Relative energy for the Korteweg theory and related Hamiltonian flows in gas dynamics

Abstract

For an Euler system, with dynamics generated by a potential energy functional, we propose a functional format for the relative energy and derive a relative energy identity. The latter, when applied to specific energies, yields relative energy identities for the Euler-Korteweg, the Euler-Poisson, the Quantum Hydrodynamics system, and low order approximations of the Euler-Korteweg system. For the Euler-Korteweg system we prove a stability theorem between a weak and a strong solution and an associated weak-strong uniqueness theorem. In the second part we focus on the Navier-Stokes-Korteweg system (NSK) with non-monotone pressure laws: we prove stability for the NSK system via a modified relative energy approach. We prove continuous dependence of solutions on initial data and convergence of solutions of a low order model to solutions of the NSK system. The last two results provide physically meaningful examples of how higher order regularization terms enable the use of the relative energy framework for models with energies which are not poly- or quasi-convex, but compensating via higher-order gradients.

Updated to Authors' Accepted Manuscript version

References in corpus (3)

• On the Finite Energy Weak Solutions to a System in Quantum Fluid Dynamics
• Existence and Nonlinear Stability of Rotating Star Solutions of the Compressible Euler-Poisson Equations
• From gas dynamics with large friction to gradient flows describing diffusion theories

Cited by in corpus (17)

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• On the compactness of weak solutions to the Navier-Stokes-Korteweg equations for capillary fluids
• Relaxation limit from the Quantum-Navier-Stokes equations to the Quantum Drift Diffusion equation
• High-friction limits of Euler flows for multicomponent systems
• Well-balanced finite volume schemes for hydrodynamic equations with general free energy
• Extended Lagrangian approach for the numerical study of multidimensional dispersive waves: applications to the Serre-Green-Naghdi equations
• The relaxation limit of bipolar fluid models
• Stability and asymptotic analysis for instationary gas transport via relative energy estimates
• On the convergence of the regularized entropy-based moment method for kinetic equations
• An asymptotic-preserving discretization scheme for gas transport in pipe networks
• Optimal transport on gas networks
• The role of Riesz potentials in the weak-strong uniqueness for Euler-Poisson systems
• Existence and spectral stability analysis of viscous-dispersive shock profiles for isentropic compressible fluids of Korteweg type
• On weak solutions to the 2D Savage-Hutter model of the motion of a gravity driven avalanche flow
• Convergence of hyperbolic approximations to higher-order PDEs for smooth solutions
• High-friction limit for bipolar Euler-Riesz systems