Abstract

In this paper, we investigate an approximation of the Euler equation by the second-grade fluid equations in . The convergence in of a sequence of solutions to the second-grade fluid equations in a uniform interval is proven as both the viscosity coefficient and filter parameter tend to zero with an initial velocity in .

1. Introduction

Let be a simply connected domain and . The second-grade fluid equations are as follows:where is the velocity, is the pressure, is the viscosity coefficient, and is the filter parameter. We impose the nonslip boundary conditions, i.e.,

It is believed that equation (1) is an interpolation between the Navier–Stokes ( = 0) and Euler- equation ( = 0). When the filter parameter and viscosity are very small, we expect the second-grade fluid system (1) to behave like the Euler system:with the following no-penetration boundary condition:where is the outward-pointing normal unit vector of .

It is well known that the zero-viscosity limit of the incompressible Navier–Stokes equations, which is expressed by (1) with a vanishing filter parameter (i.e., ), is one of the most challenging open problems in fluid mechanics (see [1–4] and the references therein). This is because of the formation of a boundary layer that appears due to the different boundary conditions between the Navier–Stokes and Euler equations. Fortunately, Lopes Filho et al. [5] proved that the solution of the 2D Euler- equation ( = 0) given by (1) with a no-slip boundary condition converges to the solution of the Euler equations (3) with no-penetration boundary conditions, despite the presence of a boundary layer. Lopes Filho et al. also obtained an approximation of the 2D Euler equation (3) using the second-grade fluid equation (1) [6]. They expected their work to shed light into the contrast between the vanishing limit of the Euler- and the vanishing viscosity limit of the Navier–Stokes system in the presence of a boundary layer. In their paper [7], Su and Zang selected a radical symmetric example such that the solution of the second-grade fluid equation converged to the solution of the Euler equation as and tended to zero in -space.

In this paper, we only examine with the vanishing and vanishing viscosity limits in the whole space, i.e., . For the case of  = 0 in (1), the inviscid limit in the whole space has been examined by several authors, e.g., Swann [8], Kato [9], Constantin [10], and Masmoudi [4]. In these previous results, they proved the convergence in space, as long as a solution of the Euler system exists. For the case of Euler- (i.e.,  = 0), Linshiz and Titi [11] verified that the convergence of the Euler- in the whole space was guaranteed using the same technique used by Masmoudi [4]. Zang [12] extended these results of the Euler- equation to periodic boundary conditions using a different method. In this paper, we focus on the uniform estimates on the viscosity and filter parameter. It is easy to see these estimates on the viscosity for Navier–Stokes equations by Swann [8] and Masmoudi [4]. However, we discuss on a priori estimates on the filter parameter by the special structure of second-grade fluid equations.

The remainder of this article is divided into two sections. In Section 2, we introduce basic notation and the existence theorem of the second-grade fluid equations, and we present a technique lemma. In Section 3, we present the main results of this paper and provide a proof.

2. Existence Theorem and Technique Lemma

First, we let and denote the scalar product and the norm in , respectively, and let and be the scalar product and the norm in , respectively. For any ,where is a multi-index derivation, .

We denote by and the closures of that are divergence free in and , respectively. By the Sobolev Theorem, is embedded in .

The well-posedness of (1) has been established previously [13, 14], and the existence and well-posedness of (3) have been determined by many mathematicians (see [15–17] and the references therein). The following theorem can be obtained.

Theorem 1. Let . There exists a such that, for any , there exists a unique solution of problem (3) with an initial velocity , where represents the class of the absolutely continuous functions on . In two dimensions, the solution exists globally in time . Similar results hold for the second-grade fluid equations (1) with a maximal interval of existence of the three-dimensional second-grade fluid equations, which are also dependent on and , and the solution satisfiesfor every .

Throughout this paper, the following lemma plays a crucial role in proving the convergence of the second-grade fluid equations to the Euler equations in .

Lemma 1. Let and both and be divergence free vectors. We can obtain the following estimates:Assuming that , the following estimates can be obtained:

Proof. First, we examine estimates (7) and (8). Since ,Integrating by parts and using a similar approach as in the proof above, the following is obtained:Thus, it is sufficient to prove that (9) is true, sinceWe divide into four parts, as follows:Integrating by parts, we observe thatAs and , it follows thatSimilarly, for , and thus,Now, we determine the bounds of and , as follows:Along with the above estimate, we have the following:For , integrating by parts yields the following:Estimating the integrals one by one in (19), it is easy to verify thatsince for , .
By the interpolation and Hölder inequalities, the following can be inferred:As in the proof of , the following can be verified:Therefore, we have shown thatFinally, integrating by parts yields the following:Therefore,

3. Main Theorem and Its Proof

We state the main result, which is as follows: the classical results of (1) will exist in the maximal interval of the solution of the Euler equation (3) and converge to the solutions of Euler equations under suitable assumptions of the initial velocity. The main results of this paper are described in detail in this section.

Theorem 2. Let and be a divergence-free and solenoidal vector , which can be approximated by as follows:  Let be the time of existence and be the solution of the Euler system (3). Then, for all , there exists such that, for all , the second-grade fluid equation (1) has a unique solution . Moreover,for all and , and C depends only on and .

Remark 1. For , from [16], if we choose a suitably small such that , then there exists a such that

Proof. We prove this theorem using the following three steps. Step 1 (energy estimates): we easily obtain the uniform existence interval of the solutions to the second-grade fluid equation (1) by using the Kato–Lai Theorem [15] with the following energy estimates (dropping and ) from Lemma 1: where . Therefore, there exists a that only depends on , and we have a uniform bound of for all : where is independent of and . Step 2 (convergence in ): the difference between (1) and (3) is given by where and is the pressure of the Euler equation (3). Multiplying both sides of (30) by and integrating in , we obtain It is evident that Similarly, Equation (31) can be rewritten as follows: It can be inferred that Letting , (35) becomes where By the Gronwall inequality, we obtain the following: By the energy estimate (29) and the interpolation inequality (38), the following can be obtained for any : since Step 3: (convergence in ): From Lemma 1, we know the following:From the energy estimates (41) and (42), we ensure the convergence in by requiring regularization of the initial data. For all , we take such thatFor any , such that , we haveas . The existence of can be constructed by the following previously reported approaches [4, 16]. The maximum existence time of the Euler equations only depends on . Thus, for any , there exists a uniform interval in which the solution of the Euler equations exists. Furthermore, , , and .
Similar to a previous report [4], for any , we haveFor equation (30), we have an energy estimate for , and we obtain the following (dropping and ):sinceFrom Step 2, converges to in , and we deduce the following:where . On contrary, from (45), we know thatfor some . Combining estimates (48) and (49) and using (46) and the Gronwall inequality, we obtainwhere is defined by (37). It follows thatwhere is chosen such that , , and are sufficiently small. Letting and , the following can be shown:Using the continuous method, we ensure convergence on any time interval for , where is the maximum existence time of the Euler equations. This completes the proof.

Data Availability

Previously reported (regular article) data were used to support this study and are available at (DOI: 10.1142/S0218202518500458). These prior studies are cited at relevant places within the text as reference [7].

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The author would like to express her sincere gratitude to Professor Aibin Zang for his careful reading of the manuscript and his fruitful suggestions. This work was supported partly by the National Natural Science Foundation of China (Grant no. 11771382) and partly by the Science and Technology Project of Education Department of Jiangxi Province (Grant no. GJJ180829).