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Existence and Incompressible Limit of Weak and Classical Solutions to the Cauchy Problem for Compressible Navier-Stokes Equations with Large Bulk Viscosity and Large Initial Data

Qinghao Lei, Chengfeng Xiong

TL;DR

This work proves global well-posedness for the 2D Cauchy problem of the barotropic compressible Navier–Stokes equations under large bulk viscosity, covering both vacuum and non-vacuum far-field states and allowing arbitrarily large initial data. It develops comprehensive a priori estimates, including density upper bounds via the effective viscous flux and Hoff's method, to extend local solutions to global ones in both vacuum and away-from-vacuum regimes. The key contribution is establishing the incompressible limit: as the bulk viscosity \\nu tends to infinity, solutions converge to those of the inhomogeneous incompressible Navier–Stokes equations, with convergence holding even when the initial velocity is not divergence-free; in the presence of additional regularity, the full sequence can converge to the unique incompressible limit. These results deepen the understanding of global dynamics and singular limits for compressible flows with large viscosity in the planar setting and provide a robust framework for large-data analysis in 2D.

Abstract

This paper investigates the Cauchy problem for the barotropic compressible Navier-Stokes equations in $\mathbb{R}^2$ with the constant state as far field, which could be vacuum or non-vacuum. Under the assumption of a sufficiently large bulk viscosity coefficient, we establish the global existence of weak, strong, and classical solutions for the non-vacuum far-field case, and long-time existence for the vacuum far-field case. It should be mentioned that this result is obtained without any restrictions on the size of the initial data. Moreover, we demonstrate that the solutions of the compressible Navier-Stokes equations converge to solutions of the inhomogeneous incompressible Navier-Stokes equations, as the bulk viscosity coefficient tends to infinity. The incompressible limit of the weak solutions holds even without requiring the initial velocity to be divergence-free.

Existence and Incompressible Limit of Weak and Classical Solutions to the Cauchy Problem for Compressible Navier-Stokes Equations with Large Bulk Viscosity and Large Initial Data

TL;DR

This work proves global well-posedness for the 2D Cauchy problem of the barotropic compressible Navier–Stokes equations under large bulk viscosity, covering both vacuum and non-vacuum far-field states and allowing arbitrarily large initial data. It develops comprehensive a priori estimates, including density upper bounds via the effective viscous flux and Hoff's method, to extend local solutions to global ones in both vacuum and away-from-vacuum regimes. The key contribution is establishing the incompressible limit: as the bulk viscosity \\nu tends to infinity, solutions converge to those of the inhomogeneous incompressible Navier–Stokes equations, with convergence holding even when the initial velocity is not divergence-free; in the presence of additional regularity, the full sequence can converge to the unique incompressible limit. These results deepen the understanding of global dynamics and singular limits for compressible flows with large viscosity in the planar setting and provide a robust framework for large-data analysis in 2D.

Abstract

This paper investigates the Cauchy problem for the barotropic compressible Navier-Stokes equations in with the constant state as far field, which could be vacuum or non-vacuum. Under the assumption of a sufficiently large bulk viscosity coefficient, we establish the global existence of weak, strong, and classical solutions for the non-vacuum far-field case, and long-time existence for the vacuum far-field case. It should be mentioned that this result is obtained without any restrictions on the size of the initial data. Moreover, we demonstrate that the solutions of the compressible Navier-Stokes equations converge to solutions of the inhomogeneous incompressible Navier-Stokes equations, as the bulk viscosity coefficient tends to infinity. The incompressible limit of the weak solutions holds even without requiring the initial velocity to be divergence-free.

Paper Structure

This paper contains 9 sections, 35 theorems, 265 equations.

Table of Contents

  1. Introduction and main results
  2. Preliminaries
  3. A Priori Estimates (I): Lower Order Estimates
  4. Far-field density is vacuum
  5. Far-field density is away from vacuum
  6. A Priori Estimates II: Higher Order Estimates
  7. Far-field density is vacuum
  8. Far-field density is away from vacuum
  9. Proofs of Theorem \ref{['th0']}--\ref{['th3']}

Key Result

Theorem 1.1

Assume the initial data $(\rho_0,u_0)$ satisfy and for some $a>1$ where (1) Far-field density is vacuum: For any $0<T<\infty$, there exists a positive constant $\nu_1$ depending only on $T$, $N_0$, $\gamma$, $\mu$, $E_0$, $\|{\bar{x}}^a \rho_0\|_{L^1}$, $\|\rho_0\|_{L^\infty}$, and $\| \nabla u_0\|_{L^2}$ such that when $\nu \ge \nu_1$, the problem ns--i3 with $\tilde and for any $1 \le p <\in

Theorems & Definitions (58)

  • Definition 1.1
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.1
  • Remark 1.2
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Remark 1.3
  • ...and 48 more