We prove weak-strong uniqueness results for the isentropic compressible Navier-Stokes system on the torus. In other words, we give conditions on a strong solution so that it is unique in a class of weak solutions. Known weak-strong uniqueness results are improved. Classical uniqueness results for this equation follow naturally.

In this paper we study the existence and asymptotic stability of periodic solutions of the differential equation ẍ+ f (x)ẋ+g(x) = h(t), where h(t) is T -periodic, f (x) is positive and g(x) is strictly monotonically increasing and has one or two weak singularities. The method of proof relies on the construction of a positively invariant region of the flux. Mathematics subject classification (20...

In this article we consider weak solutions of the three-dimensional incompressible fluid flow equations with initial data admitting a one-dimensional symmetry group. We examine both the viscous and inviscid cases. For the case of viscous flows, we prove that Leray-Hopf weak solutions of the threedimensional Navier-Stokes equations preserve initially imposed symmetry and that such symmetric flow...

Global existence of weak and strong solutions to the quasi-hydrostatic primitive equations is studied in this paper. This model, that derives from the full non-hydrostatic model for geophysical fluid dynamics in the zero-limit of the aspect ratio, is more realistic than the classical hydrostatic model, since the traditional approximation that consists in neglecting a part of the Coriolis force ...

This paper compares three popular notions of admissibility for weak solutions of the compressible isentropic Euler equations of gas dynamics: (i) the viscosity criterion, (ii) the entropy inequality (the thermodynamically admissible isentropic solutions), and (iii) the viscosity-capillarity criterion. An exact summation of the Chapman-Enskog expansion for Grad's moment system suggests that it i...

Abstract In this article, we study an initial-boundary value problem for a three-phase field model of nonisothermal solidification processes in the case two possible crystallization states. The governing equations are three phase-field coupled with nonlinear heat equation. Each equation has strong nonlinearities involving higher-order derivatives. We prove existence global-in-time weak solution...

In this paper, we prove the existence of global weak solutions for 3D compressible Navier-Stokes equations with degenerate viscosity. The method is based on the Bresch and Desjardins entropy conservation [2]. The main contribution of this paper is to derive the Mellet-Vasseur type inequality [32] for the weak solutions, even if it is not verified by the first level of approximation. This provid...

The existence of weak time-periodic solutions to Navier–Stokes equations in three dimensional whole-space with forcing terms are established. constructed such a way that the structural properties their kinetic energy obtained. No restrictions on either size or structure external force required.

We show the existence of at least one weak solution for a threepoint boundary-value problem of Kirchhoff-type. Our technical approach is based on variational methods. In addition, an example to illustrate our results is given.