Overview
A mathematical analysis has established the time-asymptotic stability of generic Riemann solutions for the one-dimensional Navier-Stokes-Fourier-Korteweg (NSFK) system. The system governs the dynamics of compressible, viscous, and heat-conducting fluids featuring internal capillarity. For adiabatic exponents satisfying $\gamma \in (1, 2]$, solutions to the NSFK system converge uniformly across space as time approaches infinity toward a composite wave structure.
Research Context
The one-dimensional NSFK equations incorporate internal capillarity alongside viscosity and thermal conductivity in compressible fluid flow. In this framework, capillary effects alter the mathematical structure of shock waves. Unlike standard viscous shocks, shocks in the NSFK system are viscous-dispersive, and their corresponding profile equations are of third order.
Approach
The stability proof accounts for asymptotic states composed of three distinct wave types: a rarefaction wave, a viscous contact wave, and a viscous-dispersive shock governed by a time-dependent shift. To manage the third-order profile equation generated by internal capillarity, the analysis employs the method of $a$-contraction with shifts, a technique introduced by Kang and Vasseur.
The mathematical framework requires specific smallness conditions on certain elements of the initial data and wave patterns:
- The initial perturbation must be sufficiently small.
- The strength of the viscous contact wave must be sufficiently small.
- The strength of the viscous-dispersive shock must be sufficiently small.
- No smallness condition is imposed on the strength of the rarefaction wave.
Findings
The authors prove that under the parameter restriction $\gamma \in (1, 2]$ and the specified smallness criteria, the solution to the one-dimensional NSFK system converges uniformly in space to the composite wave as $t \to \infty$. This derivation represents the first proof of time-asymptotic stability for a viscous-dispersive shock within the NSFK system, as well as the first such stability result for generic Riemann solutions in this system.