arXiv subjects

Florian Wendt

Publications and source records attributed to Florian Wendt.

5 recordsLinked to original sources

arXiv2026

Homogenization of a relaxed compressible viscous two-phase fluid model in a domain with very tiny holes

We study an approximate system for the compressible Navier-Stokes-Korteweg equations in a bounded domain periodically perforated by small obstacles. As the size of the obstacles decrease much faster to zero than their mutual distances, we show in spatial dimension two and three that the limiting system remains unchanged. Our result applies for a large class of monotone and non-monotone pressure functions. In particular, it holds in the physically relevant case when the approximate system is used to describe the dynamics of a compressible viscous two-phase fluid extending the corresponding homogenization results known for the compressible Navier-Stokes equations in a single-phase setting.

math.AP↗

arXiv2026

Global-in-time existence of finite energy weak solutions to a relaxed Navier-Stokes-Korteweg model

We consider a parabolic relaxation formulation of the compressible Navier-Stokes-Korteweg system and prove the global-in-time existence of finite energy weak solutions to the associated initial-boundary-value-problem. Our proof is based on a three-level approximation scheme, a weak compactness property of the effective viscous flux, and parabolic regularity estimates. Our result holds for a broad variety of non-monotone pressure functions and generalizes the corresponding results known for the compressible Navier-Stokes equations to the relaxation system.

math.AP↗

arXiv2025

Effective Equations for a Compressible Liquid-Vapor Flow Model with Highly Oscillating Initial Density

We derive and justify a new effective model for a compressible viscous liquid-vapor flow on a spray-like scale, i.e., for settings with a large number of phase boundaries. As a model on the detailed scale, we start from a parabolic relaxation of the Navier-Stokes-Korteweg system. We consider a sequence of initial data where the sequence of initial densities is assumed to be highly oscillating mimicking the high number of phase boundaries initially. Then, we consider a sequence of finite energy weak solutions corresponding to the sequence of initial data. Anticipating that the effective equations are found in the limit of infinitely many initial phase changes, we interpret the densities as Young measures and prove the convergence of the sequence of solutions to the effective model. The effective model consists of a deterministic part for the fluid's hydrodynamic quantities and a kinetic equation for the limit Young measure encoding the mixing dynamics. By characterizing the Young measure with the corresponding cumulative distribution function, we rewrite the kinetic equation for the Young measure into a kinetic equation for the cumulative distribution function such that the resulting equations are accessible by standard approximation methods.

math.AP↗

arXiv2025

Weak-Strong Uniqueness and Relaxation Limit for a Navier-Stokes-Korteweg Model

We consider a parabolic relaxation model for the compressible Navier-Stokes-Korteweg equations in the isothermal framework. This system depends on the relaxation parameters $\alpha,\beta>0$ and approximates formally solutions of the compressible Navier-Stokes-Korteweg equations in the relaxation limit $\alpha \to \infty$ and $\beta\to 0$. Introducing the class of finite energy weak solutions for the initial-boundary value problem corresponding to the relaxation model in spatial dimension three, we show that the weak-strong uniqueness principle holds. It asserts that a weak solution and a strong solution emanating from the same initial data coincide as long as the strong solution exists. Furthermore, we contribute a rigorous convergence result for the relaxation limit $\alpha \to \infty$ and $\beta\to 0$ and thus justify the relaxation model as an approximate model for the compressible Navier-Stokes-Korteweg equations from a mathematical point of view. Our results hold for general non-monotone pressure-density relations.

math.AP↗

arXiv2025

Mathematical Justification of a Baer$-$Nunziato Model for a Compressible Viscous Fluid with Phase Transition

In this work, we justify a Baer$-$Nunziato system including appropriate closure terms as the macroscopic description of a compressible viscous fluid that can occur in a liquid or a vapor phase in the isothermal framework. As a mathematical model for the two-phase fluid on the detailed scale we chose a non-local version of the Navier$-$Stokes$-$Korteweg equations in the one-dimensional and periodic setting. Our justification relies on anticipating the macroscopic description of the two-phase fluid as the limit system for a sequence of solutions with highly oscillating initial densities. Interpreting the density as a parametrized measure, we extract a limit system consisting of a kinetic equation for the parametrized measure and a momentum equation for the velocity. Under the assumption that the initial density distributions converge in the limit to a convex combination of Dirac-measures, we show by a uniqueness result that the parametrized measure also has to be a convex combination of Dirac-measures and, that the limit system reduces to the Baer$-$Nunziato system. This work extends existing results concerning the justification of Baer$-$Nunziato models as the macroscopic description of multi-fluid models in the sense, that we allow for phase transition effects on the detailed scale. This work also includes a new global-in-time well-posedness result for the Cauchy problem of the non-local Navier$-$Stokes$-$Korteweg equations.

math.AP↗