Related papers: Shock formation for the 2D rotating shallow water …
In this paper, we consider the finite-time blowup of hollow vortices. These are solutions of the two-dimensional Euler equations for which the fluid domain is the complement of finitely many Jordan curves $\Gamma_1, \ldots, \Gamma_M$, and…
The origin of a vortex structure generated during the shock-plasma interaction is investigated. A two-dimensional model based on the shock refraction mechanism successfully unifies the vortex generation with major co-processes typical for…
It is known that smooth solutions to the non-isentropic Navier-Stokes equations without heat-conductivity may lose their regularities in finite time in the presence of vacuum. However, in spite of the recent progress on such blowup…
The existence and stability of a spherical transonic shock in a hemispherical shell under the three dimensional perturbations of the incoming flows and the exit pressure is established without any further restrictions on the background…
To investigate the finite time singularity in three-dimensional (3D) Euler flows, the simplified model of 3D axisymmetric incompressible fluids (i.e., two-dimensional Boussinesq approximation equations) is studied numerically. The system…
In this paper, for compressible Euler equations in multiple space dimensions, we prove the break-down of classical solutions with a large class of initial data by tracking the propagation of radially symmetric expanding wave including…
We provide numerical evidence for a potential finite-time self-similar singularity of the 3D axisymmetric Euler equations with no swirl and with $C^\alpha$ initial vorticity for a large range of $\alpha$. We employ a highly effective…
We consider the compressible three dimensional Navier Stokes and Euler equations. In a suitable regime of barotropic laws, we construct a set of finite energy smooth initial data for which the corresponding solutions to both equations…
Models for shallow water flow often assume that the lateral velocity is constant over the water height. The recently derived shallow water moment equations are an extension of these standard shallow water equations. The extended models…
The aim of these notes is to present in a comprehensive and relatively self-contained way some recent developments in the mathematical analysis of two-dimensional viscous flows. We consider the incompressible Navier-Stokes equations in the…
We consider the two dimensional unsteady Prandtl system. For a special class of outer Euler flows and solutions of the Prandtl system, the trace of the tangential derivative of the tangential velocity along the transversal axis solves a…
The appearance of sharp vorticity gradients in two-dimensional hydrodynamic turbulence and their influence on the turbulent spectra is considered. We have developed the analog of the vortex line representation as a transformation to the…
We extend our previous studies of shock waves and shock-free solutions in thin accretion and winds in pseudo-Newtonian geometry to the case when the flow is ``two-dimensional'' and around a ``Kerr black hole''. We present equations for…
We study formation of quasi two-dimensional (thin pancakes) vortex structures in three-dimensional flows, and quasi one-dimensional structures in two-dimensional hydrodynamics. These structures are formed at high Reynolds numbers, when…
We study by a combination of analytical and numerical methods multidimensional stability and transverse bifurcation of planar hydraulic shock and roll wave solutions of the inviscid Saint Venant equations for inclined shallow-water flow,…
We study structure formation in two-dimensional turbulence driven by an external force, interpolating between linear instability forcing and random stirring, subject to nonlinear damping. Using extensive direct numerical simulations, we…
We establish a simple relation between curvatures of the group of volume-preserving diffeomorphisms and the lifespan of potential solutions to the inviscid Burgers equation before the appearance of shocks. We show that shock formation…
Whether the smooth solution of the multi-dimensional viscous compressible fluids will blow-up in finite time has always been a chanllenging problem. In the recent work\cite{FM}, Merle et al. proved that there are smooth solutions to the 2D…
We consider two cases of interaction between a planar shock and a cylindrical density interface. In the first case (planar normal shock), the axis of the gas cylinder is parallel to the shock front, and baroclinic vorticity deposited by the…
We consider the blow-up of solutions to the following parameterized nonlinear wave equation: $ u_{tt} = c(u)^{2} u_{xx} + \lambda c(u)c'(u)( u_x)^2$ with the real parameter $\lambda$. In previous works, it was reported that there exist…