Related papers: Riemann problem of Euler equations with singular s…
We consider the subsonic moving point source problem for the scalar wave equation in $\pmb{R}^3$, proving a regularity result for the direct problem, and uniqueness and stability results for the inverse problem. We then present and…
A class of semi-bounded solutions of the two-dimensional incompressible Euler equations satisfying either periodic or Dirichlet boundary conditions is examined. For smooth initial data, new blowup criteria in terms of the initial concavity…
We find an explicit form of entropy solutions to a Riemann problem for a degenerate nonlinear parabolic equation with piecewise constant velocity and diffusion coefficients. It is demonstrated that this solution corresponds to the minimum…
The relativistic full Euler equations for a Chaplygin gas are studied. The Riemann problem is solved constructively. There are two kinds of Riemann solutions, in which one consists of three contact discontinuities and the other involves a…
The Rayleigh equation, which is the linearized Euler equations near a shear flow in vorticity formulation, is a key ingredient in the study of the long time behavior of solutions of linearized Euler equations, in the study of the linear…
Let $X$ be a manifold with boundary, endowed with a metric with conic singularities at the boundary components of $X$. Let $u$ be a solution to the wave equation on $\mathbb{R} \times X$. When a singularity of $u$ strikes a cone point of…
We develop, via Arnold's geometric framework, a mechanism for constructing explicit, smooth, global-in-time, and typically non-stationary solutions of the incompressible Euler equations. The approach introduces a notion of generalized…
In this article the unique solution of the Cauchy problem is founded by the Riemann method. Some relations for given here confluent hypergeometric functions of two and three variables are used.
For 2D compressible Euler equations of isentropic gas, we prove the structural stability of mixed Riemann configurations containing centered rarefaction waves and surfaces of discontinuities (such as shock waves or vortex sheets), by…
In this paper, we are concerned with the long time behavior of the piecewise smooth solutions to the generalized Riemann problem governed by the compressible isothermal Euler equations in two and three dimensions. Non-existence result is…
We examine the blow-up claims of the incompressible Euler equations for several specific flow-fields, (1) the columnar eddies in the vicinity of stagnation; (2) a quasi-three-dimensional structure for illustrating oscillations and…
In this work we present a general strategy for constructing multidimensional Riemann solvers with a single intermediate state, with particular attention paid to detailing the two-dimensional Riemann solver. This is accomplished by…
In this paper, we present strong numerical evidences that the incompressible axisymmetric Euler equations with degenerate viscosity coefficients and smooth initial data of finite energy develop a potential finite-time locally self-similar…
We consider solutions of the 2-d compressible Euler equations that are steady and self-similar. They arise naturally at interaction points in genuinely multi-dimensional flow. We characterize the possible solutions in the class of flows…
In this paper, we consider some blow-up problems for the 1D Euler equation with time and space dependent damping. We investigate sufficient conditions on initial data and the rate of spatial or time-like decay of the coefficient of damping…
This paper proposes direct and inverse results for the Dirichlet and Dirichlet to Neumann problems for complex curves with nodal type singularities. As an application, we give a method to reconstruct the conformal structure of a compact…
The paper contains a stability analysis of the plane-wave Riemann problem for the two-dimensional hyperbolic conservation laws for an ideal compressible gas. It is proved that the contact discontinuity in the plane-wave Riemann problem is…
We consider the singular Rayleigh-Jeans equilibrium of the $4$-waves kinetic turbulence equation for the three dimensional Schr\"{o}dinger equation. We first show the formation in finite time of a Dirac measure at zero frequency in the…
The Riemann problem of one dimensional shallow water equations with discontinuous topography has been constructed recently. The elementary waves include shock waves, rarefaction waves, and the stationary wave. The stationary wave appears…
This paper addresses the inverse problem of simultaneously recovering multiple unknown parameters for semilinear wave equations from boundary measurements. We consider an initial-boundary value problem for a wave equation with a general…