Related papers: Singularity formation for the compressible Euler e…
The Newtonian Euler-Poisson equations with attractive forces are the classical models for the evolution of gaseous stars and galaxies in astrophysics. In this paper, we use the integration method to study the blowup problem of the…
We construct a new class of self-similar implosion profiles for the multi-dimensional compressible Euler equations. These profiles are smooth, genuinely non-isentropic, radially/spherically symmetric, and have explicit (closed-form)…
In this article we consider the inhomogeneous incompressible Euler equations describing two fluids with different constant densities under the influence of gravity as a differential inclusion. By considering the relaxation of the…
The primitive equations (PEs) model large scale dynamics of the oceans and the atmosphere. While it is by now well-known that the three-dimensional viscous PEs is globally well-posed in Sobolev spaces, and that there are solutions to the…
We present a preliminary study of a new phenomena associated with the Euler-Poisson equations -- the so called critical threshold phenomena, where the answer to questions of global smoothness vs. finite time breakdown depends on whether the…
We develop a general framework for studying non-uniqueness of the Riemann problem for the isentropic compressible Euler system in two spatial dimensions, and in this paper we present the most delicate result of our method: non-uniqueness of…
This work is a companion to [EJE1] and its purpose is threefold: first, we will establish local well-posedness for the axi-symmetric $3D$ Euler equation in the domains $\{(x_1,x_2,x_3) \in \mathbb{R}^3 : x_3^2 \le \mathfrak{c}(x_1^2 +…
A compactness framework is formulated for the incompressible limit of approximate solutions with weak uniform bounds with respect to the adiabatic exponent for the steady Euler equations for compressible fluids in any dimension. One of our…
This paper concerns the well-posedness theory of the motion of physical vacuum for the compressible Euler equations with or without self-gravitation. First, a general uniqueness theorem of classical solutions is proved for the three…
Solutions to a class of conservation laws with discontinuous flux are constructed relying on the Crandall-Liggett theory of nonlinear contractive semigroups~\cite{CL}. In particular, the paper studies the existence of backward Euler…
The Cauchy problem for the system of equations of two-dimensional rotational gas dynamics is considered. It is assumed that the Cauchy data are a smooth compact perturbation of a constant state. Integral conditions for the data sufficient…
Measure-valued solutions to fluid equations arise naturally, for instance as vanishing viscosity limits, yet exhibit non-uniqueness to a vast extent. In this paper, we show that some measurevalued solutions to the two-dimensional isentropic…
The ultra-relativistic Euler equations for an ideal gas are described in terms of the pressure $p$, the spatial part $\underline{u} \in \R^3$ of the dimensionless four-velocity and the particle density $n$. Radially symmetric solutions of…
Whether singularities can form in fluids remains a foundational unanswered question in mathematics. This phenomenon occurs when solutions to governing equations, such as the 3D Euler equations, develop infinite gradients from smooth initial…
We prove the finite time blow-up for $C^1$ solutions to the Euler-Poisson equations in $\Bbb R^n$, $n\geq 1$, with/without background density for initial data satisfying suitable conditions. We also find a sufficient condition for the…
In this article we consider the discretely self-similar singular solutions of the Euler equations, and the possible velocity profiles concerned not only have decaying spatial asymptotics, but also have unconventional non-decaying…
The helicity is a topological conserved quantity of the Euler equations which imposes significant constraints on the dynamics of vortex lines. In the compressible setting the conservation law only holds under the assumption that the…
We introduce a natural notion of incompressibility for fluids governed by the relativistic Euler equations on a fixed background spacetime, and show that the resulting equations reduce to the incompressible Euler equations in the classical…
We present a new hydrodynamic model consisting of the pressureless Euler equations and the isentropic compressible Navier-Stokes equations where the coupling of two systems is through the drag force. This coupled system can be derived, in…
We are concerned with the global existence theory for spherically symmetric solutions of the multidimensional compressible Euler equations with large initial data of positive far-field density. The central feature of the solutions is the…