Related papers: Singular solutions with vorticity control for a no…
The existing paradox between theory and computational experiment for weak solutions of systems of conservation laws in higher space dimensions is arguably resolved. Apparently successful computations are identified with underlying…
This paper deals with the longstanding quest of the possible existence of finite-time singularities in the equations governing the dynamics of inviscid fluids, namely, Euler equations. Here, two contributions are brought for the case of…
We study controllability issues for the 2D Euler and Navier-Stokes (NS) systems under periodic boundary conditions. These systems describe motion of homogeneous ideal or viscous incompressible fluid on a two-dimensional torus…
In this paper we use maximum principle in the far field region for the time dependent self-similar Euler equations to exclude discretely self-similar blow-up for the Euler equations of the incompressible fluid flows. Our decay conditions…
We address the question whether a singularity in a three-dimensional incompressible inviscid fluid flow can occur in finite time. Analytical considerations and numerical simulations suggest high-symmetry flows being a promising candidate…
An important question in mathematical relativity theory is that of the nature of spacetime singularities. The equations of general relativity, the Einstein equations, are essentially hyperbolic in nature and the study of spacetime…
We study a 1D transport equation with nonlocal velocity. First, we prove eventual regularization of the viscous regularization when dissipation is in the supercritical range with non-negative initial data. Next, we will prove global…
We consider the Cauchy problem with smooth data for compressible Euler equations in many dimensions and concentrate on two cases: solutions with finite mass and energy and solutions corresponding to a compact perturbation of a nontrivial…
We investigate the motion of closed smooth curves that evolve in space $\mathbb{R}^3$. The governing evolutionary equation for the evolution of the curve is accompanied by a parabolic equation for the scalar quantity evaluated over the…
We consider the three-dimensional incompressible Euler equations for helical flows without swirl. By adapting gluing techniques, we construct the first smooth multi-vortex solution in the whole space $\mathbb{R}^3$ exhibiting a cluster of…
In this paper we establish the theory on semiglobal classical solution to first order quasilinear hyperbolic systems with a kind of nonlocal boundary conditions, and based on this, the corresponding exact boundary controllability and…
A probabilistic representation formula for general systems of linear parabolic equations, coupled only through the zero-order term, is given. On this basis, an implicit probabilistic representation for the vorticity in a 3D viscous fluid…
We consider a system of nonlocal equations driven by a perturbed periodic potential. We construct multibump solutions that connect one integer point to another one in a prescribed way. In particular, heteroclinc, homoclinic and chaotic…
We consider a two-dimensional, two-layer, incompressible, steady flow, with vorticity which is constant in each layer, in an infinite channel with rigid walls. The velocity is continuous across the interface, there is no surface tension or…
For one dimensional or multidimensional compressible Euler system of polytropic gases, it is well known that the smooth solution will generally develop singularities in finite time. However, for three dimensional Chaplygin gases, due to the…
In this paper, we study the local exact boundary controllability of entropy solutions to a class linearly degenerate hyperbolic systems of conservation laws with constant multiplicity. The authors prove the two-sided boundary…
In this paper, we study desingularization of steady solutions of 3D incompressible Euler equation with helical symmetry in a general helical domain. We construct a family of steady Euler flows with helical symmetry, such that the associated…
We consider the so-called transport-Stokes system which describes sedimentation of inertialess suspensions in a viscous flow and couples a transport equation and the steady Stokes equations in the full three-dimensional space. First we…
A new rescaling of the vorticity moments and their growth terms is used to characterise the evolution of anti-parallel vortices governed by the 3D Euler equations. To suppress unphysical instabilities, the initial condition uses a balanced…
We prove long-term regularity of solutions of the one-fluid Euler-Maxwell system in 3 spatial dimensions, in the case of small initial data with nontrivial vorticity.