The key to the proof lies in using anL2 estimate for the densityρ, and us- ing the smallness of the initial velocity gradient, to obtain uniqueness for the density

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp)

EXISTENCE AND UNIQUENESS OF GLOBAL SOLUTIONS TO A MODEL FOR THE FLOW OF AN INCOMPRESSIBLE,

BAROTROPIC FLUID WITH CAPILLARY EFFECTS

DIANE L. DENNY

Abstract. We study the initial-value problem for a system of nonlinear equa- tions that models the flow of an inviscid, incompressible, barotropic fluid with capillary stress effects. We prove the global-in-time existence of a unique, clas- sical solution to this system of equations, with a small initial velocity gradient.

The key to the proof lies in using anL² estimate for the densityρ, and us- ing the smallness of the initial velocity gradient, to obtain uniqueness for the density.

1. Introduction

In this paper, we consider equations which arise from a model of the multi- dimensional flow of an incompressible, barotropic fluid with capillary stresses.

When viscosity is neglected, these equations reduce to the following system, written in terms of the densityρ, the pressurep, and velocityv:

Dv

Dt +ρ⁻¹∇p=c∇∆ρ, (1.1)

∇ ·v= 0. (1.2)

Here c is a coefficient of capillarity which is a small, positive constant, and the material derivative D/Dt = ∂/∂t+v· ∇. The termc∇∆ρ arises from capillary stresses, as described in the theory of Korteweg-type materials developed by Dunn and Serrin [4]. The fluid’s thermodynamic state is determined by the density ρ.

The pressure p is determined from the density by an equation of statep = ˆp(ρ).

In related work, the existence of a solution to a similar system of equations for the case of viscous fluid flow, which also includes a hyperbolic equation for density and a parabolic equation for temperature, has been proven by Hattori and Li [7, 8], and by Bresch, Desjardins, and Lin [2]. Anderson, McFadden and Wheeler [1] have given a review of related theories and applications to diffuse-interface modelling.

In this model, it will be shown that ∂ρ/∂t, ∇ρ, and∇v are small, for suitable initial data. Although the conservation of mass equation is only approximately satisfied, the model equations (1.1), (1.2) might be useful as an approximation in

2000Mathematics Subject Classification. 35A05.

Key words and phrases. Existence; uniqueness; capillary; incompressible;

inviscid fluid.

c

2007 Texas State University - San Marcos.

Submitted August 10, 2006. Published March 6, 2007.

1

the case of almost constant density, and nearly incompressible fluid flow with a small velocity gradient. We write equation (1.1) equivalently as

Dv

Dt +ρ⁻¹pˆ⁰(ρ)∇ρ=c∇∆ρ (1.3) The purpose of this paper is to prove the existence of a unique, global-in-time, classical solutionv,ρto equations (1.2), (1.3) with suitable initial velocity datav0∈ H^s, under periodic boundary conditions. That is, we choose for our domain the N- dimensional torusT^N, whereN= 2 or N= 3. The proof of the existence theorem is based on the method of successive approximations, in which an iteration scheme, based on solving a linearized version of the equations, is designed and convergence of the sequence of approximating solutions to a unique solution satisfying the nonlinear equations is sought. The framework of the proof follows one used, for example, by A. Majda for proving the existence of a solution to a system of conservation laws [10]. Embid [5] also uses the same general framework to prove the existence of a solution to equations for zero Mach number combustion. Under this framework, the convergence proof is presented in two steps. In the first step, we prove uniform boundedness of the approximating sequence of solutions in a high Sobolev norm.

The second step is to prove contraction of the sequence in a low Sobolev norm.

Standard compactness arguments will be used to finish the proof. The key to the proof lies in using anL² estimate for the densityρ, and using the smallness of the initial velocity gradient, to obtain uniqueness for the density.

2. A priori estimates

The main tools utilized in the existence proof are a priori estimates. We will work with the Sobolev spaceH^s(Ω) (wheres≥0 is an integer) of real-valued functions in L²(Ω) whose distribution derivatives up to ordersare inL²(Ω), with norm given by kfk²_s=P

|α|≤s

R

Ω|D^αf|²dxand inner product (f, g)_s=P

|α|≤s

R

Ω(D^αf)·(D^αg)dx.

Here, we adopt the standard multi-index notation. For convenience, we will denote derivatives byf_α=D^αf. We will let Df denote the gradient off. Also, we will denote theL² inner product by (f, g) =R

Ωf·g dx. We will use standard function spaces. L^∞([0, T], H^s) is the space of bounded measurable functions from [0, T] into H^s(Ω), with the normkfk²_s,T = ess sup_0≤t≤Tkf(t)k²_s. C([0, T], H^s) is the space of continuous functions from [0, T] intoH^s(Ω). The following technical lemmas will be needed for the proof of the existence of a classical solution to the initial-value problem for the system (1.2), (1.3).

Lemma 2.1 (Standard Calculus Inequalities).

(a) If f ∈H^s1(Ω), g∈H^s2(Ω) and s3 = min{s1, s2, s1+s2−s0} ≥0, where s₀= [^N₂] + 1, thenf g∈H^s3(Ω), and kf gks3 ≤Ckfks1kgks2. We note that s₀= 2forN = 2 orN = 3.

(b) If f ∈ H^s(Ω), g ∈ H^s−1(Ω)∩L^∞(Ω), Df ∈ L^∞(Ω), and |α| ≤ s, then kD^α(f g)−f D^αgk₀≤C(|Df|_L^∞kgk_s−1+|g|_L^∞kDfk_s−1).

In (a) the constant C depends on s1, s2, and Ω, while in (b) the constant C depends onsand Ω. These inequalities are well known. Proofs may be found, for example, in [9, 11].

Lemma 2.2 (Low-Norm Commutator Estimate). If Df ∈H^r1(Ω),g∈H^r−1(Ω), wherer₁= max{r−1, s0},s₀= [^N₂]+1, then for anyr≥1,f, gsatisfy the estimate

kD^α(f g)−f D^αgk0≤CkDfkr₁kgkr−1, wherer=|α|, and the constant C depends onr,Ω.

Proof. The proof is based on the Sobolev calculus inequalities from Lemma 2.1. We consider separately the casesr−1< s0andr−1≥s0, wherer≥1. Ifr−1< s0, we expand the termD^α(f g) using the Leibniz rule and then apply inequality (a) from Lemma 2.1 to obtain the desired estimate. If r−1≥s0, we apply the inequality (b) from Lemma 2.1 and the Sobolev inequality |h|L^∞ ≤Ckhks0 fors₀= [^N₂] + 1, to obtain the estimate for this case. Combining these two results then completes

the proof.

Lemma 2.3. If u,v,a,f, andρare sufficiently smooth in Du/Dt=−a∇ρ+c∇∆ρ+f,

∇ ·u= 0,

where u(x,0) = u0(x), ∇ ·u0 = 0, Ω = T^N, where N = 2,3, and where c is a positive constant, D/Dt=∂/∂t+v·∇, and∇ ·v= 0, then for anyr≥1,Duand usatisfy the estimates

kDuk²_r−1≤Ce^2t(1+te^2te^β(t)kDvk²_r1,T)(kDu0k²_r−1+ Z t

0

(kfk²_r+kDak²_r1k∇ρk²_r−1)dτ) and

kuk²_r≤e^2tku0k²₀+Ce^2t(1 +te^2te^β(t)kDvk²_r1,T)(kDu0k²_r−1 +

Z t

0

(kfk²_r+kDak²_r1k∇ρk²_r−1)dτ)

where β(t) =te^2tkDvk²_r1,T. Here r1 = max{r−1, s0}, and s0 = [^N₂] + 1 = 2 for N = 2,3. The constant C depends onr,Ω.

Proof. First, we obtain an L² estimate for u. Let ¯ρ = ρ− _|Ω|¹ R

Ωρdx. We then compute

1 2

d

dtkuk²₀= (ut,u)

=−(v· ∇u,u)−(a∇ρ,u) +c(∇∆ρ,u) + (f,u)

= 1

2(u∇ ·v,u) + (a¯ρ,∇ ·u) + ( ¯ρ∇a,u)−c(∆ρ,∇ ·u) + (f,u)

≤ |Da|_L^∞kuk₀kρk¯ ₀+kfk₀kuk₀

≤ kuk²₀+1

2|Da|²_L∞kρk¯ ²₀+1 2kfk²₀

(2.1)

where we used the facts that ∇ ·u = 0, and ∇ ·v = 0. And we used Cauchy’s inequality. After applying Gronwall’s inequality, we obtain the estimate

kuk²₀≤e^2tku0k²₀+e^2t Z t

0

C(|Da|²_L∞kρk¯ ²₀+kfk²₀)dτ, (2.2) Next, we obtain an estimate forkDuk²_r−1. After applying the operatorD^γ+αto the equation foru, where 0≤ |α| ≤r−1 and|γ|= 1, we obtain

Duγ+α

Dt =−a∇ρ_γ+α+c∇∆ρ_γ+α+F_γ+α (2.3)

whereFγ+α=fγ+α−[(v· ∇u)γ+α−v· ∇uγ+α]−[(a∇ρ)γ+α−a∇ργ+α]. For (2.3), estimate (2.1) becomes

1 2

d

dtku_γ+αk²₀≤ ku_γ+αk²₀+1

2|Da|²_L∞k¯ρ_γ+αk²₀+1

2kF_γ+αk²₀ (2.4) Next, we estimatekFγ+αk²₀. Using the commutator estimate from Lemma 2.2, we obtain

kF_γ+αk²₀≤Ckf_γ+αk²₀+Ck(v· ∇u)_γ+α−v· ∇u_γ+αk²₀ +Ck(a∇ρ)_γ+α−a∇ρ_γ+αk²₀

≤Ckfk²_k+CkDvk²_k

1kDuk²_k−1+CkDak²_k

1k∇ρk²_k−1

(2.5)

where k = |γ+α|, k1 = max{k−1, s0}, and s0 = [^N₂] + 1. Here, we used the triangle inequality and Cauchy’s inequality. Substituting estimate (2.5) into (2.4) yields

1 2

d

dtkuγ+αk²₀≤ kuγ+αk²₀+1

2|Da|²_L∞kρ¯γ+αk²₀+Ckfk²_k +CkDvk²_k1kDuk²_k−1+CkDak²_k1k∇ρk²_k−1 Applying Gronwall’s inequality yields the following:

kuγ+αk²₀≤e^2tk(u0)γ+αk²₀+Ce^2t Z t

0

|Da|²_L∞kρ¯γ+αk²₀+kfk²_k +kDvk²_k1kDuk²_k−1+kDak²_k1k∇ρk²_k−1

dτ

≤Ce^2tkDu0k²_k−1+Ce^2t Z t

0

|Da|²_L∞kρk¯ ²_k+kfk²_k +kDvk²_k1kDuk²_k−1+kDak²_k1k∇ρk²_k−1

dτ,

(2.6)

where|γ|= 1, and |γ+α|=k≥1, andk1= max{k−1, s0}, withs0= [^N₂] + 1.

After adding the above inequality (2.6) over allγ, where|γ|= 1, and then adding over allα, where 0≤ |α| ≤r−1, we obtain the estimate

kDuk²_r−1≤Ce^2tkDu0k²_r−1+Ce^2t Z t

0

|Da|²_L∞kρk¯ ²_r+kfk²_r +kDvk²_r1,TkDuk²_r−1+kDak²_r1k∇ρk²_r−1

dτ,

(2.7)

where r1 = max{r−1, s0}, and where s0 = [^N₂] + 1 = 2 for N = 2,3. Here, we used the fact thatP

0≤|α|≤r−1

P

|γ|=1kuγ+αk²₀=P

0≤|α|≤r−1

PN i=1

R

Ω|^∂u_∂x^α

i|²dx= P

0≤|α|≤r−1kDuαk²₀=kDuk²_r−1. After applying Gronwall’s inequality to (2.7), we get

kDuk²_r−1≤Ce^2t(1 +te^2te^β(t)kDvk²_r1,T)(kDu0k²_r−1 +

Z t

0

(|Da|²_L∞kρk¯ ²_r+kfk²_r+kDak²_r1k∇ρk²_r−1)dτ),

(2.8)

withβ(t) =te^2tkDvk²_r1,T. Adding the estimates (2.2), (2.8), we obtain kuk²_r≤(kuk²₀+CkDuk²_r−1)

≤e^2tku0k²₀+Ce^2t(1 +te^2te^β(t)kDvk²_r1,T)(kDu0k²_r−1 +

Z t

0

(|Da|²_L∞kρk¯ ²_r+kfk²_r+kDak²_r

1k∇ρk²_r−1)dτ)

≤e^2tku0k²₀+Ce^2t(1 +te^2te^β(t)kDvk²_r1,T)(kDu0k²_r−1 +

Z t

0

(kfk²_r+kDak²_r1k∇ρk²_r−1)dτ)

wherer1= max{r−1, s0}, and wheres0= [^N₂] + 1 = 2 forN = 2,3. Here, we used Poincar´e’s inequality to estimatekρk¯ ²₀≤Ck∇ρk²₀, andkρk¯ ²_r≤C(k∇ρk²_r−1+kρk¯ ²₀)≤ Ck∇ρk²_r−1. We also used Sobolev’s lemma |h|L^∞ ≤Ckhks₀ where s0 = [^N₂] + 1.

Using the estimate|Da|²_L∞kρk¯ ²_r≤CkDak²_r1k∇ρk²_r−1in the right-hand side of (2.8)

completes the proof.

Lemma 2.4. Let u, w be C¹ functions on a bounded, open, connected, convex domainΩ. And letu(x0) =w(x0)at a single, fixed pointx0∈Ω. Thenu−wand usatisfy the estimates

ku−wk²₀≤Ck∇(u−w)k²₂, kuk²₀≤C₀kwk²₀+C₀k∇wk²₂+C₀k∇uk²₂ Here C,C0 are constants which depend only on Ω.

Proof. First, we obtain an estimate for ku−wk²₀. From the mean value theorem, and sinceu−w is sufficiently smooth on the convex domain Ω, we have

(u−w)(x) = (u−w)(x₀) +∇(u−w)(x_∗)·(x−x₀)

wherex_∗ is a point on the line segment joining the pointsx0∈Ω andx∈Ω.

Since we are given thatu(x0) =w(x0), at a single fixed pointx0∈Ω, it follows that

(u−w)(x) =∇(u−w)(x_∗)·(x−x0) Taking the absolute value of both sides yields

|(u−w)(x)|=|∇(u−w)(x_∗)·(x−x₀)|

≤ |∇(u−w)|L^∞|x−x₀|

≤C|∇(u−w)|L^∞

HereCdepends only on Ω. Squaring|(u−w)(x)|and integrating over Ω, and using the above inequality, yields

Z

Ω

|(u−w)(x)|²dx≤C Z

Ω

|∇(u−w)|²_L∞dx≤Ck∇(u−w)k²₂

where we used Sobolev’s inequality |h|L^∞ ≤Ckhks0, where s₀ = [^N₂] + 1 = 2 for N = 2,3 andC depends on Ω.

Next, we obtain an estimate for kuk²₀. From using the triangle inequality and Cauchy’s inequality, and from using the previous estimate forku−wk²₀, we obtain

kuk²₀≤Ckwk²₀+Cku−wk²₀

≤Ckwk²₀+Ck∇(u−w)k²₂

≤C₀kwk²₀+C₀k∇wk²₂+C₀k∇uk²₂

HereC0 is a constant which depends only on Ω.

Lemma 2.5. Ifg is a sufficiently smooth function on the domainΩ =T^N, theng satisfies the estimate k∇gk²_r≤Ck∆gk²_r−1 where r≥1andC depends onr.

Proof. First, we integrate−∆g by parts with the function ¯g=g−_|Ω|¹ R

Ωg dxover Ω =T^N, to obtain

(∇g,∇g) =−(∆g,¯g)

≤C()k∆gk²₀+k¯gk²₀

≤C()k∆gk²₀+Ck∇gk²₀

where we used Cauchy’s inequality with . We also used Poincar´e’s inequality to estimate k¯gk²₀ ≤ Ck∇gk²₀. Then D^α is applied, yielding −∆gα, which is then integrated by parts with the functiongαover Ω =T^N, to obtain

(∇gα,∇gα) =−(∆gα, gα)

= (∆g_α−γ, g_α+γ)

≤C()k∆g_α−γk²₀+kg_α+γk²₀

≤C()k∆gk²_k−1+Ck∇gk²_k

where|γ|= 1, and|α|=k≥1. Here, we used Cauchy’s inequality with. Adding these two inequalities, for |α|=k≤r, and moving the terms containing to the

left-hand side, completes the proof.

Lemma 2.6. If u,v,a,f, andρare sufficiently smooth in the equation

∆²ρ=1

c∇ ·(a∇ρ) +1

c∇ ·(v· ∇u)−1

c∇ ·f (2.9)

where c is a positive constant, ∇ ·u = 0, ∇ ·v = 0, a(x, t) ≥ c₁, with c₁ > 0, and where Ω =T^N, N = 2,3, and r ≥ 1, we obtain the following estimates for k∆ρk²₀+k∇ρk²₀ and fork∇ρk²_r+1

k∆ρk²₀+k∇ρk²₀≤C|Dv|²_L∞kDuk²₀+Ckfk²₀, k∇ρk²_r+1≤C(k∆ρk²_r+k∇ρk²_r)

≤CkDak²_r

1k∇ρk²_r−1+Ckfk²_r−1+CkDvk²_r

2+1kDuk²_r−1

wherer₁= max{r−1, s₀},r₂= max{r−2, s₀}, withs₀= [^N₂] + 1, andN = 2,3.

Proof. First, we obtain anL²estimate. Integrating equation (2.9) by parts with ¯ρ, where ¯ρ=ρ−_|Ω|¹ R

Ωρdx, yields

(∆ρ,∆ρ) = (∆²ρ,ρ) =¯ 1

c(∇ ·(a∇ρ),ρ) +¯ 1

c(∇ ·(v· ∇u),ρ)¯ −1

c(∇ ·f,ρ)¯

=−1

c(a∇ρ,∇ρ) +1

c(∇v^T :∇u,ρ) +¯ 1 c(f,∇ρ)

≤ −c1

ck∇ρk²₀+C()|Dv|²_L∞kDuk²₀+kρk¯ ²₀+C()kfk²₀+k∇ρk²₀

≤ −c1

ck∇ρk²₀+C()|Dv|²_L∞kDuk²₀+k∇ρk²₀+C()kfk²₀+k∇ρk²₀ where we used Cauchy’s inequality with, and where we used the fact thata(x, t)≥ c1 wherec1 >0. We also used Poincar´e’s inequality to estimatekρk¯ ²₀ ≤Ck∇ρk²₀. And we used the fact that ∇ ·(v· ∇u) =∇v^T :∇u, because ∇ ·u= 0. Moving thek∇ρk²₀ terms to the left-hand side yields the estimate

k∆ρk²₀+k∇ρk²₀≤C|Dv|²_L∞kDuk²₀+Ckfk²₀ (2.10) Next, after applyingD^αto the equation (2.9), we obtain the equation:

∆²ρ_α= 1

c∇ ·(a∇ρα) +1

c∇ ·(v· ∇u)α+F_α (2.11) whereFα=−¹_c∇ ·fα+¹_c[∇ ·(a∇ρ)α− ∇ ·(a∇ρα)]. Integrating equation (2.11) by parts withρα, when|α| ≥1, yields

(∆ρα,∆ρα) = (∆²ρα, ρα)

=1

c(∇ ·(a∇ρα), ρα) +1

c(∇ ·(v· ∇u)α, ρα) + (Fα, ρα)

=−1

c(a∇ρα,∇ρα) +1

c((∇v^T :∇u)α, ρα) + (Fα, ρα)

=−1

c(a∇ρα,∇ρα)−1

c((∇v^T :∇u)α−γ, ρα+γ) + (Fα, ρα)

≤ −c₁

c (∇ρ_α,∇ρ_α) +Ck(∇v^T :∇u)_α−γk₀kρ_α+γk₀+|(F_α, ρ_α)|

≤ −c1

c (∇ρα,∇ρα) +C()k(∇v^T :·∇u)_α−γk²₀+kρα+γk²₀+|(Fα, ρα)|

(2.12) where|γ|= 1. Here, we used Cauchy’s inequality with, and we used the fact that a(x, t)≥c1 where c1 >0. Next, we estimate the terms on the right-hand side of the above inequality. First, we estimate k(∇v^T : ∇u)_α−γk²₀. When |α| = 1, we chooseγ=αand obtain

k(∇v^T :∇u)α−γk²₀=k(∇v^T :∇u)k²₀≤C|Dv|²_L∞kDuk²₀ (2.13) Next, we use the triangle inequality and the commutator estimate from Lemma 2.2 to estimatek(∇v^T :∇u)α−γk²₀, when|α|>|γ|= 1, obtaining

k(∇v^T :∇u)α−γk²₀≤C(k∇v^T :∇uα−γk²₀+k(∇v^T :∇u)α−γ− ∇v^T :∇uα−γk²₀)

≤C|Dv|²_L∞kDuα−γk²₀+CkD²vk²_k2kDuk²_k−2

≤CkDvk²_k2+1kDuk²_k−1,

(2.14) Here |γ|= 1, k=|α|, andk₂ = max{k−2, s₀}, withs₀= [^N₂] + 1. Here, we also used the Sobolev inequality|h|L^∞ ≤Ckhks0.

Next, we estimate the following term from the right-hand side of (2.12), obtaining kρα+γk²₀≤Ck∇ρk²_k (2.15) where|γ|= 1 and |α|=k.

Next, we use the commutator estimate from Lemma 2.2 to estimate |(Fα, ρα)|

from the right-hand side of (2.12), obtaining

|(F_α, ρ_α)| ≤ |1

c(∇ ·f_α, ρ_α)|+|1

c([∇ ·(a∇ρ)_α− ∇ ·(a∇ρ_α)], ρ_α)|

=|1

c(f_α−γ,∇ρ_α+γ)|+|1

c([(a∇ρ)_α−a∇ρ_α],∇ρ_α)|

≤Ckfkk−1k∇ρkk+1+CkDakk₁k∇ρkk−1k∇ρkk

≤C()kfk²_k−1+k∇ρk²_k+1+C()kDak²_k1k∇ρk²_k−1+k∇ρk²_k

(2.16)

where |γ| = 1, k =|α|, and k1 = max{k−1, s0}, with s0 = [^N₂] + 1. Again, we used Cauchy’s inequality with. Substituting (2.13)-(2.16) into (2.12), and adding (2.12) over|α|=k≤r, including theL² estimate (2.10), we obtain for r≥1 the estimate

k∆ρk²_r+k∇ρk²_r≤CkDak²_r1k∇ρk²_r−1+Ckfk²_r−1+CkDvk²_r2+1kDuk²_r−1

+Ck∇ρk²_r+1+Ck∇ρk²_r, (2.17) wherer₁= max{r−1, s₀},r₂= max{r−2, s₀}, with s₀= [^N₂] + 1. Here, we also used the Sobolev inequality|h|L^∞ ≤Ckhks0.

From Lemma 2.5, we have k∇ρk²_r+1 ≤Ck∆ρk²_r. After substituting this esti- mate into the right-hand side of (2.17), we obtain the estimate

k∆ρk²_r+k∇ρk²_r≤CkDak²_r

1k∇ρk²_r−1+Ckfk²_r−1+CkDvk²_r

2+1kDuk²_r−1, (2.18) where we have moved the terms Ck∆ρk²_r and Ck∇ρk²_r to the left-hand side.

Finally, using the estimate fork∇ρk²_r+1from Lemma 2.5, we obtain from (2.18) the estimate

k∇ρk²_r+1≤C(k∆ρk²_r+k∇ρk²_r)

≤CkDak²_r1k∇ρk²_r−1+Ckfk²_r−1+CkDvk²_r2+1kDuk²_r−1

Lemma 2.7. If u,v,a,f, andρare sufficiently smooth in

∆²ρ=1

c∇ ·(a∇ρ) +1

c∇ ·(v· ∇u)−1

c∇ ·f (2.19)

wherec is a positive constant, ∇ ·u= 0,∇ ·v= 0,a(x, t)≥c1, with c1>0, and Ω =T^N,N = 2,3, then forr≥1 ,ρsatisfies the following estimate

k∇ρk²_r+1≤C k∆ρk²_r+k∇ρk²_r

≤Ch 1 +

r

X

j=1

kDak^2j_r1i

(kfk²_r−1+kDvk²_r2+1kDuk²_r−1)

wherer₁= max{r−1, s₀},r₂= max{r−2, s₀}, withs₀= [^N₂] + 1, and C depends onr.

Proof. From Lemma 2.6 applied to equation (2.19), we have the estimate

k∆ρk²_s+k∇ρk²_s≤CkDak²_s1k∇ρk²_s−1+Ckfk²_s−1+CkDvk²_s2+1kDuk²_s−1 (2.20) wheres≥1, and wheres1= max{s−1, s0},s2= max{s−2, s0}, withs0= [^N₂] + 1.

Lettings=r in the estimate (2.20) yields

k∆ρk²_r+k∇ρk²_r≤CkDak²_r1k∇ρk²_r−1+Ckfk²_r−1+CkDvk²_r2+1kDuk²_r−1 (2.21) Applying the estimate (2.20), lettings=r−1, to the termCkDak²_r1k∇ρk²_r−1which appears on the right-hand side of (2.21) yields

k∆ρk²_r+k∇ρk²_r≤CkDak²_r

1

hkDak²_r

2k∇ρk²_r−2+kfk²_r−2+kDvk²_r

3+1kDuk²_r−2i +Ckfk²_r−1+CkDvk²_r2+1kDuk²_r−1

≤CkDak⁴_r1k∇ρk²_r−2+CkDak²_r1

kfk²_r−2+kDvk²_r2+1kDuk²_r−2 +Ckfk²_r−1+CkDvk²_r2+1kDuk²_r−1

(2.22) where r₁ = max{r−1, s₀}, r₂ = max{r−2, s₀}, and r₃ = max{r−3, s₀}, r₃ ≤ r2 ≤r1, withs0 = [^N₂] + 1 = 2 for N = 2,3. Similarly, by repeatedly applying the estimate (2.20), letting s = r−j, to the term CkDak^2j_r1k∇ρk²_r−j, for j = 2,3, . . . , r−1, which will appear on the right-hand side of (2.22) yields

k∆ρk²_r+k∇ρk²_r≤C

r−1

X

j=1

kDak^2j_r1

kfk²_r−1−j+kDvk²_r2+1kDuk²_r−1−j +CkDak^2r_r1k∇ρk²₀+Ckfk²_r−1+CkDvk²_r2+1kDuk²_r−1

≤Ch 1 +

r−1

X

j=1

kDak^2j_r1i

kfk²_r−1+kDvk²_r2+1kDuk²_r−1 +CkDak^2r_r1k∇ρk²₀

(2.23)

Substituting the estimate for k∇ρk²₀ from Lemma 2.6 into the right-hand side of (2.23) yields

k∆ρk²_r+k∇ρk²_r≤C 1 +

r−1

X

j=1

kDak^2j_r1i

kfk²_r−1+kDvk²_r2+1kDuk²_r−1 +CkDak^2r_r1

kfk²₀+|Dv|²_L∞kDuk²₀

≤Ch 1 +

r

X

j=1

kDak^2j_r1i

kfk²_r−1+kDvk²_r2+1kDuk²_r−1

This completes the proof.

3. Existence theorem

In this section, we prove the existence of a unique classical solution to the initial- value problem for equations (1.2), (1.3), on any given time interval 0≤t≤T, with periodic boundary conditions, for sufficiently small initial velocity gradient.

Theorem 3.1. Supposes > ^N₂ + 3 andΩ = T^N, N = 2,3. For any given time interval 0 ≤ t ≤ T, equations (1.2), (1.3) have a unique classical solution ρ, v, for initial data v0(x) ∈ H^s(Ω), ∇ ·v0 = 0, and for given data ρ⁰(x, t) and x0, provided Dv₀ is sufficiently small. Here ρ⁰(x, t) is a given positive function with sufficiently small gradient∇ρ⁰, andx₀ is a point in the domain Ω. The regularity of the solution is

ρ∈C([0, T], C⁵)∩L^∞([0, T], H^s+2), v∈C([0, T], C³)∩L^∞([0, T], H^s),

∂v

∂t ∈C([0, T], C²)∩L^∞([0, T], H^s−1).

andρ(x, t)≥c₁, andρ⁻¹pˆ⁰(ρ)(x, t)≥c₁, for some positive constantc₁, for x∈Ω, and0≤t≤T.

Proof. We will construct the solution of the problem for (1.2), (1.3), with Ω =T^N, through an iteration scheme. To define the iteration scheme, we will let the sequence of approximate solutions be v^k and ρ^k. Set v⁰(x, t) = v0(x), the initial velocity data, and set ρ⁰(x, t)∈ C([0, T], C⁵)∩L^∞([0, T], H^s+2) to be a positive function satisfyingk∇ρ⁰ks+1,T ≤ kDv0ks−1. Fork= 0,1,2, . . ., constructv^k+1,ρ^k+1 from the previous iteratesv^k,ρ^k by solving the linear system of equations

D^kv^k+1

Dt + (ρ^k)⁻¹pˆ⁰(ρ^k)∇ρ^k+1=c∇∆ρ^k+1, (3.1)

∇ ·v^k+1= 0, (3.2)

whereD^k/Dt=∂/∂t+v^k· ∇, and with initial datav^k+1(x,0) =v0(x). Since the solutionρ^k+1is unique up to an arbitrary function oft, we specify that the solution ρ^k+1 satisfy ρ^k+1(x0, t) = ρ^k(x0, t) at a single, fixed pointx0 ∈Ω, for all k ≥0.

Henceρ^k+1(x0, t) =ρ⁰(x0, t).

Existence of a sufficiently smooth solution to equations (3.1), (3.2) for fixed k follows from the results stated in Section 4. We proceed now to prove convergence of the iterates as k→ ∞to a unique classical solution of (1.2), (1.3). We assume thatpis a sufficiently smooth function of the thermodynamic state variableρin an open intervalG⊂R. We fix connected, bounded open setsG₀ andG₁ such that G¯₀⊂G₁ and ¯G₁⊂G, and we require that the initial iterateρ⁰ satisfiesρ⁰ ∈G₀. We fix δ= ˆδ(G0, G1) so that 0< δ <dist( ¯G0, ∂G1); therefore if|ρ^k−ρ⁰|L^∞ ≤δ, then ρ^k(x, t)∈ G1 for all x∈ T^N, and for t ∈ [0, T]. The values ofρ^k ∈ G1 are assumed to be strictly positive, bounded, and bounded away from zero. And the values of (ρ^k)⁻¹pˆ⁰(ρ^k) forρ^k∈G1are also assumed to be strictly positive, bounded, and bounded away from zero. Using a proof by induction on k, we assume that ρ^k ∈ G1, and then later we will show that ρ^k+1 ∈ G1. First, we proceed with the proof of uniform boundedness of the approximating sequence in a high Sobolev norm.

Proposition 3.2. Assume that the hypotheses of Theorem 3.1 hold. Let ρ⁰, v0

satisfy C0kρ⁰k²₀+C0k∇ρ⁰k²₂≤L²₀ ande^2Tkv0k²₀≤L²₀, whereL0 is a constant and where C0 is the constant from Lemma 2.4. There are constants 0, L1, L2, L3, where0< 0<1, such that the following estimates hold fork= 1,2,3. . ., provided kDv0ks−1 is sufficiently small:

(a) k∇ρ^kk²_s+1,T ≤₀,kρ^kk_s+2,T ≤L₁,

(b) kDv^kk²_s−1,T ≤0,kv^kks,T ≤L2, (c) k∂v^k/∂tk²_s−1,T ≤0L3.

Proof. The proof is by induction onk. We show only the inductive step. We will derive estimates for ρ^k+1 and v^k+1, and then use these estimates to prescribe₀, L1,L2,L3 a priori, independent of k, so that ifρ^k andv^k satisfy the estimates in (a)-(c), thenρ^k+1andv^k+1also satisfy the same estimates. And we will show that 0 will be small if kDv0k_s−1 is sufficiently small. In the estimates below, we use C to denote a generic constant whose value may change from one instance to the next, but is independent of0,L1,L2, andL3.

Estimate for ∇ρ^k+1: Applying the divergence operator to equation (3.1), we obtain

c∆²ρ^k+1− ∇ ·((ρ^k)⁻¹pˆ⁰(ρ^k)∇ρ^k+1) =∇ ·(v^k· ∇v^k+1) (3.3) where we used the fact that ∇ ·v^k+1 = 0, and where c is a positive constant.

Applying Lemma 2.7 we obtain from the above equation the following estimate for

∇ρ^k+1, fors >^N₂ + 3

k∇ρ^k+1k²_s+1≤C(k∆ρ^k+1k²_s+k∇ρ^k+1k²_s)

≤C[1 +

s

X

j=1

kD((ρ^k)⁻¹pˆ⁰(ρ^k))k^2j_s1]kDv^kk²_s2+1kDv^k+1k²_s−1

≤C1kDv^kk²_s−1kDv^k+1k²_s−1

(3.4)

whereC1= ˆC1(L1). Here,s1= max{s−1, s0}=s−1,s2= max{s−2, s0}=s−2, where s0= [^N₂] + 1 = 2, and s > ^N₂ + 3, sos≥5 forN = 2,3. And we used the induction hypothesis forρ^k,v^k.

Estimate for ρ^k+1: To obtain an L² estimate for ρ^k+1, we apply Lemma 2.4, where we define u = ρ^k+1 and we define w = ρ⁰ , to be used for the functions u, wappearing in Lemma 2.4. Note that since by hypothesis we haveρ^k+1(x0, t) = ρ⁰(x0, t) at a single fixed pointx0∈Ω, the hypotheses of Lemma 2.4 are satisfied, and so we obtain the estimate

kρ^k+1k²₀≤C₀kρ⁰k²₀+C₀k∇ρ⁰k²₂+C₀k∇ρ^k+1k²₂

≤L²₀+C2kDv^kk²_s0+1kDv^k+1k²₁

≤L²₀+C₂kDv^kk²_s−1kDv^k+1k²₁

(3.5)

whereC0is the constant from Lemma 2.4 (recallC0depends only on Ω), and where C2= ˆC2(L1). Here we used the estimate fork∆ρ^k+1k²₂+k∇ρ^k+1k²₂ from applying Lemma 2.7 to (3.3) with r = 2. We also used the hypothesis that C0kρ⁰k²₀+ C0k∇ρ⁰k²₂≤L²₀. Adding the estimates (3.4), (3.5) yields the following estimate for ρ^k+1,

kρ^k+1k²_s+2≤ kρ^k+1k²₀+Ck∇ρ^k+1k²_s+1

≤L²₀+C₃kDv^kk²_s−1kDv^k+1k²_s−1 (3.6) whereC₃= ˆC₃(L₁).

Estimate for v^k+1: Applying Lemma 2.3 to equations (3.1), (3.2), we obtain for Dv^k+1 the estimate

kDv^k+1k²_s−1≤Ce^2T(1 +T e^2Te^β(T)kDv^kk²_s−1,T)(kDv0k²_s−1 +

Z t

0

(kD((ρ^k)⁻¹pˆ⁰(ρ^k))k²_s1k∇ρ^k+1k²_s−1)dτ)

≤Ce^2T(1 +T e^2Te^{T e2T})(kDv₀k²_s−1+C₄ Z t

0

k∇ρ^k+1k²_s−1dτ)

≤C5kDv0k²_s−1+C5

Z t

0

kDv^kk²_s−1,TkDv^k+1k²_s−1dτ

≤C5kDv0k²_s−1+C5

Z t

0

kDv^k+1k²_s−1dτ

(3.7)

where β(T) = T e^2TkDv^kk²_s−1,T. Here, we used (3.4) to estimate k∇ρ^k+1k²_s−1, and we used the fact that by the induction hypothesis, kDv^kk²_s−1,T ≤0, where 0< 0<1. And here C4= ˆC4(L1, T),C5= ˆC5(L1, T).

Applying Gronwall’s inequality yields the estimate

kDv^k+1k²_s−1≤C6kDv0k²_s−1 (3.8) where C6 = ˆC6(L1, T). We now choose 0 to satisfy 0 =C6kDv0k²_s−1. It follows thatkDv^k+1k²_s−1,T ≤0.

Substituting the estimate (3.8) forkDv^k+1k²_s−1into the right-hand side of (3.4), (3.6) yields the following estimates fork∇ρ^k+1k²_s+1 andkρ^k+1k²_s+2:

k∇ρ^k+1k²_s+1≤C₁kDv^kk²_s−1kDv^k+1k²_s−1≤₀C₁C₆kDv0k²_s−1, kρ^k+1k²_s+2≤L²₀+C3kDv^kk²_s−1kDv^k+1k²_s−1≤L²₀+0C3C6kDv0k²_s−1 Therefore, we have kρ^k+1ks+2,T ≤ L1 and we have k∇ρ^k+1k²_s+1,T ≤ 0, pro- vided that we choose L1 large enough so that ¹₂L²₁ > L²₀, and provided that we choosekDv0k_s−1small enough so that0C3C6kDv0k²_s−1< ¹₂L²₁, and provided that we choose kDv0k_s−1 small enough so that C1C6kDv0k²_s−1 < 1. We also choose kDv0ks−1 small enough so that C6kDv0k²_s−1 <1; therefore, we have 0 < 0 <1, since ₀ = C₆kDv₀k²_s−1. (Note that ₀ will be small if kDv₀k_s−1 is sufficiently small.) This completes the proof of part (a).

Next, by applying Lemma 2.3 to (3.1), (3.2), and using the estimates (3.4), (3.7), (3.8), we have the estimate

kv^k+1k²_s≤e^2Tkv0k²₀+Ce^2T(1 +T e^2Te^β(T)kDv^kk²_s−1,T)(kDv0k²_s−1 +

Z t

0

(kD((ρ^k)⁻¹pˆ⁰(ρ^k))k²_s1k∇ρ^k+1k²_s−1)dτ)

≤L²₀+C₇kDv₀k²_s−1+C₇ Z t

0

kDv^k+1k²_s−1dτ

≤L²₀+C7kDv0k²_s−1+C8kDv0k²_s−1

withβ(T) =T e^2TkDv^kk²_s−1,T. Heres1= max{s−1, s0}=s−1,s > ^N₂ + 3, and s₀ = [^N₂] + 1 = 2 forN = 2,3. We also used the hypothesis that e^2Tkv0k²₀≤L²₀,