Related papers: Smooth and stable Euler implosions
For $1/3<K<1$, we consider the stability of two distinct families of spatially homogeneous solutions to the relativistic Euler equations with a linear equation of state $p=K\rho$ on exponentially expanding FLRW spacetimes. The two families…
Guderley's 1942 work on radial shock waves provides cases of self-similar Euler flows exhibiting blowup of primary (undifferentiated) flow variables: a converging shock wave invades a quiescent region, and the velocity and pressure in its…
Entropy stabilization of the compressible Euler system is achieved by adapting the averages that are applied to the density and internal energy variables. The approach achieves non-linear robustness despite the use of simplified symmetric…
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…
We prove finite-time vorticity blowup for smooth solutions of the 2D compressible Euler equations with smooth, localized, and non-vacuous initial data. The vorticity blowup occurs at the time of the first singularity, and is accompanied by…
We prove the existence of nonradial classical solutions to the 2D incompressible Euler equations with compact support. More precisely, for any positive integer $k$, we construct compactly supported stationary Euler flows of class…
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 study the formation of singularity for the Euler-Poisson system equipped with the Boltzmann relation, which describes the dynamics of ions in an electrostatic plasma. In general, it is known that smooth solutions to nonlinear hyperbolic…
We consider the incompressible Euler equations in $R^2$ when the initial vorticity is bounded, radially symmetric and non-increasing in the radial direction. Such a radial distribution is stationary, and we show that the monotonicity…
We study the singularity formation of smooth solutions of the relativistic Euler equations in $(3+1)$-dimensional spacetime for both finite initial energy and infinite initial energy. For the finite initial energy case, we prove that any…
This note is devoted to the linear stability of the Couette flow for the non-isentropic compressible Euler equations in a domain $\mathbb{T}\times \mathbb{R}$. Exploiting the several conservation laws originated from the special structure…
The pressureless Euler equations can be used as simple models of cosmology or plasma physics. In this paper, we construct the exact solutions in non-radial symmetry to the pressureless Euler equations in $R^{N}:$% [c]{c}%…
Strongly nonlinear dynamics, from fluid turbulence to quantum chromodynamics, have long constituted some of the most challenging problems in theoretical physics. This review describes a unified theoretical framework, the loop space…
The nonlinear Schroedinger model is a prototypical dispersive wave equation that features finite time blowup, either for supercritical exponents (for fixed dimension) or for supercritical dimensions (for fixed nonlinearity exponent). Upon…
We investigate the self-similar singularity of a 1D model for the 3D axisymmetric Euler equations, which is motivated by a particular singularity formation scenario observed in numerical computation. We prove the existence of a discrete…
In this article, we study small perturbations of the family of Friedmann-Lema\^itre-Robertson-Walker cosmological background solutions to the 1 + 3 dimensional Euler-Einstein system with a positive cosmological constant. These background…
Partial differential equations describing compressible fluids are prone to the formation of shock singularities, arising from faster upstream fluid particles catching up to slower, downstream ones. In geometric terms, this causes the…
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…
We examine the two-dimensional Euler equations including the local energy (in)equality as a differential inclusion and show that the associated relaxation essentially reduces to the known relaxation for the Euler equations considered…
We prove that standing-waves solutions to the non-linear Schr\"odinger equation in dimension one whose profiles can be obtained as minima of the energy over the mass, are orbitally stable and non-degenerate, provided the non-linear term $ G…