Related papers: Equivalence of weak formulations of the steady wat…

In this work we prove the equivalence between three different weak formulations of the steady periodic water wave problem where the vorticity is discontinuous. In particular, we prove that generalised versions of the standard Euler and…

In this paper we consider the steady water wave problem for waves that possess a merely $L_r-$integrable vorticity, with $r\in(1,\infty)$ being arbitrary. We first establish the equivalence of the three formulations--the velocity…

We prove the existence of weak solutions for the one obstacle problem associated with a class of quasilinear wave equations in one space dimension, extending previous results obtained in the linear case, and we also address the two…

In this work we investigate the existence of weak solutions for steady flows of generalized incompressible and homogeneous viscous fluids. The problem is modeled by the steady case of the generalized Navier-Stokes equations, where the…

We consider the equations governing incompressible, viscous fluids in three space dimensions, rotating around an inhomogeneous vector B(x): this is a generalization of the usual rotating fluid model (where B is constant). We prove the weak…

We study Beltrami flows in the setting of weak solution to the stationary Euler equations in $\Bbb R^3$. For this weak Beltrami flow we prove the regularity and the Liouville property. In particular, we show that if tangential part of the…

We discuss a notion of weak solution for a semilinear wave equation that models the interaction of an elastic body with a rigid substrate through an adhesive layer, relying on results in [2]. Our analysis embraces the vector-valued case in…

In this paper, we consider the conditional regularity of weak solution to the 3D Navier--Stokes equations. More precisely, we prove that if one directional derivative of velocity, say $\partial_3 u,$ satisfies $\partial_3 u \in…

We provide a connection between weak solution concepts of mean curvature flow. On the one side we have the viscosity solution which is based on the comparison principle. On the other, variational solutions, which are combined Brakke flows…

We study the unique existence of weak solutions for initial boundary value problems associated with different class of fractional diffusion equations including variable order, distributed order and multiterm fractional diffusion equations.…

The problem of propagating nonlinear acoustic waves is considered; the solution to which, both with and without damping, having been obtained to-date starting from the Navier-Stokes-Duhem equations together with the continuity and thermal…

We consider Euler's equations for free surface waves traveling on a body of density stratified water in the scenario when gravity and surface tension act as restoring forces. The flow is continuously stratified, and the water layer is…

Thermodynamically consistent models for two-phase flow in porous media have attracted significant attention in recent years. In this paper, we prove the existence, uniqueness and regularity of the weak solution to such a recent model…

We prove the Benjamin and Lighthill conjecture for all two-dimensional steady water waves with an arbitrary vorticity distribution. We show that the flow force constant of an arbitrary smooth wave is bounded by the corresponding flow force…

Our paper deals with three-dimensional nonsteady Navier-Stokes equations for non-Newtonian compressible fluids. It contains a~derivation of the relative energy inequality for the weak solutions to these equations. We show that the standard…

We derive a weak turbulence formalism for incompressible MHD. Three-wave interactions lead to a system of kinetic equations for the spectral densities of energy and helicity. We find energy spectra solution of the kinetic equations. The…

A reduced dynamical model is derived which describes the interaction of weak inertia-gravity waves with nonlinear vortical motion in the context of rotating shallow-water flow. The formal scaling assumptions are (i) that there is a…

We address the global-in-time existence, stability and long time behaviour of weak solutions of the three-dimensional compressible Navier-Stokes equations with potential force. We show the details of the $\alpha$-dependence of different…

In this paper we construct periodic capillarity-gravity water waves with an arbitrary bounded vorticity distribution. This is achieved by reexpressing, in the height function formulation of the water wave problem, the boundary condition…

A system of two-dimensional nonlinear equations of hydrodynamics is considered. It is shown that for the this system in the general case a solution with weak discontinuity-type singularity behaves as a square root of S(x,y,t), where…