Related papers: Classical Euler flows generate the strong Guderley…
A fundamental question in fluid dynamics concerns the formation of discontinuous shock waves from smooth initial data. We prove that from smooth initial data, smooth solutions to the 2d Euler equations in azimuthal symmetry form a first…
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…
In this paper, we rigorously prove the existence of self-similar converging shock wave solutions for the non-isentropic Euler equations for $\gamma\in (1,3]$. These solutions are analytic away from the shock interface before collapse, and…
From an open set of initial data, we construct a family of classical solutions to the 1D nonisentropic compressible Euler equations which form $C^{0,\nu}$ cusps as a first singularity, for any $\nu \in [1/2,1)$. For this range of $\nu$,…
Motivated by recent breakthrough on smooth imploding solutions of compressible Euler, we construct self-similar smooth imploding solutions of isentropic relativistic Euler equations with isothermal equation of state $p=\frac1\ell\varrho$…
We study the 2D isentropic Euler equations with the ideal gas law. We exhibit a set of smooth initial data that give rise to shock formation at a single point near the planar symmetry. These solutions are associated with non-zero vorticity…
We study the behavior of perturbations in a compressible one-dimensional inviscid gas with an ambient state consisting of constant pressure and periodically-varying density. We show through asymptotic analysis that long-wavelength…
In this paper, we consider the compressible Euler equations with time-dependent damping \frac{\a}{(1+t)^\lambda}u in one space dimension. By constructing 'decoupled' Riccati type equations for smooth solutions, we provide some sufficient…
Self-similar solutions to the problem of a strong imploding relativistic shock wave are calculated. These solutions represent the relativistic generalisation of the Newtonian Gouderley-Landau-Stanyukovich problem of a strong imploding…
We consider the compressible Euler system for ideal gas flow in the absence of any forces except the internal thermodynamic pressure. In this setting, and in dimensions higher 1, it is known that wave-focusing can drive Euler solutions to…
We study the Cauchy problem for the compressible Euler equations in two spatial dimensions under any physical barotropic equation of state except that of a Chaplygin gas. We prove that the well-known phenomenon of shock formation in simple…
The existence, uniqueness, and asymptotic behavior of steady transonic flows past a curved wedge, involving transonic shocks, governed by the two-dimensional full Euler equations are established. The stability of both weak and strong…
We are concerned with spherically symmetric solutions of the Euler equations for multidimensional compressible fluids, which are motivated by many important physical situations. Various evidences indicate that spherically symmetric…
Tai-Ping Liu \cite{Liu\_JJ} introduced the notion of "physical solution' of the isentropic Euler system when the gas is surrounded by vacuum. This notion can be interpreted by saying that the front is driven by a force resulting from a…
The motion of a compressible inviscid radiative flow can be described by the radiative Euler equations, which consists of the Euler system coupled with a Poisson equation for the radiative heat flux through the energy equation. Although…
We are concerned with geometric properties of transonic shocks as free boundaries in two-dimensional self-similar coordinates for compressible fluid flows, which are not only important for the understanding of geometric structure and…
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…
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…
We consider one-dimensional self-similar solutions to the isentropic Euler system when the initial data are at vacuum to the left of the origin. For $x>0$ the initial velocity and sound speed are of form $u_0(x)=u_+x^{1-\lambda}$ and…
We consider the long-time behavior of irrotational solutions of the three-dimensional compressible Euler equations with shocks, hypersurfaces of discontinuity across which the Rankine-Hugoniot conditions for irrotational flow hold. Our…