# Regularity for 3D Inhomogeneous Naiver–Stokes Equations in Vishik Spaces

Authors: Jae-Myoung Kim
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In this short paper, we consider the conditional regularity for the 3D inhomogeneous incompressible Navier–Stokes equations in Vishik spaces and give regularity criterion of strong solutions.


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## Body

## 1. Introduction

We consider the regularity issue for solutionsρ,u,Π:QT⟶ℝ×ℝ3×ℝ to 3D inhomogeneous incompressible Navier–Stokes equations forQT≔ℝ3×0,T:
(1)∂tρ+u·∇ρ=0,(2)ρut−Δu+ρu·∇u+∇Π=0,divu=0.Here,ρ is the density function of flow velocity, u is the flow velocity, and Π is the pressure. We consider the initial value problem of (1), which requires initial
(3)ρx,0=ρ0x,ux,0=u0x,x∈ℝ3.There is a very rich literature dedicated to the study of the above system. In the case of smooth data with no vacuum, Kazhikov [1] proved that the nonhomogeneous Navier–Stokes equations have at least one global weak solution in the energy space. When the initial data may contain vacuum states, Simon [2] proved the global existence of a weak solution to the equations of incompressible, viscous, nonhomogeneous fluid flow in a bounded domain of two or three spaces, under the no-slip boundary condition. Choe and Kim [3] proposed a compatibility condition and investigated the local existence of strong solutions. More precisely, under the compatibility condition,
(4)Δu0−Π0=ρ01/2ganddivu0=0,fora.e.x∈Ω,For initial data,(5)0≤ρ0∈L3/2∩L∞∩H1Ωu0∈H1∩H2Ω,they proved the local-in-time existence for solutions in the class
(6)ρ∈L∞0,T∗;H1Ω,ρt∈L∞0,T∗;L2Ω,(7)∇3u∈L20,T∗;L3/2,∇Π∈L∞0,T∗,L2Ω∩L20,T∗;L6∩W1,3/2Ω.Here,Ω⊆ℝ3 is a bounded domain or whole space. After that, Craig et al. [4] improved the above result to global strong small solutions. Very recently, without compatibility conditions, for any initial data ρ0,u0∈W1,γ∩L∞×H0,σ1 with γ>1, Li showed the existence of local strong solution for the initial-boundary value problem to the nonhomogeneous incompressible Navier–Stokes equations in the class
(8)ρ∈L∞0,T;W1,γ∩L∞∩C0,T,LγΩ,u∈L∞0,T,H0,σ1Ω∩L20,T,H2Ω,ρu∈C0,T,L2Ω,tu∈L∞0,T,H2Ω∩L20,T,W2,6,t∂tu∈L20,T,H1Ω.Moreover, ifγ≥2, then, the strong solution is unique.On the other hand, for the regularity issue to system (1)–(3), Kim [5] proved the following regularity condition:
(9)u∈Ls0,T;Lp,∞ℝ3,2s+3p=1,3<p≤∞.And Zhou and Fan [6] showed the following regularity condition:
(10)u∈L2/1−r0,T,M.2,3/rℝ3,with0<r<1.Here,M.2,3/rℝ3 stands for the homogeneous Morrey space (see Appendix).Before stating our result, we now introduce a Banach spaceV̇p,σ,θs which is larger than the homogeneous Besov space; see [7, 8].Definition 1.
Lets∈ℝ,p,σ∈1,∞,θ∈1,σ; the Vishik space V̇p,σ,θs is defined by
(11)V̇p,σ,θsℝ3≔f∈D′ℝ3:fV̇p,σ,θs<∞,with the norm
(12)fV̇p,σ,θsℝ3≔supN=1,2,⋯,∑j<N2jsθΔ̇jfLpθ1/θN1/θ−1/σ,θ≠∞,and if θ=∞, fV̇p,σ,θsℝ3≔fḂp,∞0ℝ3.Here,D′ℝ3 is the dual space of
(13)Dℝ3=f∈Sℝ3;Dαf^0=0,∀α∈ℕ3.Motivated by [7, 9], now, we are ready to state our first main result.Theorem 2.
LetT>0. Assume that the initial data ρ0,u0 satisfy the initial condition (5) and the compatibility condition (4). Let ρ,u be the corresponding unique local strong solution to system (1)–(3) with the properties stated in (6). If additionally for all t∈0,T(14)∫0tuτV̇p,σ,θ02p/p−3dτ<∞,3<p≤∞,σ,θ∈1,∞,then, the solution ρ,u can be extended smoothly beyond time t=T.Remark 3.
As mentioned in [9], we remind that the following continuous embeddings hold:
(15)Ḃp,σsℝ3=V̇p,σ,σsℝ3⊂V̇p,σ,θ1sℝ3⊂V̇p,σ,θ2sℝ3⊂V̇p,σ,1sℝ3,for s∈ℝ, p,σ∈1,∞, and θ1,θ2∈1,σ with θ1≥θ2. For this reason, (13) is a stronger condition than
(16)∫0tuτḂp,∞02p/p−3dτ<∞,3<p≤∞,σ,θ∈1,∞.Remark 4.
By the same calculations as those in [10, 11], for the initial data ρ0,u0 satisfying that ρ∈L6/5ℝ3,ρ0u02∈L1ℝ3,ρ0u0∈L1ℝ3,and∫ℝ3ρ0u0dx≠0, there exists no global-in-time smooth solution to Cauchy problem (1)–(3).


## 2. Proof of Theorem2

We first introduce some notations. LetX,· be a normed space. By Lq0,T;X, we denote the space of all Bochner measurable functions φ:0,T⟶X such that
(17)φLq0,T;X≔∫0Tφtqdt1/q<∞,1≤q<∞,φL∞0,T;X≔supt∈0,Tφt<∞,q=∞.Unless specifically mentioned, letterC is used to represent a generic constant, which may change from line to line.Proof.
By the maximum principle, we note that(18)sup0≤t≤Tρ·,tL∞≤ρ0L∞<∞.
And also, byL2-energy estimate, we know that
(19)sup0≤t≤Tρ1/2u·,tL2ℝ32+2∫0T∇u·,tL2ℝ32dt≤Cρ01/2u0L2ℝ32.
To exclude the pressure term, multiplying1.12 by ut and using Hölder’s inequality, we get
(20)12ddt∇uL22+∫ℝ3ρut2dx≤∫ℝ3ρ1/2u·∇u·ρ1/2utdx≤∫ℝ3ρ1/2∑j<−NΔju∇uρ1/2utdx+∫ℝ3ρ1/2∑j=−Nj=NΔju∇uρ1/2utdx+∫ℝ3ρ1/2∑j>−NΔju∇uρ1/2utdx≔I+II+III,where we use the decomposition of u. Let us control each term sequentially: the term (I):
(21)I≤ρ1/2L∞∑j<−NΔjuL∞∇uL2ρ1/2utL2≤C∑j<−N23/2juL2∇uL2ρ1/2utL2≤C2−3/2Nxρ1/2utL2∇uL2ρ1/2utL2≤C2−3N2uL22∇uL22+132ρ1/2utL22,the term (II):
(22)II≤∑j=−Nj=Nρ12ΔjuLp∇uLp−2/2pρ1/2utL2≤C∑j=−Nj=NΔjuLp∇uL21−3/p∇2uL23/pρ1/2utL2≤CN1−1/σsupN=1,2,⋯∑j=−Nj=NΔ̇juLpN1−1/σ∇uL21−3/p∇2uL23/pρ1/2utL2≤CN1−1/σsupN=1,2,⋯∑j=−Nj=NΔ̇juLpN1−1/σ2∇uL22−6/p∇2uL26/p+132ρ1/2utL2≤CNuV̇p,σ,102p/p−3∇uL22+132ρ1/2utL2+∇2uL22,and the term (III):
(23)III≤∑j>Nρ1/2ΔjuL3∇uL6ρ1/2utL2≤C∇uL6∑j>N21/2juL2ρ1/2utL2≤C2−N/2uL2∇2uL2ρ1/2utL2≤2−N2uL22∇2uL22+132ρ1/2utL22.
Summing up the estimate above with the energy estimate, we get(24)ddtρ1/2uL22+∇uL22+∫ℝ3∇u2+ρut2dx≤C2−3N2uL22∇uL22+CNuV̇p,σ,102p/p−3∇uL22+C2−N2uL22∇2uL22+116∇2uL22.
On the other hand, we note that(25)∇2uL2ℝ32≤CρutL2ℝ32+ρu·∇uL2ℝ32≤CρutL2ℝ32+C2−3N2uL22∇uL22+C2−N2uL22∇2uL22+CNuV̇p,σ,102p/p−3∇uL22.
Collecting (23) and (24), we have
(26)ddtρ1/2uL22+∇uL22+∫ℝ3∇u2+∇2u2+ρut2dx≤CρutL2ℝ32+C2−3N2uL22∇uL22+C2−N2uL22∇2uL22+CNuV̇p,σ,102p/p−3∇uL22.
Now, choosingN>0 sufficiently large such that C2−N2uL22≤1/128, (indeed, the constant C>0 is also depending on ρ01/2u0L22), the estimate (25) becomes
(27)ddtρ1/2uL22+∇uL22+∫ℝ3∇u2+∇2u2+ρut2dx≤CNuV̇p,σ,102p/p−3∇uL22.
By Grönwall’s inequality under assumption (13), we obtain
(28)ρ1/2u,∇u∈L∞0,T;L2ℝ3,∇u,∇2u,ρ1/2ut∈L20,T;L2ℝ3.
Lastly, according to the arguments in [6], Lemma 2.3, differentiating 12 with respect to time t and multiplying the equations by ut, we can obtain
(29)ρ1/2ut∈L∞0,T;L2ℝ3,∇ut∈L20,T;H1ℝ3.
This is the desired result, and thus, the proof in Theorem2 is completed.



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*Source: 7061004-2022-03-12.xml*