Related papers: On the instability of a nonlocal conservation law
We prove the existence and uniqueness of strong solutions to the equation $u u_x - u_{yy} = f$ in the vicinity of the linear shear flow, subject to perturbations of the source term and lateral boundary conditions. Since the solutions we…
We introduce a new class of nonlocal nonlinear conservation laws in one space dimension that allow for nonlocal interactions over a finite horizon. The proposed model, which we refer to as the nonlocal pair interaction model, inherits at…
We complete a full classification of non-degenerate traveling waves of scalar balance laws from the point of view of spectral and nonlinear stability/instability under (piecewise) smooth perturbations. A striking feature of our analysis is…
We prove that smooth solutions of non-ideal (viscous and resistive) incompressible magnetohydrodynamic equations satisfy a stochastic law of flux conservation. This property involves an ensemble of surfaces obtained from a given, fixed…
The boundary conditions at the deformable interface between two contacting fluids are derived for the general case of the large-amplitude perturbations. The interface is modeled as perturbed free boundary that evolves in time, and the…
In this expository note we discuss our recent work [arXiv:1306.5028] on the nonlinear asymptotic stability of shear flows in the 2D Euler equations of ideal, incompressible flow. In that work it is proved that perturbations to the Couette…
The well-known Stokes waves refer to periodic traveling waves under the gravity at the free surface of a two dimensional full water wave system. In this paper, we prove that small-amplitude Stokes waves with infinite depth are nonlinearly…
We study the dynamics of solutions of nonlinear Schr\"odinger equation near unstable ground states. The existence of the local center stable manifold around ground states and the asymptotic stability for the solutions on the manifold is…
We prove a non-mixing property of the flow of the 3D Euler equation which has a local nature: in any neighbourhood of a "typical" steady solution there is a generic set of initial conditions, such that the corresponding Euler flows will…
This paper is devoted to the study of the nonlinear instability of shear layers and of Prandtl's boundary layers, for the incompressible Navier Stokes equations. We prove that generic shear layers are nonlinearly unstable provided the…
Inviscid laminar flow is a stationary solution of the incompressible Euler equations whose streamlines foliate the fluid domain. Their structure on symmetric domains is rigid: all laminar flows occupying straight periodic channels are shear…
A thin and narrow rectangular plate having the two short edges hinged and the two long edges free is considered. A nonlinear nonlocal evolution equation describing the deformation of the plate is introduced: well-posedness and existence of…
Flows of granular media down a rough inclined plane demonstrate a number of nonlocal phenomena. We apply the recently proposed nonlocal granular fluidity model to this geometry and find that the model captures many of these effects.…
We give an overview on the solution of the stationary Navier-Stokes equations for non newtonian incompressible fluids established by G. Dias and M.M. Santos (Steady flow for shear thickening fluids with arbitrary fluxes, J. Differential…
This study investigates unsteady boundary layer phenomena in electrically conducting fluids subjected to static magnetic fields. Using a semi-explicit similarity transformation method, the momentum equation associated with the Stokes stream…
A theoretical and numerical analysis of the linear stability of the boundary layer flow under a solitary wave is presented. In the present work, the nonlinear boundary layer equations are solved. The result is compared to the linear…
Assessment of the degree of boundedness/stability of multidimensional nonlinear systems with time-dependent and nonperiodic coefficients is an important problem in various applied areas which has no adequate resolution yet. Most of the…
This article considers non-stationary incompressible linear fluid equations in a moving domain. We demonstrate the existence and uniqueness of an appropriate weak formulation of the problem by making use of the theory of time-dependent…
Ideal systems like MHD and Euler flow may develop singularities in vorticity (w = curl v). Viscosity and resistivity provide dissipative regularizations of the singularities. In this paper we propose a minimal, local, conservative,…
In this article, we study small perturbations of the family of Friedmann-Lema\^itre-Robertson-Walker cosmological background solutions to the 1 + 3 dimensional Euler-Einstein system with a positive cosmological constant. These background…