Related papers: Self-similar solutions for dyadic models of the Eu…
The study of spherically symmetric motion is important for the theory of explosion waves. In this paper, we construct rigorously self-similar solutions to the Riemann problem of the spherically symmetric Euler equations for general…
It is shown that if the system of the Euler equations has a special global in time smooth solution with the linear profile of velocity, then another solutions with Cauchy data, close in the Sobolev norm to the initial data of the given…
We are concerned with a family of dissipative active scalar equation with velocity fields coupled via multiplier operators that can be of high-order. We consider sub-critical values for the fractional diffusion and prove global…
Dyadic models of the Euler equations were introduced as toy models to study the behaviour of an inviscid fluid in turbulence theory. In 1974 Novikov proposed a generalized mixed dyadic model that extends both Katz-Pavlovic and Obukhov…
Self-similar Euler singularities may be useful for understanding some aspects of Navier-Stokes turbulence. Here, a causal explanation for intermittency is given, based on the control of the sudden growth of the gradients by the Euler…
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 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)…
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…
We establish rigorously the existence of a three-parameter family of self-similar,globally bounded, and continuous weak solutions in two space dimensions to the compressible Euler equations with axisymmetry for gamma-law polytropic gases…
This paper presents the vortical and self-similar solutions for 2D compressible Euler equations using the separation method. These solutions complement Makino's solutions in radial symmetry without rotation. The rotational solutions provide…
The aim of this note is to present the recent results in [Buckmaster, Cao-Labora, G\'omez-Serrano, arXiv:2208.09445, 2022], concerning the existence of "imploding singularities" for the 3D isentropic compressible Euler and Navier-Stokes…
Master character of the multidimensional homogeneous Euler equation is discussed. It is shown that under restrictions to the lower dimensions certain subclasses of its solutions provide us with the solutions of various hydrodynamic type…
We present a dark fluid model described as a non-viscous, non-relativistic, rotating, and self-gravitating fluid. We assumed that the system has spherical symmetry and the matter can be described with the polytropic equation of state. The…
We study exactly self-similar blow-up profiles fot the generalized De Gregorio model for the three-dimensional Euler equation: $w_t + auw_x = u_xw, \quad u_x = Hw$ We show that for any $\alpha \in (0, 1)$ such that $|a\alpha|$ is…
The purpose of this paper is to investigate the well-posedness of several linear and nonlinear equations with a parabolic forward-backward structure, and to highlight the similarities and differences between them. The epitomal linear…
We consider the focusing nonlinear Schr\"odinger equation in three spatial dimensions with powers close to three and prove the existence of a self-similar solution. This generalizes a previous result on the cubic case and shows that…
We study the long-time behavior of solutions to the compressible Euler equations with frictional damping in the whole space, where we prescribe direction-dependent values for the density at spatial infinity. To this end, we transform the…
We are concerned on the study of the unique continuation type property for the 3D incompressible Euler equations in the self-similar type form. Discretely self-similar solution is a generalized notion of the self-similar solution, which is…
A new discrete-velocity model is presented to solve the three-dimensional Euler equations. The velocities in the model are of an adaptive nature---both the origin of the discrete-velocity space and the magnitudes of the discrete-velocities…
We consider the barotropic Euler equations in dimension d>1 with decaying density at spatial infinity. The phase portrait of the nonlinear ode governing the equation for spherically symmetric self-similar solutions has been introduced in…