arXiv subjects

Misha Vishik

Publications and source records attributed to Misha Vishik.

6 recordsLinked to original sources

arXiv2020

Instability for Axisymmetric Blow-up Solutions to Incompressible Euler Equations

It is still not known whether a solution to the incompressible Euler equation, endowed with a smooth initial value, can blow-up in finite time. In [{\em Comm. Math. Phys.}, 378:557--568, 2020] it has been shown that, if it exists, such a solution becomes linearly unstable close to the blow-up time. In this paper, we show that the same phenomenon holds even in the more rigid axisymmetric case. To obtain this result, we first prove a blow-up criterion involving only the toroidal component of the vorticity. The instability of blow-up profiles is also investigated.

math.AP↗

arXiv2019

Blow-up solutions to 3D Euler are hydrodynamically unstable

We study the interaction between the stability, and the propagation of regularity, for solutions to the incompressible 3D Euler equation. It is still unknown whether a solution with smooth initial data can develop a singularity in finite time. This article explains why the prediction of such a blow-up, via direct numerical experiments, is so difficult. It is described how, in such a scenario, the solution becomes unstable as time approaches the blow-up time.

math.AP↗

arXiv2018

Instability and non-uniqueness in the Cauchy problem for the Euler equations of an ideal incompressible fluid. Part I

In Part I of the paper, we prove non-uniqueness of the solution to the Cauchy problem of the Euler equations of an ideal incompressible fluid in dimension two with vorticity in some Lebesgue space. The radially symmetric external force is locally integrable with values in the same Lebesgue space.

math.AP↗

arXiv2018

Instability and non-uniqueness in the Cauchy problem for the Euler equations of an ideal incompressible fluid. Part II

In Part II of the paper, we prove linear instability of a certain class of radially symmetric flows of an ideal incompressible fluid in dimension two used in Part I

math.AP↗

arXiv2003

Long time Evolution of Quantum Averages Near Stationary Points

We construct explicit expressions for quantum averages in coherent states for a Hamiltonian of degree 4 with a hyperbolic stagnation point. These expressions are valid for all times and "collapse" (i.e., become infinite) along a discrete sequence of times. We compute quantum corrections compared to classical expressions. These corrections become significant over a time period of order C log 1/\hbar.

quant-ph↗

arXiv2002

Asymptotic Theory for Quantum Bose Systems with Many Degrees of Freedom

We construct asymptotic expansions of Laplace type for the time-dependent quantum averages for Bose systems with many degrees of freedom, initially populated in coherent states. These solutions are localized in phase space, and they are different from the usual oscillating asymptotics for the quasi-classical wave functions. These expansions are valid on any fixed time interval, and caustics do not contribute to the asymptotics.

quant-ph↗