Dapeng DU

Abstract The author proposes a two-dimensional generalization of Constantin-Lax-Majda model.Some results about singular solutions are given.This model might be the first step toward the singular solutions of the Euler equations.Along the same line(vorticity formulation),the author presents some further model equations.He possibly models various aspects of difficulties related with the singular solutions of the Euler and Navier-Stokes equations.Some discussions on the possible connection between turbulence and the singular solutions of the Navier-Stokes equations are made.

Keywords Euler equations,Navier-Stokes equations,Singular solutions,Turbulence

1 Introduction

The incompressible Navier-Stokes equations describe the motion of incompressible viscous flow.Whether singular solutions exist in 3D is one of the famous seven millennium prize problems.In terms of singular solutions,the study of incompressible Euler equations looks most probably to be the first major step.For the sake of convenience,we will omit incompressible from now on.The first nonlocal model was constructed by Constatin,Lax and Majda[2].They constructed a one-dimensional model and got the singular solution explicitly.The motivation is the vorticity formulation.There are many developments after this model(see[1,3,4,5,8,14]and others).Shortly before the publication of the current paper,the author found out that Kiselev[9,Problem 6]has proposed to study the two dimensional analog of[2]in 2016.Essentially Model 1 and 1’in section two are some special cases of his proposal.

In this paper,we give some high dimensional generalizations of the Constantin-Lax-Majda model.The study of them might help the understanding of singular solutions to the Euler and Navier-Stokes equations.The motivation is still vorticity formulation.

We first present a two-dimensional zero order scalar model.One may think of it as a nonlocal ODE.The good understanding of it is possibly the first step toward singular solutions of the Euler and Navier-Stokes equations.

Then we give further models.In some sense,the vorticity formulation provides an explanation why the singular solutions to the Navier-Stokes equations are so hard.Roughly speaking,the zero order term is pro-singularity term.The first and second order terms are perturbations.To be able to construct original solution from vorticity,we need the initial vorticity to be divergence-free.Any of them is hard to handle.The combination of them composes one of the most difficult problems in contemporary mathematics.

One potential major application of singular solutions of the Navier-Stokes equations is about turbulence.There is a prevailling viewpoint that the turbulent theory is deeply connected with the singular solutions of the Navier-Stokes equations.Actually one can interpret the behaviour of turbulence by the guessed properties of singular solutions to the Navier-Stokes equations.

This paper is organized as follows.In Section 2,we discuss the two-dimensional zero order scalar models.In Section 3,further models are given.The possible connection between singular solutions of the Navier-Stokes equations and turbulence theory is presented in Section 4.The notations we use are standard ones.

2 Zero Order Scalar Models

Then the equation forQis

So under the assumptions of the current proposition,Lais coercive:

So

3 Further Models

In 2D,u=(−∂x2∆−1w,∂x1∆−1w).

Roughly speaking,(3.2)is a simple situation when the model is a system(The matrix in(3.2)is symmetric,and the equation also has simple singular solutions similar with Theorem 2.1.).(3.3)models the pro-ingularity effect of zero order term in the vorticity formulation.(3.4)-(3.6)model the effects of first order perturbation,second order perturbation,first and second order perturbation combined for scalar equations.

Remark 3.1Different from zero order model,it seems that for the first and second order models,the whole space case is inclined to be first considered.One reason is that self-similar singular solutions for PDEs only occur in whole space.

Remark 3.2It is known that in the whole space situation,the vorticity formulation with divergence initial vorticity is equivalent to the original Euler/Navier-Stokes equations(see[11,p.78,Proposition 2.21]).In general,the requirement of divergence-free initial data will make the situation harder.For instance,most probably the self-similar singular solutions would not exist if we add the divergence-free initial data requirement.

If the dimension is higher than three,it is convenient to think the velocity as 1-form and vorticity as 2-form.In this case,the divergence-free requirement becomes dw0=0,where d is the exterior differential andw0is initial vorticity.We refer to[15]for more details in the case ofn,n>3.

The proposition is proven.

The Proposition 3.1 above suggests that the zero order term has certain algebraic skewsymmetry,which may cause some more trouble in the study of singular solutions.

Possible steps toward Euler equationsIn the luckiest scenario,the study of model equations might lead to the existence of singular solutions of the Euler equations and even Navier-Stokes equations.The following are possible steps toward Euler equations:

(1)Model 1,

(2)(3.3),

(3)(3.4),

(4)the whole Euler equations.

Remark 3.3It was suggested in[6,p.3]that the degree of difficulty for singular solutions to Navier-Stokes equations may decrease a lot in higher dimensions.Probably this scenario will also hold true for certain second order model.

Remark 3.4For zero order models,if there is no divergence-free requirement on the initial data,the self-similar singular solutions probably exist.But for more complicated situations,one might have to work on singular solutions with general form.One evidence is that the Navier-Stokes equations do not have self-similar singular solutions at any dimensions(see[12,16]).There were also no reliable numerical evidence that Euler equations have self-similar singular solutions.

4 Possible Connection with Turbulence

It is well accepted that the main features of turbulence is irregular,random and chaotic.Based on what is known on Navier-Stokes equations and the features of turbulence,it seems reasonable to make the following guess.

Conjecture 4.1The singular solutions of three-dimensional Navier-Stokes equations generically are fluctuated.

Using the conjecture above,we could interpret the turbulence in the following way.Since the solutions are fluctuatedly singular,the average of them are irregular.The randomness comes from the infinite amplifying effect of fluctuatedly singular solutions over arbitrarily small experimental error.The chaotic behavior could be explained in the similar way.

Remark 4.1The results on model equations(see[7,13])and numerical simulation for Euler equations suggest that,in dimension five and higher the usual singular solutions possibly are also typical for Navier-Stokes equations.There is no information in dimension four so far.

The model system constructed byandin[13]has the form:

wherea ∈[0,1].The system(4.1)has the similar mathematical structures with the Navier-Stokes equations.It is possible that Conjecture 4.1 also holds true for(4.1).The phenomenon controlled by the system(4.1)might be calledturbulence.The advantage of this turbulence is that it would be a lot easier since the governing system is local.

At this stage little is known regarding the singular solutions of the Navier-Stokes equations.Therefore the application in the turbulence theory is not much.With the development of the mathematical theory on the singular solutions,more and more applications could be expected.To some degree,the good understanding of turbulence may depend on the good understanding of singular solutions to the three-dimensional Navier-Stokes equations.

AcknowledgementThe author would like to thank Hongjie Dong and Vladimirfor valuable comments.The author would also thank to Yipeng Shi for wonderful discussions on turbulence.