Related papers: The relativistic Euler equations: ESI notes on the…
In his 2007 monograph, D. Christodoulou proved a breakthrough result giving a detailed description of the formation of shocks in solutions to the relativistic Euler equations in three spatial dimensions. He assumed that the data have small…
The ultra--relativistic Euler equations describe gases in the relativistic case when the thermal energy dominates. These equations for an ideal gas are given in terms of the pressure, the spatial part of the dimensionless four-velocity, and…
The question of well- and ill-posedness of entropy admissible solutions to the multi-dimensional systems of conservation laws has been studied recently in the case of isentropic Euler equations. In this context special initial data were…
We study the relativistic Euler equations on the Minkowski spacetime background. We make assumptions on the equation of state and the initial data that are relativistic analogs of the well-known physical vacuum boundary condition, which has…
One of the most profound questions of mathematical physics is that of establishing from first principles the hydrodynamic equations in large, isolated, strongly interacting many-body systems. This involves understanding relaxation at long…
Geometrical shock dynamics, also called CCW theory, yields approximate equations for shock propagation in which only the conditions at the shock appear explicitly; the post-shock flow is presumed approximately uniform and enters implicitly…
We present a numerical method to solve the equations of general relativistic hydrodynamics in a given external gravitational field. The method is based on a generalization of Roe's approximate Riemann solver for the non relativistic Euler…
In this paper, we show the shock formation of the solutions to the 3-dimensional (3D) compressible isentropic and irrotational Euler equations with damping for the initial short pulse data which was first introduced by…
This paper concerns the existence and location of three-dimensional axisymmetric transonic shocks with large swirl velocity for shock solutions of the steady compressible full Euler system in a cylindrical nozzle with prescribed receiver…
In this paper, we show the shock formation to the compressible Euler equations with time-dependent damping $\frac{a\p u}{(1+t)^{\lam}}$ in three spatial dimensions without any symmetry conditions. It's well-known that for $\lam>1$, the…
For the $2D$ Euler equation in vorticity formulation, we construct localized smooth solutions whose critical Sobolev norms become large in a short period of time, and solutions which initially belong to $L^\infty \cap H^1$ but escapes $H^1$…
In this article, we investigate the two-dimensional pressureless Euler equations with three constant Riemann initial data. Our primary focus is on the wave interactions involving contact discontinuities and delta shocks. A distinguishing…
We study the Cauchy problem for the $3D$ compressible Euler equations under an arbitrary equation of state with positive speed of sound, aside from that of a Chaplygin gas. For open sets of smooth initial data with non-trivial vorticity and…
Whether the 3D incompressible Euler equations can develop a singularity in finite time from smooth initial data is one of the most challenging problems in mathematical fluid dynamics. This work attempts to provide an affirmative answer to…
We present a new, first-order, flux-conservative formulation of relativistic viscous hydrodynamics in the BDNK framework, applicable to conformal and nonconformal fluids at zero chemical potential. Focusing on the conformal case in 1+1…
We consider the Riemann problem composed of two shocks for the 1D Euler system. We show that the Riemann solution with two shocks is stable and unique in the class of weak inviscid limits of solutions to the Navier-Stokes equations with…
A class of three-dimensional initial data characterized by uniformly large vorticity is considered for the Euler equations of incompressible fluids. The fast singular oscillating limits of the Euler equations are studied for parametrically…
This paper reports several new classes of weakly unstable recurrent solutions of the 2+1-dimensional Euler equation on a square domain with periodic boundary conditions. These solutions have a number of remarkable properties which…
Radial similarity flow offers a rare instance where concrete inviscid, multi-dimensional, compressible flows can be studied in detail. In particular, there are flows of this type that exhibit imploding shocks and cavities. In such flows the…
A second-order PDE is derived from Euler's equaitons under certain assumptions. It is shown that this PDE admits shock and rarefaction waves, and that a single point gradient blow-up admits a unique similarity extension after blow-up that…