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Remark on the local well-posedness of compressible non-Newtonian fluids with initial vacuum

Hind Al Baba, Bilal Al Taki, Amru Hussein

TL;DR

The paper proves local-in-time existence and uniqueness of strong solutions to the compressible non-Newtonian Navier–Stokes equations on the 3D torus with initial vacuum. It extends Newtonian results by incorporating a nonlinear elliptic regularity framework for the velocity and imposing a $W^{2,p}$ regularity assumption for the nonlinear elliptic system, enabling vacuum at $t=0$. The authors construct approximate solutions, obtain uniform a priori estimates, and pass to limits to handle vacuum and nonlinearity, finally proving a blow-up criterion that ties the maximal existence time to certain Sobolev norms of the density and velocity. This work provides a robust local well-posedness theory for non-Newtonian compressible fluids in a periodic setting, bridging gaps between Newtonian and non-Newtonian analyses with vacuum.

Abstract

We discuss in this short note the local-in-time strong well-posedness of the compressible Navier-Stokes system for non-Newtonian fluids on the three dimensional torus. We show that the result established recently by Kalousek, Mácha, and Nečasova in \doi{10.1007/s00208-021-02301-8} can be extended to the case where vanishing density is allowed initially. Our proof builds on the framework developed by Cho, Choe, and Kim in \doi{10.1016/j.matpur.2003.11.004} for compressible Navier-Stokes equations in the case of Newtonian fluids. To adapt their method, special attention is given to the elliptic regularity of a challenging nonlinear elliptic system. We show particular results in this direction, however, the main result of this paper is proven in the general case when elliptic $W^{2,p}$-regularity is imposed as an assumption. Also, we give a finite time blow-up criterion.

Remark on the local well-posedness of compressible non-Newtonian fluids with initial vacuum

TL;DR

The paper proves local-in-time existence and uniqueness of strong solutions to the compressible non-Newtonian Navier–Stokes equations on the 3D torus with initial vacuum. It extends Newtonian results by incorporating a nonlinear elliptic regularity framework for the velocity and imposing a regularity assumption for the nonlinear elliptic system, enabling vacuum at . The authors construct approximate solutions, obtain uniform a priori estimates, and pass to limits to handle vacuum and nonlinearity, finally proving a blow-up criterion that ties the maximal existence time to certain Sobolev norms of the density and velocity. This work provides a robust local well-posedness theory for non-Newtonian compressible fluids in a periodic setting, bridging gaps between Newtonian and non-Newtonian analyses with vacuum.

Abstract

We discuss in this short note the local-in-time strong well-posedness of the compressible Navier-Stokes system for non-Newtonian fluids on the three dimensional torus. We show that the result established recently by Kalousek, Mácha, and Nečasova in \doi{10.1007/s00208-021-02301-8} can be extended to the case where vanishing density is allowed initially. Our proof builds on the framework developed by Cho, Choe, and Kim in \doi{10.1016/j.matpur.2003.11.004} for compressible Navier-Stokes equations in the case of Newtonian fluids. To adapt their method, special attention is given to the elliptic regularity of a challenging nonlinear elliptic system. We show particular results in this direction, however, the main result of this paper is proven in the general case when elliptic -regularity is imposed as an assumption. Also, we give a finite time blow-up criterion.
Paper Structure (10 sections, 7 theorems, 82 equations)

This paper contains 10 sections, 7 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Second order $L^{p}$-estimates for the non-linear elliptic system
  4. Proof of Theorem \ref{['Main-Theorem']}
  5. Construction of approximate solutions
  6. Uniform estimates of approximate solutions
  7. Convergence of approximate solutions as $k\to\infty$
  8. Conclusion of the existence proof: $\delta\to 0$
  9. Uniqueness and continuous dependence on the data
  10. Blow-up criterion

Key Result

Theorem 2.2

Let $d=3$, Suppose that $\mu\in C^1([0,\infty),\mathbb{R})$ and $\lambda\in C^1(\mathbb{R},\mathbb{R})$ are functions satisfying eq:ellipticity2--eq:ellipticity such that Assumption assumption-elliptic-reg holds for $p=2$ and $p=q_0$, and let $p(\cdot)\in C^{1}([0, \infty),\mathbb{R}^+)$. Assume that the data and the compatibility condition Then there exist a time $T_{\ast}\in (0,T]$ and a uniq

Theorems & Definitions (15)

  • Theorem 2.2: Main result
  • Remark 2.3: Notion of strong solutions
  • Proposition 3.1: Second order $L^p$-estimate under smallness conditions
  • Example 3.2
  • proof : Proof of Proposition \ref{['elliptic-estimates_vega']}
  • Proposition 3.3: Second order $L^p$-estimate in the 1-dimensional case
  • proof
  • Proposition 3.4: Second order $L^2$-estimate in the general case
  • proof
  • Lemma 4.1
  • ...and 5 more