On the Cahn-Hilliard-Brinkman Equations in R4:Global Well-Posedness

Bing Li,Fang Wang,Ling Xue,Kai Yangand Kun Zhao

1School of Mathematical Sciences,Tiangong University,Tianjin 300387,China

2School of Mathematics and Statistics,Changsha University of Science and Technology,Changsha,Hunan 410114,China

3CollegeofMathematicalSciences,HarbinEngineeringUniversity,Harbin,Heilongjiang 150001,China

4School of Mathematics,Southeast University,Nanjing,Jiangsu 211189,China

5DepartmentofMathematics,TulaneUniversity,NewOrleans,LA 70118,USA

Abstract.We study the global well-posedness of large-data solutions to the Cauchy problem of the energy critical Cahn-Hilliard-Brinkman equations in R4.By developing delicate energy estimates,we show that for any given initial datum in H5(R4),there exists a unique global-in-time classical solution to the Cauchy problem.As a special consequence of the result,the global wellposedness of large-data solutions to the energy critical Cahn-Hilliard equation in R4 follows,which has not been established since the model was first developed over 60 years ago.The proof is constructed based on extensive applications of Gagliardo-Nirenberg type interpolation inequalities,which provides a unified approach for establishing the global well-posedness of large-data solutions to the energy critical Cahn-Hilliard and Cahn-Hilliard-Brinkman equations for spatial dimension up to four.

Key words:Cahn-Hilliard-Brinkman equations,energy criticality,Cauchy problem,classical solution,global well-posedness.

1 Introduction

1.1 Background and object

The Cahn-Hilliard equation[7,8]:

is one of the fundamental models in mathematical physics,describing the evolutionary process of phase separation of binary fluids.Here,the unknown function

?

denotes the relative difference of fluid concentrations,and

μ≡?ε

?

?

+

ε

(

?

??

)stands for the chemical potential which can be derived from a coarse-grained study of the free energy of the fluid(c.f.[20]).The positive parameters

M

and

ε

model the mobility and diffusive interface thickness,respectively.

Besides phase separation,the Cahn-Hilliard equation also appears in modeling many other phenomena,including the evolution of two components of intergalactic material[45],dynamical interaction of two populations[10],modeling of bacterial film[25],and thin film problems[42,44].Since its initiation in the 1950’s,the Cahn-Hilliard equation has been serving as a foundation for the mathematical modeling of phase separation.The success of the model is demonstrated through its capability of capturing the essential features of spinodal decomposition(anti-nucleation).

Because of its physical background and mathematical features,the qualitative and quantitative behaviors of the Cahn-Hilliard equation have been analyzed in the mathematics literature to a great extent.We refer the reader to[1,2,9,14,16,28,37–39,41,47]for analytical investigations,to[22,23]for numerical simulations,and to[27,29–34]for important progress recently made on the numerical analysis of the model.

Meanwhile,because phase separation appears in many fluid-related problems,of great interest to researchers in applied sciences is the coupling of the Cahn-Hilliard equation with fluid dynamics equations.

For example,the Cahn-Hilliard-Navier-Stokes(CHNS)system:

and its variants have been utilized to study phase separation in general incompressible fluid flows(c.f.[4,17,21,35]).

Recently,a related model:

which is referred to as the Cahn-Hilliard-Hele-Shaw(CHHS)system,was developed in[18,19]to understand the spinodal decomposition of binary fluids in a Hele-Shaw cell.This model has been utilized in the study of morphological growth of solid tumors(c.f.[12,13]).Some qualitative properties,such as global well-posedness and long-time behavior,have been established in the literature,see e.g.,[36,46,48].

In this paper,we are interested in a mathematical model which describes the motion of phase separation of incompressible binary fluid flows in porous media.The motion of the flow of a fluid through a porous medium can be described by the Darcy’s law,which states that the velocity of the fluid flow is proportional to the negative of the gradient of the pressure.One of the modified versions of the Darcy’s law is the Brinkman equation:

which was introduced by Brinkman in 1947[5]to account for transitional flow between boundaries.Here,the unknown functions u and

π

denote the fluid velocity and pressure,respectively,and the positive parameters

η

and

ν

model the fluid permeability and kinematic viscosity,respectively.

To understand the process of phase separation of incompressible binary fluid flows in porous media,the following model was proposed in[40]:

x

∈

R,

t

>0,which is referred to as the Cahn-Hilliard-Brinkman(CHB)system.This model is derived by making a coupling of the Cahn-Hilliard equation(1.1)with the Brinkman equation(1.4).Here,the positive parameter

γ

models the capillarity(surface tension)of the fluid.Note that(1.5)reduces to(1.3)when the kinematic viscosity

ν

=0.On the other hand,(1.5)becomes the CHNS system,(1.2),if the velocity

η

,u is replaced by the inertial term:

?

u+u·

?

u.

For mathematical interests,we consider the Cauchy problem of(1.5)supplemented with the following initial condition:

Next,we point out the fact that motivates the current work.

1.2 Motivation and goal

As one of the fundamental questions in the studies of nonlinear partial differential equation models,the global well-posedness(GWP)of large-data solutions is of central interest to applied analysts.For mathematical models involving the Cahn-Hilliard equation,a free energy functional is oftentimes verified under appropriate conditions.Similar to(1.1)–(1.3),one can check that(1.5)enjoys the free energy functional with respect to(roughly)the

H

-metric:

under appropriate boundary conditions at the far fields as|x|

→∞

.Next,we provide some heuristic arguments to illustrate the idea behind the motivation.

To simplify the presentation and since we are interested in the GWP of largedata solutions to(1.5)for fixed values of the parameters,we take all the positive system parameters to be equal to one,resulting in the simpler form of the model:

Note that the definition of

μ

implies

and we can rewrite(1.8)as

Our preliminary study shows that because of the presence of the biharmonic operator,the negative Laplacian term in(1.9a)does not have a significant impact on the GWP of large-data solutions to the model.Similarly,the velocity term in(1.9b)plays a less important role than the viscosity term.Hence,as far as the GWP of large-data solutions is concerned,the qualitative behavior of(1.9)is dictated by that of the following system of equations:

One can check that the free energy functional associated with(1.10)reads

Meanwhile,we can verify that(1.10)holds its form under the rescaling transformations:

Using(1.12),we can show that

From the point of view of energy criticality,(1.11)and(1.13)indicate that the GWP of large-data solutions to(1.10)is

sub

-

critical

when

d

=2,3,

critical

when

d

=4,and

super

-

critical

when

d

>4.Since(1.10)is obtained by removing minor terms from(1.9),the latter one carries the same qualitative property.Moreover,applying the same argument,one can verify the same feature for the Cahn-Hilliard equation(1.1).Our heuristic argument partially explains why large-data solutions to(1.9)is globally well-posed in the three-dimensional space[3],since such a problem lies in the energy sub-critical regime.However,the critical case,i.e.,when

d

=4,remains widely open.Also,the GWP of large-data solutions to the Cahn-Hilliard equation in the four-dimensional space is still unknown,even though the model was first developed over 60 years ago.The current work is directly motivated by the lack of such information in the knowledge base.The primary goal of this paper is to establish the GWP of large-data solutions to the Cauchy problem(1.5)-(1.6)in the critical case when

d

=4 for given initial data satisfying appropriate conditions at the far fields as|x|

→∞

.The situation encountered in this paper resembles a similar character as those came across in the studies of classical PDE models,such as the 2D incompressible Navier-Stokes equations,4D defocusing nonlinear Schr?dinger equation[43],and 2D surface quasigeostrophic(SQG)equation[6,11,24],whose studies have inspired the invention of many new analytical methods.

1.3 Result and remarks

Now we state the main result of this paper.We first introduce some notations for convenience.

Notation 1.1.

Throughout this paper,we use

∥

·

∥

,

∥

·

∥∞

,and

∥

·

∥

s

to denote the norms of the Lebesgue spaces

L

(R),

L

(R),and Hilbert space

H

(R),respectively.Unless otherwise specified,we use

C

to denote a generic constant which is independent of the unknown functions.The value of the constant varies line by line according to the context.

Our main result is recorded in the following theorem.

Theorem 1.1.

Consider

the

Cauchy

problem

(1.8)

and

(1.6).

Suppose

that

the initial

function

satis

fi

es

?

?

1

∈H

(R).

Then

there

exists

a

unique

solution

(

?

,u),

such

that

for

any

T

>0,

We have several remarks regarding Theorem 1.1.

Remark 1.1.

The readers will see from the proof of Theorem 1.1 that the result is also valid if the initial function satisfies

?

+1

∈H

(R),which generates a solution spatially decaying to

?

1 at the far fields as|x|

→∞

.

Remark 1.2.

As a direct consequence of Theorem 1.1,one obtains the GWP of large-data solutions to the Cahn-Hilliard equation in the critical case when

d

=4,which has not appeared in the literature.Moreover,the proof of Theorem 1.1 provides a unified approach for obtaining the GWP of large-data solutions to the CHB system and Cahn-Hilliard equation in spatial dimension up to four.

Remark 1.3.

In the physical setting when the spatial dimension is either two or three,the energy method developed in this paper can be adapted to study the initial-boundary value problem(IBVP)of the CHB system subject to the no-flux boundary conditions:

2 Proof of Theorem 1.1

For the reader’s convenience,we restate the Cauchy problem as follows:

x

∈

R,

t

>0,where

?

?

1

∈H

(R).First,we remark that the local well-posedness of(2.1)can be established by utilizing standard arguments,such as iteration technique and fixed point argument.For the convenience of the reader,we state the result in the following lemma.

Lemma 2.1.

Consider

the

Cauchy

problem

(2.1).

Suppose

that

the

initial

function satis

fi

es

?

?

1

∈H

(R).

Then

there

exists

a

fi

nite

time

T

>0,

such

that

a

unique solution

(

?

,u)

to

(2.1)

exists

and

satis

fi

es

In the following subsections,we focus on deriving the

a

priori

estimates of the local solution,in order to extend it to a global one.We begin the proof with the free energy functional associated with(2.1)and collect some by-products from it.

2.1 Free energy

Taking the

L

inner products of(2.1a)with

μ

,(2.1c)with u,and using the incompressibility condition(2.1d),we arrive at

Upon integrating(2.2)with respect to

t

,we have

Next,we derive some consequences from the above energy estimates.

2.2 By-products of free energy

This subsection contains the crucial estimates for overcoming the energy criticality of the problem.After such estimates are established,the subsequent ones for the higher order derivatives of the solution can be obtained by extensive applications of various Gagliardo-Nirenberg type interpolation inequalities.

which will be utilized in the subsequent subsections.

We remark that the energy estimates derived in this subsection are solely for the function

?

based on the free energy functional(2.2).Since the classical Cahn-Hilliard equation enjoys a similar free energy functional(i.e.,(2.2)without the fluid component),the estimate(2.15)is also valid for the Cahn-Hilliard equation.In addition,by utilizing(2.15),the subsequent energy estimates can also be carried through for the Cahn-Hilliard equation.

2.3 and estimates of

2.4 and estimates of

2.5 and estimates of u

2.6 and estimates of

2.7 and estimates of

2.8 and estimates of u

3 Concluding remarks

We have established the global well-posedness of large-data solutions to the Cauchy problem of the Cahn-Hilliard-Brinkman equations in R,which is an energy critical problem.As a special case of the result,the global well-posedness of large-data solutions to the Cauchy problem of the Cahn-Hilliard equation in Ris established.The proof is constructed by deriving delicate energy estimates based on the Gagliardo-Nirenberg interpolation inequalities,which provides a unified approach for establishing the global well-posedness of large-data solutions to the Cahn-Hilliard and Cahn-Hilliard-Brinkman equations in any spatial dimension less than or equal to four.

Meanwhile,we propose some new problems for future studies in this area.

?

The energy method developed in this paper can be adapted to study the global well-posedness of large-data solutions to the following model of hydrodynamic two-phase fluid flows:

The method utilized in this paper might need to be adjusted to a substantial extent in order to fit into such a complicated situation.

We leave the investigation of proposed problems in forthcoming papers.

Acknowledgements

The authors would like to thank the anonymous referee(s)for valuable comments and suggestions which helped improve the quality of the paper.Support for this work came in part from a National Natural Science Foundation of China Award 12001064(F.Wang),a Hunan Education Department Project 20B006(F.Wang),a Double First-Class International Cooperation Expansion Project 2019IC39(F.Wang),a National Natural Science Foundation of China Award 12171116(L.Xue),a Fundamental Research Funds for Central Universities of China Award 3FT2020CFT2402(L.X-ue),a Natural Science Foundation of Jiangsu Province of China Award BK20200346(K.Yang),and from Simons Foundation Collaboration Grant for Mathematicians Award 413028(K.Zhao).K.Yang also gratefully acknowledges funding from the Shuang Chuang Doctoral Plan of Jiangsu Province of China.