Hana Petzeltová

On the Existence of Globally Defined Weak Solutions to the Navier—Stokes Equations

Abstract. We prove the existence of globally defined weak solutions to the Navier—Stokes equations of compressible isentropic flows in three space dimensions on condition that the adiabatic constant satisfies

Long Time Behavior of Solutions to the Caginalp System with Singular Potential

\gamma > 3/2

ANALYSIS OF A PHASE-FIELD MODEL FOR TWO-PHASE COMPRESSIBLE FLUIDS

.

A nonlocal phase-field system with inertial term

We consider a nonlinear parabolic system which governs the evolution of the (relative) temperature θ and of an order parameter χ. This system describes phase transition phenomena like, e.g., melting-solidification processes. The equation ruling χ is characterized by a singular potential W which forces χ to take values in the interval [−1, 1]. We provide reasonable conditions on W which ensure that, from a certain time on, χ stays uniformly away from the pure phases 1 and −1. Combining this separation property with the Lojasiewicz-Simon inequality, we show that any smooth and bounded trajectory uniformly converges to a stationary state and we give an estimate of the decay rate. ∗This work was partially supported by the Italian MIUR PRIN Research Projects Modellizzazione Matematica ed Analisi dei Problemi a Frontiera Libera and Aspetti Teorici e Applicativi di Equazioni a Derivate Parziali, and by the Italian MIUR FIRB Research Project Analisi di Equazioni a Derivate Parziali, Lineari e Non Lineari: Aspetti Metodologici, Modellistica, Applicazioni †The work of H.P. was supported by the Grant A1019302 of GA AV CR ‡The work of G.S. was partially supported by the HYKE Research Training Network

FRONT PROPAGATION IN NONLINEAR PARABOLIC EQUATIONS

A model describing the evolution of a binary mixture of compressible, viscous, and macroscopically immiscible fluids is investigated. The existence of global-in-time weak solutions for the resulting system coupling the compressible Navier–Stokes equations governing the motion of the mixture with the Allen–Cahn equation for the order parameter is proved without any restriction on the size of initial data.

On a class of physically admissible variational solutions to the Navier-Stokes-Fourier system

We study a phase-field system where the energy balance equation has the standard (parabolic) form, while the kinetic equation ruling the evolution of the order parameter X is a nonlocal and nonlinear second-order ODE. The main features of the latter equation are a space convolution term which models long-range interactions of particles and a singular configuration potential that forces X to take values in (-1,1). We first prove the global existence and uniqueness of a regular solution to a suitable initial and boundary value problem associated with the system. Then, we investigate its long time behavior from the point of view of ω-limits. In particular, using a nonsmooth version of the Lojasiewicz-Simon inequality, we show that the ω-limit of any trajectory contains one and only one stationary solution, provided that the configuration potential in the kinetic equation is convex and analytic.

Global existence for a quasi-linear evolution equation with a non-convex energy

We study existence and stability of travelling waves for nonlinear convection diffusion equa- tions in the 1-D Euclidean space. The diffusion coefficient depends on t gradient in analogy with the p-Laplacian and may be degenerate. Unconditional stability is established with respect to initial data perturbations in L 1 (R). Running head: Poincare inequality and P.-S. condition

On the domain dependence of solutions to the two-phase Stefan problem

The main objective of the present paper is to introduce a class of admissible variational solutions to the Navier-Stokes-Fourier system of equations based on the second law of thermodynamics. We also show that the solutions exist globally in time regardless the size of the initial data. Finally, the question of the long-time behaviour of these solutions is being addressed.

Asymptotic spatial homogeneity of solutions to conservation laws with memory in several space dimensions

We establish the existence of global in time weak solutions to the initial-boundary value problem related to the dynamics of coherent solid-solid phase transitions in viscoelasticity. The class of the stored energy functionals includes the double well potential, and a general convolution damping term is considered.

On the Motion of Chemically Reacting Fluids Through Porous Medium

We prove that solutions to the two-phase Stefan problem defined on a sequence of spatial domains Ώn ⊂ ℝN converge to a solution of the same problem on a domain Ω where Ω is the limit of Ωn in the sense of Mosco. The corresponding free boundaries converge in the sense of Lebesgue measure on ℝN.