Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A well-posedness result for the compressible two-fluid model with density-dependent viscosity

Sagbo Marcel Zodji

TL;DR

This work addresses local-in-time well-posedness for a compressible two-fluid model with density-dependent viscosity in the whole space, featuring a sharp, moving interface and piecewise Hölder regular densities. The authors recast the problem in Lagrangian coordinates to fix the domain, develop a linear theory with precise a priori estimates, and then implement a nonlinear fixed-point argument to obtain a unique solution with controlled interface regularity. A key novelty is handling density-dependent viscosities across a discontinuous interface using piecewise Hölder and Hoff-type estimates for the velocity and its derivatives, together with quantitative piecewise Hölder bounds for even-order Riesz transforms. The results extend prior Hölder-interface work (notably Tani) to rougher initial data and interface regularity, laying a rigorous groundwork for potential global-in-time analysis under small data and providing a robust framework for the dynamics of density patches in multi-fluid systems.

Abstract

In this paper, we study a system of PDEs describing the motion of two compressible viscous fluids occupying the whole space $\mathbb R^d\;(d\in \{2,3\}$). The two phases of the mixture are separated by a $\mathscr{C}^{1+α}$-regular sharp interface $\mathcal{C}$ across which the density can experience jumps. We prove the existence of a unique local-in-time solution assuming that the initial density is $α$-Hölder continuous on both sides of $\mathcal{C}$. The initial velocity belongs to the Sobolev space $H^1(\mathbb R^d)$, and the divergence of the initial stress tensor belongs to $L^2(\mathbb R^d)$. The later assumption expresses somehow the continuity of the stress tensor. This result is more general than the one by Tani [32], as it allows for less regular initial data and furthermore it can serve as a building block for the construction of global-in-time solutions.

A well-posedness result for the compressible two-fluid model with density-dependent viscosity

TL;DR

This work addresses local-in-time well-posedness for a compressible two-fluid model with density-dependent viscosity in the whole space, featuring a sharp, moving interface and piecewise Hölder regular densities. The authors recast the problem in Lagrangian coordinates to fix the domain, develop a linear theory with precise a priori estimates, and then implement a nonlinear fixed-point argument to obtain a unique solution with controlled interface regularity. A key novelty is handling density-dependent viscosities across a discontinuous interface using piecewise Hölder and Hoff-type estimates for the velocity and its derivatives, together with quantitative piecewise Hölder bounds for even-order Riesz transforms. The results extend prior Hölder-interface work (notably Tani) to rougher initial data and interface regularity, laying a rigorous groundwork for potential global-in-time analysis under small data and providing a robust framework for the dynamics of density patches in multi-fluid systems.

Abstract

In this paper, we study a system of PDEs describing the motion of two compressible viscous fluids occupying the whole space ). The two phases of the mixture are separated by a -regular sharp interface across which the density can experience jumps. We prove the existence of a unique local-in-time solution assuming that the initial density is -Hölder continuous on both sides of . The initial velocity belongs to the Sobolev space , and the divergence of the initial stress tensor belongs to . The later assumption expresses somehow the continuity of the stress tensor. This result is more general than the one by Tani [32], as it allows for less regular initial data and furthermore it can serve as a building block for the construction of global-in-time solutions.
Paper Structure (40 sections, 18 theorems, 367 equations)

This paper contains 40 sections, 18 theorems, 367 equations.

Table of Contents

  1. Introduction and main result
  2. Presentation of the model
  3. Piecewise regularity
  4. Reformulation of the system
  5. Review of known results
  6. Main result
  7. Change of variables in Lagrangian coordinates
  8. Study of the linearised system
  9. Statement of the main result
  10. A priori estimates
  11. Proof of $\mathscr{E}\neq \emptyset$
  12. Sketch of the proof
  13. Detailed proof of \ref{['th2']} and \ref{['lemA3']}
  14. A priori estimates
  15. Closing of the homotopy argument
  16. ...and 25 more sections

Key Result

Theorem 1.1

[theo]thlocal Let $(c_0, \rho_0, u_0)$ be initial data associated with the system ep1.1, satisfying the conditions c4.36, epq2 and epq1. Additionally, assume: There exists $[\mu]>0$ depending only on the dimension $d\in \{2,3\}$, $\alpha$ and $\widetilde{\mu}$ such that if We refer to c3.34-c3.55 for the definition of $\ell_{\varphi_0}$ and $\mathfrak{P}_{\mathcal{C}_0}$. then, there exist a time

Theorems & Definitions (24)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Corollary 1.5: Blow-up criterion
  • Remark 1.6
  • Remark 1.7
  • Theorem 2.2
  • Theorem 3.1
  • Remark 3.2
  • ...and 14 more