmic Sobolev inequality of Theorem 2.3, (ii).

As for Theorem 2.3, (i), related results will be stated in Corollary 3.8.

Proof ⊳ Letf be a smooth function onM,m=m(f) a median off with respect toµ, and Ω₊:={|f|> m}, Ω₋:={|f|< m}. As in [BR03], we can write the dual formulation

Ent_µ f^2q 6sup

Z

(|f|^q−m^q)²₊h dµ : h>0, Z

e^hdµ6e²+ 1

+ sup Z

(|f|^q−m^q)²₋h dµ : h>0, Z

e^hdµ6e²+ 1

. (2) Such an inequality follows from Rothaus’ estimate, [Rot85],

Entµ g²

6Entµ (g−a)²

+ 2µ (g−a)² for any a∈R, and the fact that, according to Lemma 5 in [BR03],

Ent_µ (g−a)²₊

+ 2µ((g−a)²₊)6sup Z

(g−a)²₊h dµ : h>0, Z

e^hdµ6e²+ 1

. Estimates for the positive and the negative part are exactly the same, so we will give details only forF₊:= (|f| −m)₊. Using the fact that (t^q−1)² <(t−1)^2qfor anyt >1, forq ∈(0,1), we get

Z

(|f|^q−m^q)²₊h dµ6 Z

F₊^2qh dµ . Letρ∈(0,1).

Z

F₊^2qh dµ= Z +∞

0

Z

F+>t

h dµ d(t^2q) =X

k∈Z

Z ρ^k ρ^k+1

Z

F+>t

h dµ d(t^2q)

6X

k∈Z

ρ^2qk−ρ^2q(k+1)Z

F+>ρ^k+1

h dµ= 1−ρ^2q ρ^2q

X

k∈Z

ρ^2qk Z

F+>ρ^k

h dµ .

Using Lemma 6 of [BR03], which asserts that Z

F+>ρ^k

h dµ6µ(Ω_k) log

1 + e² µ(Ω_k)

where Ω_k:=

F+> ρ^k , we obtain sup

Z

F₊^2qh dµ : h>0, Z

e^hdµ6e²+ 1

6 1−ρ^2q ρ^2q

X

k∈Z

ρ^2qkµ(Ω_k) log

1 + e² µ(Ω_k)

.

By H¨older’s inequality, it follows that Ent_µ

F₊^2q

6 1−ρ^2q ρ^2q

X

k∈Z

ρ^2qkµ(Ω_k) log

1 + e² µ(Ω_k)

61−ρ^2q ρ^2q





 X

k∈Z

h

µ(Ω_k) log

1 + _µ(Ω^e2

k)

i1/(1−q)

[Cap_ν(Ω_k,Ω_k+1)]^q/(1−q)







1−q

X

k∈Z

ρ^2kCap_ν(Ω_k,Ω_k+1)

!q

6 1−ρ^2q

ρ^2q β^q_LS X

k∈Z

ρ^2kCap_ν(Ω_k,Ω_k+1)

!q

6 1−ρ^2q ρ^2q β^q_LS

Z

|∇F₊|²dµ q

. The same computation shows that

Ent_µ F₋^2q

6 1−ρ^2q ρ^2q β^q_LS

Z

|∇F₋|²dµ q

. Summing both contributions in Inequality (2) completes the proof with

κ_LS = 2^1−qq 1−ρ^2q1/q

ρ⁻².

⊲

3 Weak inequalities and explicit criteria

The goal of this section is to provide tractable criteria to establish L^q-Poincar´e and the L^q-logarithmic Sobolev inequalities. The strategy here is to adapt results which have been obtained for weak Poincar´e inequalities by Barthe, Cattiaux and Roberto in [BCR05]. Two important results stated in this paper are extended to measuresµandνwhich are not supposed to be absolutely continuous with respect to the volume measure, and given with proofs in Section 5.

3.1 L^q Poincar´e and weak Poincar´e inequalities

Even if the constantsβP andβLSprovide an estimate of the best constant of theL^q-Poincar´e and theL^q-logarithmic Sobolev inequalities, their expressions in terms of suprema taken over infinite sequences of sets are a priori difficult to use. In this section, we look for simpler criteria and establish upper and lower bounds on the constants.

The first idea is relate theL^q-Poincar´e inequality and the weak Poincar´e inequality introduced by R¨ockner and the fourth author in [RW01]. Let us define the oscillation of a bounded functionf byOsc_µ(f) := supess_µf−infess_µf. Ifµis absolutely continuous with respect to the volume measure andf is continuous, we can therefore define such a quantity as (sup ˜f−inf ˜f) where ˜f is the restriction off to the support ofµ. Our definition slightly differs from the one of [RW01], which is based on supess_µ|f−R

f dµ|.