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Global weak solutions of the Navier-Stokes-Korteweg Equations in one dimension

Paolo Antonelli, Didier Bresch, Stefano Spirito

TL;DR

This work proves the global existence of finite-energy weak solutions to the one-dimensional Navier–Stokes–Korteweg equations with density-dependent viscosity and capillarity, allowing strong degeneracy (no upper bound on the viscosity exponent). Central to the result is a generalized strong coercivity condition and a truncation-based compactness framework that preserves the BD entropy and energy structure. An approximating system with mollified coefficients is shown to satisfy uniform energy and BD bounds, enabling passage to the limit and construction of a global weak solution. The approach extends prior results by relaxing the degeneracy constraints and provides a robust toolkit for NSK in 1D, with potential extensions to broader coefficient families via the generalized SCC.

Abstract

We prove the global existence of weak solutions of the one-dimensional Navier-Stokes-Korteweg (NSK) equations when the viscosity and the capillarity coefficients are power functions of the density, which may be zero on a set with positive measure. The proofs are based on a truncation argument combined with the Energy estimate and BD Entropy. Notably, we do not require any upper bound on the exponent of the power of the viscosity coefficient. In particular, we are able to consider very degenerate viscosity coefficient and to substantially improve previous results.

Global weak solutions of the Navier-Stokes-Korteweg Equations in one dimension

TL;DR

This work proves the global existence of finite-energy weak solutions to the one-dimensional Navier–Stokes–Korteweg equations with density-dependent viscosity and capillarity, allowing strong degeneracy (no upper bound on the viscosity exponent). Central to the result is a generalized strong coercivity condition and a truncation-based compactness framework that preserves the BD entropy and energy structure. An approximating system with mollified coefficients is shown to satisfy uniform energy and BD bounds, enabling passage to the limit and construction of a global weak solution. The approach extends prior results by relaxing the degeneracy constraints and provides a robust toolkit for NSK in 1D, with potential extensions to broader coefficient families via the generalized SCC.

Abstract

We prove the global existence of weak solutions of the one-dimensional Navier-Stokes-Korteweg (NSK) equations when the viscosity and the capillarity coefficients are power functions of the density, which may be zero on a set with positive measure. The proofs are based on a truncation argument combined with the Energy estimate and BD Entropy. Notably, we do not require any upper bound on the exponent of the power of the viscosity coefficient. In particular, we are able to consider very degenerate viscosity coefficient and to substantially improve previous results.

Paper Structure

This paper contains 9 sections, 15 theorems, 193 equations.

Table of Contents

  1. Introduction
  2. The strong coercivity condition
  3. Finite energy weak solutions
  4. The approximating system
  5. Proof of the main result
  6. Uniform bounds
  7. Convergence lemma
  8. Proof of Theorem \ref{['teo:main']}
  9. Proof of Theorem \ref{['teo:appex']}

Key Result

Theorem 1.2

Let $\mu$ and $k$ satisfies eq:def1 and $p(\rho):=\rho^{\gamma}$ with $2\gamma>\alpha$. Assume that $\rho_0$ and $u_0$ are such that Let $\alpha$ and $\beta$ such that eq:abrange is satisfied. Then, if there exists a least a finite energy global in time weak solutions of eq:nsk-eq:id.

Theorems & Definitions (32)

  • Definition 1.1: Strong Coercivity Condition
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Lemma 2.1: Bernis inequality
  • Remark 2.2
  • Proposition 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 22 more