Global Regularity of 2-D Density Patches for Viscous Inhomogeneous Incompressible Flow with General Density: High Regularity Case

Xian Liao and Ping Zhang

1 Institute for Analysis, Karlsruhe Institute for Technology, Englerstrasse 2, 76131 Karlsruhe,Germany

2 Academy of Mathematics & Systems Science and Hua Loo-Keng Key Laboratory of Mathematics, Chinese Academy of Sciences, China, and School of Mathematical Sciences,University of Chinese Academy of Sciences,Beijing 100049,China

Received 7 September 2017;Accepted(in revised version)8 June 2018

Abstract. This paper is a continuation work of [26] and studies the propagation of the high-order boundary regularities of the two-dimensional density patch for viscous inhomogeneous incompressible flow. We assume the initial density ρ0=η11Ω0+η21Ωc0,where(η1,η2)is any pair of positive constants and Ω0 is a bounded,simply connected domain with Wk+2,p(R2) boundary regularity. We prove that for any positive time t, the density function ρ(t)=η11Ω(t)+η21Ω(t)c, and the domain Ω(t) preserves the Wk+2,p-boundary regularity.

Key Words:Inhomogeneous incompressible Navier-Stokes equations,density patch,striated distributions,Littlewood-Paley theory.

1 Introduction

We consider the two-dimensional density-dependent incompressible Navier-Stokes system:

Here the unknowns (ρ,v)∈R+×R2represent the density and the velocity field of the two-dimensional fluid at time t and point x respectively,and π designates the unknown pressure which ensures the incompressibility of the fluid.

This system(1.1)can describe the dynamics of a viscous fluid which is incompressible but with variable density,e.g.,mixtures of incompressible and non-reactant components,fluids containing a melted substance. In the simple case when ρ0≡1, the system (1.1)reduces to the classical incompressible Navier-Stokes system:

There is a substantial amount of literature devoted to the study of the well-posedness issue of the system(1.1),e.g.,[28,32]in the weak solution framework,[1-3,12,24,29]in the strong solution framework. If the density function has a jump across some hypersurface which is of interest in this paper, [13,14,16,22,30] have also established some well-posedness results(see[26]for more detailed introduction to these references).

We are interested in the propagation of regularities of the interface between fluids with different densities,for which we take the assumptions as follows.Let Ω0be a simply connected bounded domain with Wk+2,p(R2)-boundary regularity,k≥1,p∈(2,4),that is,we can parametrize ∂Ω0as

For any η1,η2＞0,we take the initial density ρ0and the initial velocity v0as

for some

Definition 1.1. Consider a smooth radial function φ on R,supported in[3/4,8/3] such that ∑j∈Zφ(2-jτ)=1 for any τ ＞0. We denote

Let(p,r,κ)∈[1,+∞]3and s∈R. The Besov norms are defined as

The tangential vector field X(t,·) of the boundary of the domain Ω(t) is transported by this flow in this following way

or equivalently,

and then recursively for any ℓ≥1,

, we ask ourselves whether or not

The original question was proposed by P.-L.Lions in[28]with(η1,η2)=(1,0). As a first progress toward this question, we [25] confirmed the propagation of the boundary regularity of any order under the small jump condition: |η1-η2| is sufficiently small. Forany positive numbers(η1,η2),we[26]considered the low regularity case k=1 and in this paper we continue to study the high regularity case with k≥2.

Recently [17,19] studied the propagation of the low boundary regularity of H¨older type. Danchin and Mucha [15] investigated the vacuum case and hence the aforementioned question has a complete answer in the low regularity regime.

We consider this regularity propagation problem in the framework of conormal distributions or striated distributions(with respect to the regularities only in the tangential direction in(1.3),(1.4),(1.8)). This idea was used by J.-Y.Chemin[8,9]to prove the global regularities of the two-dimensional vortex patch for ideal flow(see also[6,7,10,11,18,20,21,31]).

In the remaining part of the introduction,we shall introduce

· The main result in Subsection 1.1,

· A series of equations in Subsection 1.2,

· Two useful lemmas in Subsection 1.3,

· The outline of the proof in Subsection 1.4,

and we will supply the detailed proofs in Section 2, following the outline in Subsection 1.4.

1.1 Main results

then Eq.(1.7)implies that the two operators Dtand ∂Xcommute:

In order to show Xℓ-1(t,·)∈(W2,p)2, we need vℓ∈L1([0,t];(W2,p)2) and this can be obtained from the H1-energy estimates for vℓand(Dtvℓ)①We consider Dtvℓ instead of ∂tvℓ since we would like to benefit from the density equation Dtρ=0 while∂tρ=-v·▽ρ has singularities since the density ρ is discontinuous in space variable. The idea was used by D.Hoff in[23]for the study of the compressible Navier-Stokes system.§We derive from H1-energy estimate for Dtvℓ and Sobolev embedding that Dtvℓ ∈(Lp) which controls▽2vℓ ∈(Lp)up to lower order terms,see Eq.(1.29)below.§. To this end,we will consider the ℓ-th unknowns(vℓ,Dtvℓ,▽Dtπℓ,Xℓ-1)together and aim to get the estimate for Aℓ(t)defined below

where

We [26] have studied the Cauchy problem (1.1)-(1.4) and proved (1.8) for k=1 by considering the coupled system (1.1)-(1.7) (see also (1.26)-(1.27) below) with the initial data (1.3)-(1.4) for k=1. Let us recall the global well-posedness and the low regularity propagation results therein where the estimates for v also play an important role here:

where(recalling the time weight σ(t)in(1.15))

and(recalling the definition(1.13)of Aℓ)

Here the notation a⊗b means: (a⊗b)ij=aibjfor a=(ai)iand b=(bj)j.

In this paper we will generalize k=1 in Theorem 1.1 to any positive integer and the main result of this paper is the following:

Theorem 1.2. Let k ≥1 be any integer. Let the initial data (ρ0,v0,X0) be given by (1.3)-(1.4).Then the coupled system(1.1)-(1.7)has a unique solution(ρ,v,▽π,X)such that the density patch(1.5)persists the initial boundary regularity(1.8).

Furthermore, there exists a constant Cℓ(depending on the initial data (1.3)-(1.4)) such that(recalling the definition(1.13)of Aℓand〈t〉=max{1,t})

Remark 1.1. The initial density assumption in(1.4)can be relaxed to more general case:

1.2 A series of equations

Let(ρ,v,▽π,X)be the solution of the Cauchy problem(1.1)-(1.7)and(1.3)-(1.4)given by Theorem 1.1. Recall the notations Dt,vℓ,▽πℓ,Xℓdefined in (1.9). In this subsection we will derive the system satisfied by(vℓ,▽πℓ),which will be of the following form:

where (u,▽Π) are the unknowns and (f,g,u0) are known data. While (Dtvℓ,▽Dtπℓ)satisfies a linear system reading as below

where(Dtw,▽q)are the unknowns and(F,a,b)are known data. Indeed,we will derive below the precise formulations for

And we will make use of Lemmas 1.1,1.2 below to get the energy estimates for vℓ,Dtvℓrespectively and finally derive(1.8)for Xℓ-1which satisfies the transport equation(1.12).

We first deduce from the transport equation (1.7) and divv=0 that X’s divergence divX satisfies also the free transport equation ∂t(divX)+v·▽(divX)=0. We then conclude from the initial assumption (1.3) on X0that the divergence-free condition always holds true for the vector field X(t):

It is hence convenient to define the operator

such that due to divX=0,

1.2.1 Equations for(v,▽π)&(Dtv,▽Dtπ)

It is straightforward to see from(1.1)that(v,▽π)satisfies(1.21)with(f,g,u0)=(0,0,v0).

Here and in what follows,repeated indices of α means summation of α from 1 to 2.

1.2.2 Equations for(v1,▽π1)&(Dtv1,▽Dtπ1)

We apply ∂Xto the v-equation in (1.1) to get the Eq. (1.21) for v1with data (f1,g1) (see also[26])

We apply the operator Dtto the v1equation of(1.26)to get(see also[26])

where F0(·,·), f1(·,·)and PXare given in(1.25),(1.26)and(1.24)respectively.

1.2.3 Equations for(vℓ,▽πℓ)&(Dtvℓ,▽Dtπℓ)with general ℓ≥2 Recall the definition of the operator PXin(1.24). Recall the definition of g1in(1.26). We derive gℓby induction:

We apply the operator Dtto (1.29) to get the Eq. (1.22) for (Dtvℓ,▽Dtπℓ) with data(Fℓ,aℓ,bℓ):

where F0(·,·)is given by(1.25)and hence

1.3 Two useful lemmas

In this subsection we recall two useful lemmas in[26],which give the H1estimates for u,Dtw of the linear equations(1.21),(1.22)respectively.

Lemma 1.1. Let(ρ,v)be the solution of (1.1)-(1.4). Let(u,▽Π)be a smooth enough solution of(1.21)with initial data u0=0.Then for any s∈(0,1)and δ∈(s,1),one has

Lemma 1.2. Let(ρ,v)be the solution of (1.1)-(1.4). Let(w,▽q)be a smooth enough solution of the system(1.22). Then for any s∈(0,1),we have

and in particular,for any ε＞0,there exists Cεsuch that

1.4 Outline of the proof

We prove Theorem 1.2 by an inductive argument:By Theorem 1.1,we can assume inductively that

and it suffices to show(1.19) Aℓ(t)≤Hℓ(t). Recall the definition(1.13)of Aℓand

and the constant Cldepending on the initial data(1.3)-(1.4)until l-th order.

We apply Lemmas 1.1, 1.2 to the Eqs. (1.29), (1.32) respectively to get H1estimates for vℓ,Dtvℓwhere we take use of the inductive assumption(1.37)to control the ℓ-th data(fℓ,gℓ),(Fℓ,aℓ,bℓ). More precisely we follow the following procedure to prove(1.19)and detailed proofs are provided in Section 2:

· We derive more estimates from the inductive assumption(1.37)in Subsection 1.4.1,

· We derive the estimates for the data(fℓ,gℓ)in Subsection 1.4.2,

· We derive the H1estimate for vℓand then some consequent estimates for Dtvℓin Subsection 1.4.3,

· We derive the estimates for the data(Fℓ,aℓ,bℓ)in Subsection 1.4.4,

· We derive H1energy estimate of Dtvℓand finally(1.19)in Subsection 1.4.5.

1.4.1 Deductive estimates from inductive assumptions

Observe from Subsection 1.2.3 that for ℓ≥1,0≤i≤ℓ-1,i1+···+ir=i,

· fℓcontains terms such as ▽2Xi⊗▽vℓ-1-i,▽Xi·▽2vℓ-1-i,▽Xi·▽πℓ-1-i,and gℓcon

tains terms such as vℓ-1-i·▽Xi1···▽Xir;

· Fℓcontains terms such as

Since the operators ∂X,Dtdo not commute with ▽,∂t:

Lemma 1.3 (Inductive assumptions). Let (ρ,v,▽π,X) be the solution of the coupled system(1.1)-(1.7)with the initial data(1.3)-(1.4),such that the inductive assumption(1.37)holds. Then one has

In the above,for ℓ≥1,

For

and

and

where B0(t)is given by(1.17b).

1.4.2 Estimates for(fℓ,gℓ)

We derive in this step the estimates for the data(fℓ,gℓ)from those estimates(1.38).

Lemma 1.4 (Estimates for fℓ). Let ℓ=2,···,k, and r ∈{2,p}. Under the same hypotheses in Lemma 1.3,there hold

Lemma 1.5(Estimates for gℓ). Under the same hypotheses in Lemma 1.3,one has

1.4.3 H1energy estimate of vℓ

In this step we derive the time-weighted H1estimate of vℓ. We first decompose(vℓ,▽πℓ)as where(vℓ1,▽pℓ1)solves the linear system(1.21)with the data(f,g)=(0,0):

and(vℓ2,▽pℓ2)solves the linear system(1.21)with zero-initial data:

It follows the Besov estimate and H1estimate for vℓ1directly from Theorem 1.1(noticing that (v,▽π) and (vℓ1,▽pℓ1) both satisfy the linear system (1.21) with data (f,g)=(0,0)):

Then we apply Lemma 1.1 to bound the time-weighted H1-norm of vℓ2by

Proposition 1.1(The bound for Aℓ1). Let Aℓ1(t)be given by(1.14a). Let Cℓ(X)be given by(1.50). Under the assumptions of Lemma 1.3,we have

Corollary 1.1(Preliminary estimates for Dtvℓ). Under the assumptions of Proposition 1.1,one has

1.4.4 Estimates for(Fℓ,aℓ,bℓ)

We now get the estimates for the ℓ-th data(Fℓ,aℓ,bℓ)defined in Subsection 1.2.3.

Lemma 1.6 (Estimate for Fℓ). Under the assumptions of Lemma 1.3, we have the following estimates for Fℓ,ℓ=2,···,k:

for B0(t)given by(1.17b).

Lemma 1.7(Estimate for aℓ). Under the hypotheses of Lemma 1.6,we have the following estimates for aℓ,ℓ=2,···,k,

Here and in all that follows,for Bℓ(t)determined by(1.43),we always designate

Lemma 1.8(Estimate for bℓ). Under the hypotheses of Lemma 1.7,for any ε＞0,there exists a constant Cεsuch that for bℓ,ℓ=2,···,k we have

1.4.5 Proof of(1.19)

We apply Lemma 1.2 and the estimates for the data in Lemmas 1.6-1.7-1.8 to the equation(1.32) to get the H1-estimate for Dtvℓ, which will be used to get the W2,p-estimate for Xℓ-1:

Proposition 1.2 (Bounds for Aℓ1,Aℓ2,Xℓ-1). Let Aℓ1(t),Aℓ2(t) be given by (1.14a). Then under the assumptions of Lemma 1.6,we have

The proof of (1.57) follows exactly the same argument as those in proofs of the Besov estimates in Theorem 1.1 (see [26]). It is delicate but does not contain substantial new ideas. We skip the details here.

2 Proofs

We follow the procedure in Subsection 1.4 to prove the estimate(1.19)Aℓ(t)≤Hℓ(t)from the inductive assumption(1.37) Al(t)≤Hl(t),∀l ≤ℓ-1.

2.1 Deductive estimates from the inductive assumption

In this subsection we shall derive more estimates (1.38) from the inductive assumption(1.37),which will be used constantly in the following context.

First of all,we state the lemma concerning the commutator estimates:

Lemma 2.1 (Commutator estimates). Let ℓ ∈{1,···,k} and (i,j) be any pair of nonnegative integers with i+j ≤ℓ. Then for r1,r,r3satisfying (1.40), there exists a positive constant C such that

and

This lemma follows from the same argument as that in the proof of Lemma 4.1 of[25].It is elementary and we skip the details. Interested readers are referred to Appendix A of[27]for details.

We derive(1.38)for ℓ=1 immediately from Theorem 1.1 and the proof can be found in Section 5.2[27]:

Proposition 2.1. Under the hypotheses of Theorem 1.1,we have(1.38)for l=1:

By view of(2.3)we can derive more general inductive assumptions(1.38)in Lemma 1.3 by an inductive argument.

Proof of Lemma 1.3. By(2.3),we can assume that

We shall show the estimate(2.4)for κ+1.Indeed,it follows from(1.37)that

which together with the commutator estimate(2.1)for κ and(2.4)ensures that

Similarly, due to the estimate (2.5) and the commutator estimates (2.1b)-(2.2a), we can prove(2.4)for κ+1 from the assumption(1.37).We skip the details and interested readers may check the proof of Lemma 6.2 in[27]for details.

2.2 Estimates for(fℓ,gℓ)

In this subsection we prove the estimates (1.44)-(1.45a) in Lemmas 1.4-1.5 for the data(fℓ,gℓ)defined in Subsection 1.2.3 respectively,by use of the general inductive assumption(1.38).

Proof of Lemma 1.4. Recall the definition of fℓin(1.29). For r=2 or p,we deduce

which together with(2.1)ensures that for r=2 or r=p

As a result,we deduce from(1.38)that for r=2 or r=p

Recall in Subsection 1.2.3 that

and hence

from which,we infer for r=2 or r=p

Then we deduce from the commutator estimate(2.1)that for r=2,p,

By virtue of(1.38)and(2.7),we obtain the first inequality of(1.44).

On the other hand,we rewrite the vℓ-equation(1.29)as

so that for any r∈(1,∞),it follows from classical estimates for Stokes operator that

which together with the first inequality of(1.44)gives rise to the second one of(1.44).

Proof of Lemma 1.5. Recall the definition of gℓin Subsection 1.2.3:

It is obvious to observe from Lemmas 2.1 and 1.3 that

where

and

from which and the commutator estimate(2.1),we infer

Along the same line,we deduce from(2.12)that

from which and Lemma 1.3,we infer that

While by separating the case when i1=i=ℓ-1 from others and taking into account of the fact that: ∂tX=-v·▽X+v1one has

which yields

that is

2.3 H1 energy estimate of vℓ

In this subsection,we prove the time-weighted H1estimate of vℓin Proposition 1.1 and the consequent estimates for Dtvℓin Corollary 1.1.

Proof of Proposition 1.1. We first apply Lemma 1.1 to the vℓ2-equation(1.48)that

In view of the formulation(2.10)of gℓ,we get,by applying the law of product in Besov spaces(see[5]for instance),that

While it follows from Lemma 1.4 and θ0=sℓ-1-sℓthat

And Lemma 1.5 ensures that

Inserting the above estimates together with the H1estimate(1.49)for vℓ1into(2.16)and applying Gronwall’s inequality results in

for Cℓ(X)given by(1.50).

This above estimate implies the L2energy estimate of vℓ2. Indeed, we take L2(R2)inner product of(1.48)with vℓ2to get

Hence we apply Gronwall’s inequality to get

We sum up the above H1estimates for vℓ2and the H1estimate(1.49)for vℓ1to get A2ℓ1(t)≤Cℓ(X).

Proof of Corollary 1.1. We will derive the estimates (1.52) from the bound (1.16) in Theorem 1.1,the bound A2ℓ1≤Cℓ(X),the commutator estimates in Lemma 2.1 and the inductive assumptions in Lemma 1.3.

and

Indeed,we know

and hence we derive from Lemmas 2.1 and 1.3,and Proposition 1.1 that

and estimate the difference by

we have the estimate

Thus,we complete the proof.

2.4 Estimates for(Fℓ,aℓ,bℓ)

In this subsection we show the estimates for the data(Fℓ,aℓ,bℓ)in Lemmas 1.6-1.7-1.8,by use of the known estimates(1.16)-(1.38)-(1.51)-(1.52).

Proof of Lemma 1.6. Recall the definition of Fℓin(1.32). We denote

such that

Hence we derive from(1.38)that

As a result,we deduce from(1.38)again that

By virtue of(1.38)-(1.51)-(1.52),we complete the proof of Lemma 1.6.

and by virtue of(1.24)such that div(PXu)=∂Xdivu,

Let

Then we deduce from(1.38)and the formulation(2.27)of aℓabove that

and hence

To handle the estimate of ▽zℓ,we write

where

This gives rise to

from which and(1.38),we infer that

Taking into account of(2.28),we deduce from(2.30)and(2.32)that

Together with(1.16)-(1.51)-(1.52),we conclude

Finally let us turn to the estimate of ▽divaℓ:

Then, by virtue of Eqs. (1.16), (1.38), (1.51), (1.52) and (2.33), we complete the proof of(1.53).

Proof of Lemma 1.8. Recall the formulation bℓgiven in Subsection 1.2.3:

where

We calculate directly

Noticing the commutator

we obtain

from which,(1.16),(1.38)and(1.52),we infer

Recall the estimate(1.36)in Lemma 1.2,we know from the Dtvℓ-equation(1.32)that

Inserting the above estimate into(2.35)and using Lemmas 1.6 and 1.7,we achieve

Finally,we notice that

Substituting the above inequality and(2.37)into(2.34)and using(1.16),(1.38)and(1.52),we conclude the proof of(1.55).

2.5 Proof of Proposition 1.2

We prove in this subsection Proposition 1.2.Proof of Proposition 1.2. Recall the definition of Aℓ2in (1.14a). We first get, by applying Lemma 1.2 to the linear system(1.32),that

from which,Proposition 1.1,Corollary 1.1 and Lemmas 1.6,1.7 and 1.8,we infer

which together with the fact: p＜2/(1-sℓ),ensures that

Recall the Eq.(1.12): DtXℓ-1=vℓ=X·▽vℓ-1. We first get the Lptype energy estimate for Xℓ-1

We then apply Δ to(1.12)to obtain the Lptype estimate for ΔXℓ-1

We sum up the above two inequalities and then apply Gronwall’s inequality to obtain

which together with the second inequality of(1.44)ensures that

Acknowledgements

We would like to thank Professor Jean-Yves Chemin and Professor Rapha¨el Danchin for inspiring discussions of this topic. Part of this work was done when we were visiting Morningside Center of Mathematics (MCM), Chinese Academy of Sciences. We thank MCM for the hospitality and the financial support. X. Liao is supported by SFB 1060,Universit¨at Bonn during the last part of the work.P.Zhang is partially supported by NSF of China under Grants Nos.11371347 and 11688101,and innovation grant from National Center for Mathematics and Interdisciplinary Sciences.