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Non-existence of classical solutions to a two-phase flow model with vacuum

Hai-Liang Li, Yuexun Wang, Yue Zhang

TL;DR

This work demonstrates that classical solutions in the inhomogeneous Sobolev setting cannot persist for the 1D pressureless Euler–Navier–Stokes two-phase flow with vacuum, under suitable initial data near vacuum. By transforming to Lagrangian coordinates and analyzing a degenerate parabolic momentum operator via Hopf's lemma and the strong maximum principle, the authors derive a contradiction that rules out $C^1([0,T];H^m)$ solutions for any $T>0$. The results hinge on the geometric arrangement of the initial densities' supports and sign conditions on the initial velocities, providing two complementary nonexistence regimes. The techniques rely on maximum-principle arguments for degenerate operators and extend to related coupled fluid models, clarifying ill-posedness in inhomogeneous Sobolev spaces for multi-phase flows with vacuum.

Abstract

In this paper, we study the well-posedness of classical solutions to a two-phase flow model consisting of the pressureless Euler equations coupled with the isentropic compressible Navier-Stokes equations via a drag forcing term. We consider the case that the fluid densities may contain a vacuum, and the viscosities are density-dependent functions. Under suitable assumptions on the initial data, we show that the finite-energy (i.e., in the inhomogeneous Sobolev space) classical solutions to the Cauchy problem of this coupled system do not exist for any small time.

Non-existence of classical solutions to a two-phase flow model with vacuum

TL;DR

This work demonstrates that classical solutions in the inhomogeneous Sobolev setting cannot persist for the 1D pressureless Euler–Navier–Stokes two-phase flow with vacuum, under suitable initial data near vacuum. By transforming to Lagrangian coordinates and analyzing a degenerate parabolic momentum operator via Hopf's lemma and the strong maximum principle, the authors derive a contradiction that rules out solutions for any . The results hinge on the geometric arrangement of the initial densities' supports and sign conditions on the initial velocities, providing two complementary nonexistence regimes. The techniques rely on maximum-principle arguments for degenerate operators and extend to related coupled fluid models, clarifying ill-posedness in inhomogeneous Sobolev spaces for multi-phase flows with vacuum.

Abstract

In this paper, we study the well-posedness of classical solutions to a two-phase flow model consisting of the pressureless Euler equations coupled with the isentropic compressible Navier-Stokes equations via a drag forcing term. We consider the case that the fluid densities may contain a vacuum, and the viscosities are density-dependent functions. Under suitable assumptions on the initial data, we show that the finite-energy (i.e., in the inhomogeneous Sobolev space) classical solutions to the Cauchy problem of this coupled system do not exist for any small time.
Paper Structure (9 sections, 11 theorems, 122 equations)

This paper contains 9 sections, 11 theorems, 122 equations.

Table of Contents

  1. Introduction
  2. main results
  3. Results
  4. Remarks
  5. Lagrangian formulation
  6. Proof of Theorem \ref{['thm1']}
  7. The Hopf's lemma
  8. The strong maximum principle
  9. Proof of Theorem \ref{['thm2']}

Key Result

Theorem 2.1

Assume that the initial data $(\rho_0,u_0,n_0,w_0)$ satisfies support and inda0. When $\Omega_1\subsetneqq\Omega_2$, if there exist some constants $p_0>0$, $d_0\in(0,1)$ such that or When $\Omega_1=\Omega_2$, except inda1--inda2, if the initial data additionally satisfies Then main0--initial does not admit any solution $(\rho,u,n,w)\in C^1(0,T;H^m(\mathbb{R}))$ with $m>2$ for any positive time

Theorems & Definitions (23)

  • Theorem 2.1
  • Theorem 2.2
  • Remark 1
  • Remark 2
  • Remark 3
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • ...and 13 more