Linghai Zhang

(Dept.of Math.,Lehigh University,14 East Packer Avenue, Bethlehem,Pennsylvania 18015 USA)

THE IMPROVED FOURIER SPLITTING METHOD AND DECAY ESTIMATES OF THE GLOBAL SOLUTIONS OF THE CAUCHY PROBLEMS FOR NONLINEAR SYSTEMS OF FLUID DYNAMICS EQUATIONS?

Linghai Zhang?

(Dept.of Math.,Lehigh University,14 East Packer Avenue, Bethlehem,Pennsylvania 18015 USA)

Dedicated to Professor Boling Guo on the occasion of his eightieth birthday!

Consider the Cauchy problems for an n-dimensional nonlinear system of fluid dynamics equations.The main purpose of this paper is to improve the Fourier splitting method to accomplish the decay estimates with sharp rates of the global weak solutions of the Cauchy problems.We will couple together the elementary uniform energy estimates of the global weak solutions and a well known Gronwall’s inequality to improve the Fourier splitting method. This method was initiated by Maria Schonbek in the 1980’s to study the optimal long time asymptotic behaviours of the global weak solutions of the nonlinear system of fluid dynamics equations.As applications,the decay estimates with sharp rates of the global weak solutions of the Cauchy problems for n-dimensional incompressible Navier-Stokes equations,for the n-dimensional magnetohydrodynamics equations and for many other very interesting nonlinear evolution equations with dissipations can be established.

nonlinear systems of fluid dynamics equations;global weak solutions;decay estimates;uniform energy estimates;Fourier transformation; Plancherel’s identity;Gronwall’s inequality;improved Fourier splitting method

2000 Mathematics Subject Classification 35Q20

Ann.of Appl.Math.

32:4(2016),396-417

1　Introduction

1.1The mathematical model equations

First of all,consider the Cauchy problems for the n-dimensional incompressible Navier-Stokes equations

The real vector valued function u=u(x,t)represents the velocity of the fluid at position x and time t.The real scalar function p=p(x,t)represents the pressure of the fluid at x and t.

Secondly,consider the Cauchy problems for the n-dimensional magnetohydrodynamics equations

In this system,the real vector valued function u=u(x,t)represents the velocity of the fluid at position x and time t,the real vector valued function A=A(x,t) represents the magnetic field at position x and time t.The real scalar functionrepresents the total pressure,where the real scalar function p=p(x,t)represents the pressure of the fluid andrepresents the magnetic pressure.Additionally,M ＞0 represents the Hartman constant, RE represents the Reynolds constant and RM represents the magnetic Reynolds constant.

Now let us consider the Cauchy problems for the following n-dimensional nonlinear system of fluid dynamics equations

In this system,α＞0 is a positive constant,x=(x1,x2,···,xn)represents the spatial variable,the dimension n≥3.Moreover,u(x,t)=(u1(x,t),u2(x,t),···,um(x,t)) represents the unknown function,f(x,t)=(f1(x,t),f2(x,t),···,fm(x,t))represents the external force,and N(u,?u)=(N1(u,?u),N2(u,?u),···,Nm(u,?u))represents the nonlinear function,which is sufficiently smooth,m≥n is an integer.

The general system(7)-(8)contains the n-dimensional incompressible Navier-Stokes equations(1)-(2)and the n-dimensional magnetohydrodynamics equations (3)-(6)as particular examples.The general system also contains many other interesting nonlinear evolution equations with dissipations as examples.

Many mathematicians have accomplished the existence of the global weak solutions of the Cauchy problems for the n-dimensional incompressible Navier-Stokesequations.They have also established the existence of the global smooth solutions with small initial functions and small external forces.However,the uniform energy estimates of all order derivatives of the global weak solutions with large initial functions or large external forces have been open,see[4,8,9].For the n-dimensional magnetohydrodynamics equations,there hold very similar results.Very recently, Lei and Lin[1],Lei,Lin and Zhou[2],Peng and Zhou[5]established the existence of large global smooth solution of three-dimensional incompressible Navier-Stokes equations for special cases.

1.2The main purpose

The decay estimates with sharp rates of the global weak solutions of the Cauchy problems for the n-dimensional incompressible Navier-Stokes equations(1)-(2),the global weak solutions of the Cauchy problems for the n-dimensional magnetohydrodynamics equations(3)-(6)and the global weak solutions of the Cauchy problems for many other interesting nonlinear evolution equations with dissipations are of great interests and importance in applied mathematics.The Fourier splitting method was developed by Maria Schonbek[6,7]to accomplish the optimal long time asymptotic behaviors of the global weak solutions.Today,it has become a very popular tool to study the asymptotic behaviors and thus has been widely used,see[3,10–12]for closely related results.To obtain the optimal decay rate,one must iterate some of the most important steps in the Fourier splitting method for finitely many times, see[6,7].We will make complete use of the uniform energy estimates(see Lemmas 3,4 and 8 in Section 2)and the Gronwall’s inequality to avoid the iteration process for n-dimensional problems,where n≥3.Therefore,the main purpose of this paper is to improve the Fourier splitting method so that it may become the most powerful tool to accomplish the decay estimates with sharp rates for the global weak solutions of many nonlinear evolution equations with dissipations.

1.3The main results—decay estimates with sharp rates

First of all,let us make several reasonable assumptions needed for the main results.

(A1)Suppose that the initial function u0∈L1(?n)∩L2(?n)and the external force f∈L1(?n×?+)∩L1(?+,L2(?n)).

(A2)Suppose that there exist real scalar functions φkl∈C1(?n)∩L1(?n)and ψkl∈C1(?n×?+)∩L1(?n×?+),such that

and

for all k=1,2,···,m and l=1,2,···,n.

(A3)Suppose that there holds the condition

(A4)Suppose that there holds the condition

(A5)Suppose that the nonlinear function satisfies

for all u∈L∞(?+,L2(?n))with?u∈L2(?+,L2(?n)).

(A6)Suppose that the nonlinear function satisfies the condition

for all u∈C1(?n),where C1＞0 is a positive constant,independent of u and?u.

(A7)Suppose that the nonlinear function satisfies

for all u∈L∞(?+,L2(?n))with?u∈L2(?+,L2(?n))and for all(ξ,t)∈?n×?+, where C2＞0 is a positive constant,independent of u,?u and(ξ,t).

(A8)Suppose that there exists a global weak solution to the Cauchy problems for the nonlinear system of fluid dynamics equations(7)-(8):

Theorem 1(I)Suppose that assumptions(A1),(A3),(A5)-(A8)hold.There holds the decay estimate

for all time t＞0,where C3＞0 is a positive constant,independent of u and(x,t).

(II)Suppose that assumptions(A1)-(A8)hold.There holds the decay estimate with sharp rate

for all time t＞0,where C4＞0 is a positive constant,independent of u and(x,t).

The global weak solutions of the Cauchy problems for the n-dimensional incompressible Navier-Stokes equations(1)-(2)and for the n-dimensional magnetohydrodynamics equations(3)-(6)satisfy these conditions and results.

2　The Mathematical Analysis and the Proofs of the Main Results

We will couple together the elementary uniform energy estimates,the Fourier transformation,the Plancherel’s identity and the Gronwall’s inequality to improve the Fourier splitting method to accomplish the decay estimates with sharp rates for the global weak solutions of the Cauchy problems for the nonlinear systems of fluid dynamics equations(7)-(8).The improved Fourier splitting method involves the splitting of the frequency space into two time-dependent subspaces(a small ball with radius proportional to(1+t)?1/2and the exterior of the small ball)and the delicate estimates of the Fourier transformation of the global weak solutions.The key point of the improvement is that for many nonlinear evolution equations with dissipations,we may apply the improved Fourier splitting method to accomplish the decay estimates with sharp rates for the global weak solutions.

2.1The uniform energy estimates

The main purpose is to use traditional ideas,methods and techniques to establish some uniform energy estimates.

Lemma 1(The Cauchy-Schwartz’s inequality)Let the functions f∈L2(?n) and g∈L2(?n).There holds the following Cauchy-Schwartz’s inequality

Lemma 2(The H?lder’s inequality)Let f∈Lp(?n)and g∈Lq(?n),where p≥ 1 and q≥ 1 are positive constants,such thatThere holds the following estimate

Lemma 3 Suppose that the initial function u0∈L2(?n)and the external force f∈L1(?+,L2(?n)).There holds the following uniform energy estimate

In particular,if the initial function u0∈L2(?n)and the external force f=0,then there holds the following uniform energy estimate

Proof Multiplying system(7)by 2u and integrating the result with respect to x over ?nyield the following energy equation

where

By using Cauchy-Schwartz’s inequality,there hold the following estimates

Now the above energy equation becomes the differential inequality

Simplifying the inequality gives

Integrating this inequality with respect to time t leads to the desired energy estimate

If f=0,then the uniform energy estimate

follows immediately.The proof of Lemma 3 is finished.

2.2The Fourier transformation of the global weak solutions

Lemma 4(I)There holds the following Fourier representation

for all(ξ,t)∈?n×?+.

(II)There holds the following estimate

for all(ξ,t)∈?n×?+,where C5＞0 is a positive constant,independent ofand (ξ,t).

(III)There holds the following estimate

where C6＞0 is a positive constant,independent ofand(ξ,t),if

for all time t＞0 and for another positive constant C7＞0,independent of u and (x,t).

Proof Performing the Fourier transformation to(7)leads to

Multiplying this equation by the integrating factor exp(α|ξ|2t)gives

Integrating with respect to time t yields

Finally,we have the representation.

Now let us make estimates aboutFirst of all,there hold the following estimates

In particular,for the n-dimensional incompressible Navier-Stokes equations(1)-(2), there hold the following estimates

for all(ξ,t)∈?n×?+.

Therefore,there hold the following estimates

Moreover,if there exist real scalar functions φkl∈C1(?n)∩L1(?n)and ψkl∈C1(?n×?+)∩L1(?n×?+),such that

for all k=1,2,···,m and l=1,2,···,n,then we have

By applying Cauchy-Schwartz’s inequality to the Fourier transformations,we get the following estimates

Recall that there exists a positive constant C2＞0,independent ofand(ξ,t), such that

for all(ξ,t)∈?n×?+.Now we obtain the following estimate

for all(ξ,t)∈?n×?+.The proof of Lemma 4 is finished.

2.3The Fourier splitting method

Now let us review the Fourier splitting method developed in[6,7]to establish the decay estimates for the global weak solutions of system(7)-(8).

Lemma 5(The Plancherel’s identity)There holds the following Plancherel’s identity for all real vector valued functions f∈L2(?n)

Lemma 6(The Gronwall’s inequality)Suppose that the nonnegative continuous functions f≥0,g≥0 and h≥0 satisfy the inequality

for all t＞0,where the derivative f′≥0.Then

for all t＞0.

Lemma 7 Let t＞0 and define

There holds the following estimate

Proof Multiplying system(7)by 2u and integrating the result with respect to x over ?nyield

where

Applying the Plancherel’s identity to this equation gives

Multiplying it by(1+t)2nto get the energy equation

By applying Cauchy-Schwartz’s inequality,we have

Now the above energy equation becomes the inequality

Recall that ?(t)={ξ∈?n:α|ξ|2(1+t)≤2n}.Then we have the following estimates

Now the energy inequality

becomes the new differential inequality

The proof of Lemma 7 is finished.

2.4The improved Fourier splitting method

Lemma 8(I)There holds the following decay estimate for the global weak solutions of the Cauchy problems(7)-(8)if assumptions(A1),(A3),(A5)-(A8)hold:

for all time t＞0,where C11＞0 is a positive constant,independent of u and(x,t).

(II)There holds the following decay estimate with sharp rate for the global weak solutions of the Cauchy problems for the nonlinear system of fluid dynamics equations(7)-(8)if assumptions(A1)-(A8)hold:

for all time t＞0,where C12＞0 is a positive constant,independent of u and(x,t).

Proof(I)Recall that there exists a positive constant C5＞0,independent ofand(ξ,t),such that

for all(ξ,t)∈?n×?+.

Therefore,by using Lemma 7,we have

Integrating this inequality with respect to time t yields

That is

where C17＞0 and C18＞0 are positive constants,independent ofand (ξ,t).Note that

Recall that there holds the following uniform energy estimate

By using Gronwall’s inequality,it follows that

where C19＞0 and C20＞0 are positive constants,independent of b u(ξ,t)and(ξ,t).

(II)Recall that

for all(ξ,t)∈?n×?+.Now we have the following estimates

Integrating the inequality in time t yields the estimate

That is

The proof of Lemma 8 is finished.

Therefore,the proofs of the main results stated in Theorem 1 are finished.

RemarkBoth the global weak solutions of the Cauchy problems for the ndimensional incompressible Navier-Stokes equations(1)-(2)and the global weak solutions of the Cauchy problems for the n-dimensional magnetohydrodynamics equations(3)-(6)enjoy the decay estimates.

3　Conclusions and Remarks

3.1Summary

The main purpose of this paper is to improve the Fourier splitting method to simplify the mathematical analysis to accomplish the decay estimates with sharp rates.

We considered the Cauchy problems for the following n-dimensional nonlinear system of fluid dynamics equations

The general system contains the n-dimensional incompressible Navier-Stokes equations(1)-(2)and the n-dimensional magnetohydrodynamics equations(3)-(6)as particular examples.There holds the following decay estimate with sharp rate

for all time t＞0,where C25＞0 is a positive constant,independent of u and(x,t). The uniform energy estimate played a key role in the mathematical analysis.

3.2Summary about the n-dimensional incompressible Navier-Stokes equations

Consider the Cauchy problems for the n-dimensional incompressible Navier-Stokes equations

Suppose that the initial function u0∈L1(?n)∩L2(?n)and the external force f∈L1(?n×?+)∩L1(?+,L2(?n)).There exists a global weak solution u∈L∞(?+,L2(?n)),such that?u∈L2(?+,L2(?n)).

There holds the following uniform energy estimate

Due to the divergence free conditions?·u0=0 and?·f=0,it is necessarily true that∫

?nu0(x)d x=0 and∫?nf(x,t)d x=0,for all t＞0.

Suppose that there exist real scalar functions φkl∈C1(?n)∩L1(?n)and ψkl∈C1(?n×?+)∩L1(?n×?+),such that

and that

for all k=1,2,···,n and l=1,2,···,n.

There holds the following decay estimate for the global weak solutions of the Cauchy problems for the n-dimensional incompressible Navier-Stokes equations

for all time t＞0,where C26＞0 is a positive constant,independent of u and(x,t).

Let us review some important open problems about the global smooth solutions and their influences.Suppose that the initial function u0∈L1(?n)∩H2m+1(?n) and the external force f∈L1(?n×?+)∩L1(?+,L2(?n))∩L2(?+,H2m(?n)).The following uniform energy estimates have been open

for all positive integers m ≥1 and for all time t＞0,where C27＞0,C28＞0, C29＞0 and C30＞0 are positive constants,independent of u and(x,t).Therefore, the existence of the global smooth solution u∈L∞(?+,H2m+1(?n))such that?u∈L2(?+,H2m+1(?n))has been open.

Suppose that the initial function u0∈L1(?n)∩H2m+1(?n)and the external force f∈L1(?n×?+)∩L1(?+,L2(?n))∩L2(?+,H2m(?n)).Suppose that there exists a global smooth solution to the Cauchy problems for the n-dimensional incompressible Navier-Stokes equations(1)-(2):u∈L∞(?+,H2m+1(?n)),such that?u∈L2(?+,H2m+1(?n)),where m ≥ 1 is a positive integer.There hold the following decay estimates with sharp rates

and

for all positive integers m ≥1 and for all time t＞0,where C31＞0,C32＞0, C33＞0,C34＞0,C35＞0,C36＞0,C37＞0,C38＞0 are positive constants, independent of u and(x,t).

Now let us consider a very interesting question for the global smooth solutions of the Cauchy problems for the n-dimensional incompressible Navier-Stokes equations (1)-(2).As t→∞,how do the following exact limits

depend on the nonlinear function,the initial function,the nonhomogeneous function,the dimension n and the order of the derivative(that is,the integer m)?Do the global smooth solutions carry initial information(e.g.mass,energy and momentum of physical objects)to the very end?In another word,can the global smooth solutions“remember the very beginning”at“the very end”?We will couple together existing ideas,methods,results and new ideas to generate a very different method to solve these complicated mathematical problems and accomplish very general results.

Again consider the Cauchy problems for the n-dimensional incompressible Navier-Stokes equations

The following estimate has been open

for all(ξ,t)∈?n×?+,where κ1=κ1(t)and κ2=κ2(t)are positive,continuous, increasing functions defined on(0,∞),0＜ε?1 is a constant.

If this estimate is true,then there exists a unique global smooth solution u∈C∞(?n×?+)to(1)-(2)and there hold the following decay estimates with sharp rates

for all positive integers m ≥1 and for all time t＞0,where C39＞0 is a positive constant,independent of u and(x,t).

If there exists a global smooth solution u∈C∞(?n×?+)to the Cauchy problems for the n-dimensional incompressible Navier-Stokes equations(1)-(2),then there holds the following solution representation

3.3Summary about the n-dimensional magnetohydrodynamics equations

Consider the Cauchy problems for the n-dimensional magnetohydrodynamics equations

Suppose that the initial functions

Suppose that the external forces

There exists a global weak solution

such that

There holds the following uniform energy estimate

Suppose that there exist real scalar functions

such that

and that

for all k=1,2,···,n and l=1,2,···,n.

There holds the following decay estimate with sharp rate

for all time t＞0,where C40＞0 is a positive constant,independent of(u,A)and (x,t).

Let us review some open problems and their influences about system(3)-(6). Suppose that the initial functions

Suppose that the external forces

The following uniform energy estimates have been open

for all positive integers m ≥1 and for all time t＞0,where C41＞0,C42＞0, C43＞0,C44＞0 are positive constants,independent of(u,A)and(x,t).

The existence of the global smooth solution of the Cauchy problems for the n-dimensional magnetohydrodynamics equations(3)-(6):

such that

has been open.

Suppose that there exists a global smooth solution to the Cauchy problems for the n-dimensional magnetohydrodynamics equations:

such that

where m≥1 is a positive integer.

For the global smooth solution of the n-dimensional magnetohydrodynamicsequations(3)-(6),there hold the following decay estimates with sharp rates

and

for all positive integers m≥1 and for all time t＞0,where C45＞0,C46＞0,C47＞0,C48＞0,C49＞0,C50＞0,C51＞0,C52＞0 are positive constants,independent of(u,A)and(x,t).The proofs follow from the Fourier splitting method.

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(edited by Mengxin He)

?Manuscript June 16,2016

?.E-mail:liz5@lehigh.edu.