# Vanishing viscosity limit of Navier-Stokes

Equations in Gevrey class

###### Abstract.

In this paper we consider the inviscid limit for the periodic solutions to Navier-Stokes equation in the framework of Gevrey class. It is shown that the lifespan for the solutions to Navier-Stokes equation is independent of viscosity, and that the solutions of the Navier-Stokes equation converge to that of Euler equation in Gevrey class as the viscosity tends to zero. Moreover the convergence rate in Gevrey class is presented.

###### Key words and phrases:

Gevrey class, Incompressible Navier Stokes equation, Vanishing viscosity limit###### 2010 Mathematics Subject Classification:

35M30, 76D03, 76D05## 1. Introduction

The Navier-Stokes equations for incompressible viscous flow in read

(1.1) |

where is the unknown velocity vector function at point and time , is the unknown scalar pressure function, is the kenematic viscosity, is the given initial data.

If the viscosity , the equations (1.1) become the Euler equations for ideal flow with the same given initial data ,

(1.2) |

where we denote the unknown vector velocity function to be and the unknown scalar pressure function to be .

The existence and uniqueness of solutions to (1.1) and (1.2) in Sobolev space for , on a maximal time interval is classical in [4, 15, 30]. There are abundant studies on the analyticities of solutions to (1.1) and (1.2) in various methods, for reference in [2, 3, 5, 12, 33]. The Gevrey regularity of solutions to Navier-Stokes equations was started by Foias and Temam in their work [9], in which the authors developed a way to prove the Gevrey class regularity by characterizing the decay of their Fourier coefficients. And later [17, 18, 19, 20, 21] developed this method to study the Gevrey class regularity of Euler equations in various conditions.

The subject of inviscid limits of solutions to Navier-Stokes equations has a long history and there is a vast literature on it, investigating this problem in various functional settings, cf. [16, 31] and references therein. Briefly, convergence of smooth solutions in or torus is well developed (cf. [15, 32] for instance). Much less is known about convergence in a domain with boundaries. In fact the vanishing viscosity limit for the incompressible Navier-Stokes equations, in the case where there exist physical boundaries, is still a challenging problem due to the appearance of the Prandtl boundary layer which is caused by the classical no-slip boundary condition. So far the rigorous verification of the Prandtl boundary layer theory was achieved only for some specific settings, cf. [1, 7, 11, 13, 22, 28, 34] for instance, not to mention the convergence to Prandtl’s equation and Euler equations. Several partial results on the inviscid limits, in the case of half-space, were given in [33] by imposing analyticity on the initial data, and in [26] for vorticity admitting compact support which is away from the boundary.

On the other hand, the Prandtl boundary layer equation is ill-posed in Sobolev space for many case (see [7, 10, 25]), while the Sobolev space is the suitable function space for the energy theory of fluid mechanic. Since the verification of the Prandtl boundary layer theory meet the major obstacle in the setting of the Sobolev space, it will be interesting to expect the vanishing viscosity limit for the incompressible Navier-Stokes equations in the setting of Gevery space as sub-space of Sobolev space, see a series of works in this direction [11, 22, 23]. In fact, Gevrey space is an intermediate space between the space of analytic functions and the Sobolev space. On one hand, Gevrey functions enjoy similar properties as analytic functions, and on the other hand, there are nontrivial Gevrey functions having compact support, which is different from analytic functions. As a preliminary attempt, in this work we study the vanishing viscosity limit of the solution of Navier Stokes equation to the solution of Euler equation in Gevrey space. Here we will concentrate on the torus, we hope this may give insights on the case when the domain has boundaries, which is a much more challenging problem.

We introduce the functions spaces as follows. We usually suppress the vector symbol for functions when no ambiguity arise. Let be the vector function space

where is the th order Fourier coefficient of , . The condition means in the weak sense, so it is the standard space with the divergence free condition. Let be the vector periodic Sobolev space : for ,

Here the condition means , so it is the standard Sobolev space with the divergence free condition. Denote the inner product of two vector functions. Let us define the fractional differential operator and the exponential operator as follows,

The vector Gevrey space for is

where the condition means , so it is sub-space of the Sobolev space .

The following theorem is the main result of this paper.

###### Theorem 1.1.

Let . Assume that the initial data , then there exists and is a decreasing function such that, for any , the Navier-Stokes equations (1.1) admit the solutions

and the Euler equations (1.2) admit the solution

Furthermore, we have the following convergence estimates : for any

(1.3) |

where is a constant depending on and .

###### Remark 1.1.

The uniform lifespan is where is the maximal lifespan of solutions. The uniform (with respect to ) Gevrey radius of the solution is

(1.4) |

where are constants depending on .

###### Remark 1.2.

## 2. Premilinary lemmas

We first recall the following classical result of Kato in [15].

###### Theorem 2.1.

###### Remark 2.1.

The time in Theorem 2.1 is actually depending on and , specifically

where is a constant depending on . In fact, the constant was created by using the Leibniz formula and Sobolev embedding inequality when estimating the nonlinear term. So, if the initial data , then we have , , because there exists a constant such that . But we can’t directly obtain an uniform bound for by the Gevrey norm of when is very large. Then we can’t say that, if goes to infinity, has a positive lower bound. In this paper, we will pay many attention to the uniform lifespan that depends on .

###### Remark 2.2.

Compared with the known results Theorem 2.1, the additional difficulty arises on the estimate of the convecting term in Gevrey class setting. We need to use the decaying property of the radius of Gevrey class regularity to cancel the growth of the convecting term.

We will use the following inequality, for any , we have

The proof is a simple result of triangle inequality which we omit the details here. And we will give two Lemmas which will be used in the proof of Theorem 1.1.

###### Lemma 2.2.

Given two real numbers and , then the following inequality holds

(2.1) |

where is a positive constant depending only on .

###### Proof.

With the use of Lemma 2.2, we have the following estimate about the nonlinear term.

###### Lemma 2.3.

Let and is a constant. Then for any , the following estimate holds,

(2.2) |

where is a constant depending only on and .

###### Proof.

By the definition of the vector function space , we have and . Using Fourier series convolution property, one have

Applying the operator on , one have

And . Now we take the inner product of with over . The orthogonality of the exponentials in implies

The cancellation property of the convecting term implies

Then we have

where

and

Before we come to the estimate of and , we recall the following mean value theorem, for , there exists a constant such that

Then there exists a constant depending only on such that

From the inequality that holds for all , we can bounded the exponential by . Then can be bounded by

With application of discrete Hölder inequality and Minkowski inequality, one can obtain the following estimates. For example, we give the details for , and the rest can be estimated in the same way,

where is a constant depending on and for , the summation in the above is bounded by some constant depending on . Similarly with , we have

and

Note that in the summation, and

and

Noting that , then . Thus we obtain

As for , we have

We note that the inequality holds for . Then

Since , we have

Then we actually have

By Lemma 2.2, we have

Then can be bounded by the inequality

where

We have used the inequality and for in the estimation of . With application of Hölder inequality and Minkowiski inequality, we have for ,

Symmetrically, one has a same bound for , then for ,

Then we obtain

which finishes the proof of Lemma 2.3. ∎

## 3. Uniform existence of solutions

In this section, we will first show the existence of Gevrey class solutions to Navier Stokes equations (1.1). And the existence of Gevrey class solution to Euler equations (1.2) can be obtained similarly. The method of the proof are based on Galerkin approximation. Before that, we first consider the following equivalent equation for Navier-Stokes equation,

(3.1) | ||||

where is the well-known Stokes operator and is the Leray projector which maps a vector function into its divergence free part , such that and , is a scalar function and . Similarly for Euler equation, we have the following equivalent form,

(3.2) | ||||

We then recall some properties of the Stokes operator , which are known in [Chapter 4 in [6]].

###### Proposition 3.1.

The Stokes operator is symmetric and selfadjoint, moreover, the inverse of the Stokes operator, , is a compact operator in . The Hilbert theorem implies there exists a sequence of positive numbers and an orthonormal basis of , which satisfies

Moreover, in the case of , the sequence of eigenvector functions and eigenvalues are the sequences of functions and numbers ,

where , , and are the canonical basis in . So we know that each are not only in , but also in for . Now we will show that there exists a solution to equation (3.1) for with , and is a differentiable decreasing function of . To this end, we first prove a priori estimate in the following Proposition.

###### Proposition 3.2.

Let and is a differentiable decreasing function of defined on with , where and is the maximal time of solution to (3.1) with respect to the initial data . Let be the solution to (3.1), then the following a priori estimates holds,

With the same assumptions as above, let be the solution to (3.2), we also have

Furthermore the uniform radius is given by

where are constants depending on .

###### Proof.

Applying on both sides of (3.1) and taking the inner product of both sides with , one has