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Shinbrot Type Criteria for Energy Conservation of the Compressible Navier-Stokes Equations

Ruxuan Chen, Qi Zhang, Zhikang Zhang, Xiongbo Zheng

TL;DR

This work proves that weak solutions to the compressible Navier–Stokes equations satisfy the energy equality under Shinbrot-type regularity criteria, extending to both constant and degenerate viscosity. The authors develop a two-step framework: first, a local energy equality via a novel mollification-based test function and a weak-type temporal commutator to manage time irregularities; second, a global energy balance obtained through spatial cutoffs and careful limiting. The results relax previous Serrin- and Besov-type conditions, showing energy conservation under weaker integrability for the velocity, while accommodating boundary effects and degeneracy in viscosity. The methods advance the Onsager-type understanding in the compressible setting and provide a robust tool for establishing energy conservation in bounded domains and periodic settings, with potential implications for turbulence theory and numerical analysis of compressible flows.

Abstract

We prove that weak solutions to the compressible Navier-Stokes equations satisfy the energy equality under a Shinbrot-type regularity criterion. Our method applies to the fluids with both constant and degenerate viscosity and relies on a novel weak-type commutator estimate. These criterion are strictly weaker than those required in prior works [Arch. Ration. Mech. Anal., 225 (2017)] and [SIAM J. Math. Anal. 52 (2020)].

Shinbrot Type Criteria for Energy Conservation of the Compressible Navier-Stokes Equations

TL;DR

This work proves that weak solutions to the compressible Navier–Stokes equations satisfy the energy equality under Shinbrot-type regularity criteria, extending to both constant and degenerate viscosity. The authors develop a two-step framework: first, a local energy equality via a novel mollification-based test function and a weak-type temporal commutator to manage time irregularities; second, a global energy balance obtained through spatial cutoffs and careful limiting. The results relax previous Serrin- and Besov-type conditions, showing energy conservation under weaker integrability for the velocity, while accommodating boundary effects and degeneracy in viscosity. The methods advance the Onsager-type understanding in the compressible setting and provide a robust tool for establishing energy conservation in bounded domains and periodic settings, with potential implications for turbulence theory and numerical analysis of compressible flows.

Abstract

We prove that weak solutions to the compressible Navier-Stokes equations satisfy the energy equality under a Shinbrot-type regularity criterion. Our method applies to the fluids with both constant and degenerate viscosity and relies on a novel weak-type commutator estimate. These criterion are strictly weaker than those required in prior works [Arch. Ration. Mech. Anal., 225 (2017)] and [SIAM J. Math. Anal. 52 (2020)].
Paper Structure (7 sections, 6 theorems, 76 equations, 1 figure)

This paper contains 7 sections, 6 theorems, 76 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Methodology
  3. Main results
  4. Preliminaries
  5. Local energy equality of the compressible Navier-Stokes equations with constant viscosity
  6. Global energy equality of the compressible Navier-Stokes equations with constant viscosity
  7. The energy conservation for the compressible Navier-Stokes equations with degenerate viscosity

Key Result

Theorem 1.1

Let $(\rho, u)$ be a weak solution of eq:comNS in the sense of Definition defcomNS. Assume that the density satisfies and the initial velocity satisfies If the velocity satisfies then the energy equality eq:energy_comNS holds for any $t \in[0, T]$.

Figures (1)

  • Figure 1: The shaded region illustrates the exponent ranges covered by Theorem \ref{['theoremcomNS']}.The dashed and grid subregions represent new ranges of exponents not obtained before in the literature, where the grid area only corresponds to bounded domain with no-slip boundary and the dashed area corresponds to both bounded domain and torus.

Theorems & Definitions (17)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Definition 2.1: adams2003sobolev
  • Definition 2.2
  • Lemma 2.3
  • Lemma 2.4: mellet2007barotropic
  • Lemma 2.5
  • Remark 2.6
  • ...and 7 more