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

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,


Introduction
We consider the regularity issue for solutions ðρ, u, ΠÞ: Q T ⟶ ℝ × ℝ 3 × ℝ to 3D inhomogeneous incompressible Navier-Stokes equations for Q T ≔ ℝ 3 × ½0, TÞ: 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 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, Δu 0 − Π 0 = ρ 1/2 0 g and div u 0 = 0, for a:e: For initial data, they proved the local-in-time existence for solutions in the class Here, Ω ⊆ ℝ 3 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 ðρ 0 , u 0 Þ ∈ ðW 1,γ ∩ L ∞ Þ × H 1 0,σ 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 Moreover, if γ ≥ 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: And Zhou and Fan [6] showed the following regularity condition: Here, M : 2,3/r ðℝ 3 Þ stands for the homogeneous Morrey space (see Appendix).
Before stating our result, we now introduce a Banach space _ V s p,σ,θ which is larger than the homogeneous Besov space; see [7,8].
with the norm Motivated by [7,9], now, we are ready to state our first main result. Theorem 2. Let T > 0. Assume that the initial data ðρ 0 , u 0 Þ satisfy the initial condition (5) and the compatibility condition (4). Let ðρ, uÞ be the corresponding unique local strong solution to system (1)-(3) with the properties stated in (6). If additionally for all t ∈ ½0, TÞ then, the solution ðρ, uÞ can be extended smoothly beyond time t = T.

Proof of Theorem 2
We first introduce some notations. Let ðX, k·kÞ be a normed space. By L q ð0, T ; XÞ, we denote the space of all Bochner measurable functions φ : ð0, TÞ ⟶ X such that Unless specifically mentioned, letter C is used to represent a generic constant, which may change from line to line.
Proof. By the maximum principle, we note that And also, by L 2 -energy estimate, we know that To exclude the pressure term, multiplying ð1:1Þ 2 by u t and using Hölder's inequality, we get 2 Journal of Function Spaces where we use the decomposition of u. Let us control each term sequentially: the term (I): the term (II): and the term (III): Summing up the estimate above with the energy estimate, we get On the other hand, we note that Collecting (23) and (24), we have Now, choosing N > 0 sufficiently large such that C2 −N 2 kuk 2 L 2 ≤ 1/128, (indeed, the constant C > 0 is also depending on kρ 1/2 0 u 0 k 2 L 2 ), the estimate (25) becomes By Grönwall's inequality under assumption (13), we obtain ρ 1/2 u, ∇u ∈ L ∞ 0, T ; L 2 ℝ 3 À Á À Á ,∇u, ∇ 2 u, ρ 1/2 u t ∈ L 2 0, T ; L 2 ℝ 3 À Á À Á : Lastly, according to the arguments in [6], Lemma 2.3, differentiating ð1Þ 2 with respect to time t and multiplying the equations by u t , we can obtain ρ 1/2 u t ∈ L ∞ 0, T ; L 2 ℝ 3 À Á À Á , ∇u t ∈ L 2 0, T ; This is the desired result, and thus, the proof in Theorem 2 is completed.