# H

Authors: Hui Zhou
Journal: Discrete Dynamics in Nature and Society
 (2020)

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## Abstract

In this paper, we investigate an approximation of the Euler equation by the second-grade fluid equations inℝdd=2,3. The convergence in Hm of a sequence of solutions to the second-grade fluid equations in a uniform interval is proven as both the viscosity coefficient ν and filter parameter α tend to zero with an initial velocity in Hm.


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## Body

## 1. Introduction

LetΩ⊂ℝdd=2,3 be a simply connected domain and T>0. The second-grade fluid equations are as follows:(1)vt−νΔu+u⋅∇v+∑i=1dvi∇ui+∇p=0,inΩ×0,T,v=u−α2Δu,∇⋅u=0,inΩ×0,T,ut=0=u0α,ν,in,Ω,where u is the velocity, p is the pressure, ν is the viscosity coefficient, and α is the filter parameter. We impose the nonslip boundary conditions, i.e.,(2)u=0,in∂Ω×0,T.It is believed that equation (1) is an interpolation between the Navier–Stokes (α = 0) and Euler-α equation (ν = 0). When the filter parameter α and viscosity ν are very small, we expect the second-grade fluid system (1) to behave like the Euler system:(3)∂tu¯+u¯⋅∇u¯+∇p¯=0,inΩ×0,T,∇⋅u¯=0,inΩ×0,T,u¯t=0=u0,inΩ,with the following no-penetration boundary condition:(4)u¯⋅n^=0,in∂Ω×0,T,where n^ is the outward-pointing normal unit vector of ∂Ω.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., α=0), 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-α equation (ν = 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 α limit of the Euler-α 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 α and ν tended to zero in L2-space.In this paper, we only examine with the vanishingα and vanishing viscosity limits in the whole space, i.e., Ω=ℝdd=2,3. For the case of α = 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 Hm space, as long as a solution of the Euler system exists. For the case of Euler-α (i.e., ν = 0), Linshiz and Titi [11] verified that the convergence of the Euler-α in the whole space was guaranteed using the same technique used by Masmoudi [4]. Zang [12] extended these results of the Euler-α 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 α by the special structure of second-grade fluid equations.The remainder of this article is divided into two sections. In Section2, 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 L2ℝd, respectively, and let ⋅,⋅m and ⋅m be the scalar product and the norm in Hmℝd, respectively. For any f,g∈Hmℝd,(5)f,gm=∑α≤mDαf,Dαg,where Dα is a multi-index derivation, α=α1,…,αd.We denote byH and Vm the closures of C0∞ℝd that are divergence free in L2ℝd and Hmℝdm≥1, respectively. By the Sobolev Theorem, Vm is embedded in H.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.
Letu0∈Vm,m>s0=d/2+1. There exists a T∗=T∗u0s0>0 such that, for any T<T∗, there exists a unique solution u∈C0,T;Vm∩AC0,T;Vm−1 of problem (3) with an initial velocity u0, where AC0,T represents the class of the absolutely continuous functions on 0,T. In two dimensions, the solution exists globally in time T∗=∞. 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 α and ν, and the solution uα,ν satisfies(6)uα,νt2+α2∇uα,νt2+ν∫0t∇uα,ντ2dτ=u02+α2∇u02,for every t∈0,T.Throughout this paper, the following lemma plays a crucial role in proving the convergence of the second-grade fluid equations to the Euler equations inHm.Lemma 1.
Letm>s0 and both u∈Hm+3 and v∈Hm+3 be divergence free vectors. We can obtain the following estimates:(7)u⋅∇Δu,vm≤Cum∇umvm+3,(8)∑j=1dΔujΔuj,vm≤Cum∇umvm+2.
Assuming thatm>s0+1, the following estimates can be obtained:(9)u⋅∇Δu,um+∑j=1dΔujΔuj,um≤CuHm∇um2.Proof.
First, we examine estimates (7) and (8). Since m≥s0,(10)∑j=1dΔuj∇uj,vm≤∑j=1d∑β=2Δuj∇uj,∂2βvm−2+∑j=1dΔuj∇uj,v1≤∑j=1dΔuj∇ujm−2vm+2+∇u1Δu1v2+u22v1≤C∇umΔum−2vm+2+∇u1Δu1v2+u22v1≤Cum∇umvm+2.
Integrating by parts and using a similar approach as in the proof above, the following is obtained:(11)u⋅∇Δu,vm=u⋅∇v,Δum≤uΔum−2∇vm+2+uΔu2∇v1≤Cum∇um∇vm+2.
Thus, it is sufficient to prove that (9) is true, since(12)u⋅∇Δu,uHm=∑α=0m∫ℝdDαu⋅∇ΔuDαudx=−∑α=0m∑β≤ααβ∫∑l=1dDβ∂lu⋅∇Dα−β∂luDαudx−∑α=0m∑0<β≤ααβ∫∑l=1dDβu⋅∇Dα−β∂luDα∂ludx=I1+I2.
We divideI1 into four parts, as follows:(13)I1=∑α>0m∑0<β≤m/2αβ∫ℝd∑l=1dDβ∂lu⋅Dα−β∂lu∇Dαudx+∑α>0m∑β≤m/2+1m−1αβ∫ℝd∑l=1dDβ∂lu⋅Dβ−αDlu∇Dαudx+∑α>0m∑l=1d∫ℝd∂luDα∂lu∇Dαudx+∑l=1d∑β=m∫ℝdDβ∂lu∂lu∇Dαudx=I11+I12+I13+I14.
Integrating by parts, we observe that(14)I11≤C∑α>0m∑0<β≤m/2Dβ∂luL4Dα−β∂luL4∇um≤C∑α>0m∑0<β≤m/2Dβ+1uL4Dα−β∇uL4∇um.
Asm≥s0+1 and β+2≤m/2+2≤m/2+2≤m, it follows that(15)I11≤Cum∇um2.
Similarly,α−β+2≤m for β≥m/2+1, and thus,(16)I12≤Cum∇um2.
Now, we determine the bounds ofI13 and I14, as follows:(17)I13=∑α=0m∑l=1d∫ℝd∂luDα∂lu∇Dαudx≤C∇uL∞∇uHm2≤CuH3∇uHm2≤CuHm∇uHm2.
Along with the above estimate, we have the following:(18)I14=∑l=1d∑β=m∫ℝdDβ∂lu∂lu∇Dαudx≤CuHm∇uHm2.
ForI2, integrating by parts yields the following:(19)I2=−∑α=0m∑β>0αβ∫ℝdDβu⋅∇Dα−β∂lu⋅Dα∂ludx≤∑α=0m∑0<β≤m/2αβ∫ℝdDβu⋅∇Dα−β∂lu⋅Dα∂ludx+∑α=0m∑0≤α−β<m−3αβ∫ℝdDβu⋅∇Dα−β∂lu⋅Dα∂ludx+∑α=0m∑m−3<α−β≤m/2αβ∫ℝdDβu⋅∇Dα−β∂lu⋅Dα∂ludx=I21+I22+I23.
Estimating the integrals one by one in (19), it is easy to verify that(20)I21≤C∑α=0m∑0<β≤m/2DβuL∞Dα−β+1∇uL2∇uHm≤CuHm∇uHm2,since for β<m/2, DβuL∞≤CDβuH2uHmm≥4.
By the interpolation and Hölder inequalities, the following can be inferred:(21)I22≤C∑m−3<α−β<m/2DβuL4∇Dα−β+1uL4∇uHm≤CuHm∇uHm2.
As in the proof ofI21, the following can be verified:(22)I23≤C∑α−β≤m−3DβuL2∇Dα−β+1uL∞∇uHm≤CuHm∇uHm2.
Therefore, we have shown that(23)u⋅∇Δu,uHm≤CuHm∇uHm2.
Finally, integrating by parts yields the following:(24)∑j=1dΔuj∇uj,uHm≤∑j=1d∑β=1Δuj∇uj,∂β+1uHm−1+∑j=1dΔuj∇uj,uL2≤∑j=1dΔuj∇ujHm−1∇2uHm−1+uL∞ΔuL2∇uL2≤C∇uHm−1ΔuHm−1∇2uHm−1+uH3∇uH12≤CuHm∇uHm2.
Therefore,(25)u⋅∇Δu,uHm+∑j=1dΔuj∇uj,uHm≤CuHm∇uHm2.


## 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.
Letm>d/2+2 and u0∈Hmℝd be a divergence-free and solenoidal vector u0α,ν∈Hm+1ℝd, which can be approximated by u0 as follows:u0α,ν−u0m⟶0,asα,ν⟶0α∇u0α,νm⟶0,asα,ν⟶0
LetT∗ be the time of existence and u¯∈C0,T∗;Hm be the solution of the Euler system (3). Then, for all 0<T0<T∗, there exists α0,ν0>0 such that, for all α<α0,ν<ν0, the second-grade fluid equation (1) has a unique solution uα,ν∈C0,T0;Hm+1ℝd. Moreover,(26)α∇uα,νL∞0,T0;Hm+uα,ν−u¯L∞0,T0;Hm⟶0,asα,ν⟶0,αuα,νL∞0,T;L2+uα,ν−u¯L∞0,T0;L2≤Cνt1/2+αt1/2+u0α,ν−u0L2+α∇u0α,νL2,α∇uα,νL∞0,T;HS0+uα,ν−u¯L∞0,T;HS0≤Cνtm−S0/2+αtm−S0/2+u0α,ν−u0Hm+αu0α,νHm,for all 0<t<T0 and d/2≤S0≤m−1, and C depends only on d and T0.Remark 1.
Foru0∈Hmℝd, from [16], if we choose a suitably small ε>0 such that α/ε⟶0, then there exists a u0ε∈Hm+1ℝn such that(27)u0−u0εHm⟶0,α∇u0εHm⟶0,asα⟶0,ν⟶0.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 α and ν) from Lemma 1:(28)ddtuHm2+α2∇uHm2+ν∇uHm2≤Cus0uHm2+α2us0∇uHm2,wherem+1>s0>d/2+1. Therefore, there exists a T∗>0 that only depends on u0α,νs0, and we have a uniform bound of uα,ν for all t∈0,T∗:(29)uL∞0,t;Hm≤K,whereK is independent of α,ν and 0<α<α0,0<ν<ν0.Step 2 (convergence inL2): the difference between (1) and (3) is given by(30)u−u¯t−α2Δut−νΔu+u⋅∇u−u¯⋅∇u¯−α2u⋅∇Δu−α2∑j=1dΔuj∇uj=−∇p+12u2−p¯,whereu=uα,ν and p¯ is the pressure of the Euler equation (3).Multiplying both sides of (30) by u−u¯ and integrating in ℝd, we obtain(31)ddtu−u¯2+α2∇u2−α2∇u,∇u¯+α2∇u,∇u¯t+ν∇u2−ν∇u,∇u¯+∫ℝdu−u¯∇u¯u−u¯−α2∫ℝdu⋅∇Δuu−u¯−α2∑j=1d∫ℝdΔuj∇uju−u¯=0.It is evident that(32)α2∫ℝdu⋅∇Δuu¯dx=−α2∑i=1d∑j=1d∫ℝduiΔuj∂iu¯j=α2∑i=1d∑j=1d∑l=1d∫ℝd∂lui∂luj∂iu¯j+∫ℝdui∂luj∂l∂iu¯j≤α2∇u2∇u¯L∞+α2uL4∇uL2∇2u¯L4≤Cu¯H3α2∇u2.Similarly,(33)α2∑l=1d∫ℝn∇ui∇uju¯=α2∑j=1d−∫ℝd∂luj∇∂luju¯+∫ℝd∂luj∇uj∂lu¯≤α2∑l=1d∫ℝd∂luj∇uj∂lu¯≤C∇u¯L∞α2∇u2≤Cu¯H3α2∇u2.Equation (31) can be rewritten as follows:(34)ddtu−u¯2+α2∇u2−α2∇u,∇u¯+ν∇u2≤α2∇u∇u¯t+ν∇u∇u¯+∇u¯L∞u−u¯2+Cα2u¯H3∇u2.It can be inferred that(35)u−u¯2+α2∇u2+ν∫0t∇u2ds≤α22∇u2+α22∇u02+α22∇u0α,ν2+α22∇u¯2+ν2∫0t∇u2+ν2∫0t∇u¯2+C∫0tνH3u−u¯2+∫0tCu¯H3+1α2∇u2+α22∫0t∇u¯t2ds+u0α,ν−u¯02+α2∇u02.Lettinggt=u−u¯2+α2∇u2, (35) becomes(36)gt≤∫0tgtds+fα,ν,u0,ν0,where(37)fα,ν,u0,ν0=Cα2t+Cνt+u0α,ν−u02+α2∇u0α,ν2⟶0,asα⟶0,ν⟶0.By the Gronwall inequality, we obtain the following:(38)u−u¯2+α2∇u2≤Cα2t+Cνt+u0α,ν−u02+α2∇u02.By the energy estimate (29) and the interpolation inequality (38), the following can be obtained for any 0<s<m,0<t<T0:(39)u−u¯L∞Hs+α∇uL∞Hs≤Cαs/mts/2m+Cνts/2m+u0α,ν−u0Hm+α∇u0α,νHm,since(40)u−u¯Hs≤Cu−u¯s/mu−u¯Hm1−s/m,α∇uHs≤Cα∇us/mα∇uHm1−s/m.Step 3: (convergence inHm): From Lemma 1, we know the following:(41)u⋅∇∇u,u¯Hm≤CuHm∇uHm∇u¯Hm+2,(42)∑j=1dΔuj∇uj,u¯Hm≤CuHm∇uHmu¯Hm+2.
From the energy estimates (41) and (42), we ensure the convergence in Hm by requiring regularization of the initial data. For all δ>0, we take u0δ such that(43)u0δHm≤Cu0Hm,u0δHm+k≤Cδk,k∈ℤ+.
For anys, such that d/2<s<m−1, we have(44)u0δ−u0Hs≤Cδm−s,u0δ−u0Hm⟶0,as δ⟶0. The existence of u0δ can be constructed by the following previously reported approaches [4, 16]. The maximum existence time T0 of the Euler equations only depends on u0s0s0=d/2+1. Thus, for any δ>0, there exists a uniform interval 0,T0 in which the solution u¯δ of the Euler equations exists. Furthermore, u¯δHm≤C, u¯δHm+k≤C/δkk=1,2…, and u¯tδHm+1≤C/δ.
Similar to a previous report [4], for any d/2<s<m−1, we have(45)u¯δ−u¯L∞Hm≤Cu0δ−u0Hm+δm−s−1T.
For equation (30), we have an energy estimate for wν,α,δ=uν,α−u¯δ, and we obtain the following (dropping ν,α and δ):(46)ddtwHm2+α2∇uHm2−α2∇u,∇u¯δHm+ν∇uHm2+α2∇u,∇u¯tδHm≤ν∇u,∇u¯δHm+CwL∞u¯δHm+1wHs+u¯δHm+uHmwHm2+Cα2uHm∇uHm∇u¯δHm+2+u¯δHm+2,since(47)wL∞u¯δHm+1wHm≤uα,ν−u¯L∞u¯δHm+1wHm+u¯−u¯δL∞u¯δHm+1wHm.
From Step 2,uα,ν converges to u¯ in Hm−1, and we deduce the following:(48)uα,ν−u¯L∞≤Cuα,ν−u¯L∞Hm+1≤Cαm−1/mtm−1/2m+Cνtm−1/2m+g1/20,where g0=u0α,ν−u0m2+α∇u0α,νm2. On contrary, from (45), we know that(49)u¯−u¯δL∞≤Cu0δ−u0Hm+δm−s′T:=f˜δ,for some s0<s′<m−1. Combining estimates (48) and (49) and using (46) and the Gronwall inequality, we obtain(50)wHm2+α2∇uHm2≤α2u¯δH3+ν∫0T∇u¯δHm2+α2∫0T∇uHm+α2∫0T∇u¯Hmds+∫0Tfα,ν,u0,ν0δwHm+∫0TCα2∇uHmu¯Hm+3+∫0Tf˜δwHm+∫0TCwHm2,where fα,ν,u0,ν0 is defined by (37). It follows that(51)u−u¯L∞0,T0;Hm2+α2∇uL∞0;T0;Hm2≤CT0α2δ+νδ2+fα,ν,u0,ν0δ+α2δ3+u0δ−u0Hm+f2δ,where δ is chosen such that α2/δ3, ν/δ2, and u0α,ν−u0Hm/δ are sufficiently small. Letting α,ν⟶0 and δ⟶0, the following can be shown:(52)limα,ν⟶0uα,β−u¯Hm2+α2∇uHm2=0.
Using the continuous method, we ensure convergence on any time interval0,T0 for T0<T∗, where T∗ is the maximum existence time of the Euler equations. This completes the proof.



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