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We concern with the global existence and large time behavior of compressible fluids (including the inviscid gases, viscid gases, and Boltzmann gases) in an infinitely expanding ball. Such a problem is one of the interesting models in…

We establish global existence of solutions to the compressible Euler equations, in the case that a finite volume of ideal gas expands into vacuum. Vacuum states can occur with either smooth or singular sound speed, the latter corresponding…

Considering the isentropic Euler equations of compressible fluid dynamics with geometric effects included, we establish the existence of entropy solutions for a large class of initial data. We cover fluid flows in a nozzle or in spherical…

We establish the existence and uniqueness of smooth solutions with large vorticity and weak solutions with vortex sheets/entropy waves for the steady Euler equations for both compressible and incompressible fluids in arbitrary infinitely…

In this paper, we provide a much simplified proof of the main result in [Lin and Zhang, Comm. Pure Appl. Math.,67(2014), 531--580] concerning the global existence and uniqueness of smooth solutions to the Cauchy problem for a 3D…

We prove the existence of a large class of global-in-time expanding solutions to vacuum free boundary compressible Euler flows without relying on the existence of an underlying finite-dimensional family of special affine solutions of the…

Some classical and recent results on the Euler equations governing perfect (incompressible and inviscid) fluid motion are collected and reviewed, with some small novelties scattered throughout. The perspective and emphasis will be given…

In this paper we consider the local and global well-posedness to the density-dependent incompressible viscoelastic fluids. We first study some linear models associated to the incompressible viscoelastic system. Then we approximate the…

This paper studies the global existence and uniqueness of strong solutions and its large-time behavior for the compressible isothermal Euler equations with a nonlocal dissipation. The system is rigorously derived from the kinetic…

Local stay of the solutions to the Euler equations for an ideal incompressible fluid in the end-point Triebel-Lizorkin spaces $F^s_{1, \infty}(\mathbb{R}^d)$ with $s \geq d + 1$ is clarified.

This paper is a continuation of the works in \cite{Euler} and \cite{NS}, where the authors have established the global existence of smooth compressible flows in infinitely expanding balls for inviscid gases and viscid gases, respectively.…

It has been known since work of Lichtenstein [42] and Gunther [29] in the 1920's that the $3D$ incompressible Euler equation is locally well-posed in the class of velocity fields with H\"older continuous gradient and suitable decay at…

The compressible Euler-Riesz equations are fundamental with wide applications in astrophysics, plasma physics, and mathematical biology. In this paper, we are concerned with the global existence and nonlinear stability of finite-energy…

In three space dimensions, we consider the compressible inviscid model describing the time evolution of two fluids sharing the same velocity and enjoying the algebraic pressure closure. By employing the technique of convex integration, we…

We present a local existence result for the three dimensional incompressible Euler equations. The solution is constructed using a formulation of the equations as an active vector system in Eulerian coordinates. The formulation employs the…

We show that H\"{o}lder continuous incompressible Euler flows that satisfy the local energy inequality ("globally dissipative" solutions) exhibit nonuniqueness and contain examples that strictly dissipate kinetic energy. The collection of…

This paper deals with the evolution of the Einstein gravitational fields which are coupled to a perfect fluid. We consider the Einstein--Euler system in asymptotically flat spacestimes and therefore use the condition that the energy density…

In this note we consider the motion of a solid body in a two dimensional incompressible perfect fluid. We prove the global existence of solutions in the case where the initial vorticity belongs to $L^p$ with $p>1$ and is compactly…

We prove the existence of infinite energy global solutions of the cubic wave equation in dimension greater than 3. The data is a typical element on the support of suitable probability measures.

We study the infinite-energy solutions of the Cahn-Hilliard equation in the whole 3D space in uniformly local phase spaces. In particular, we establish the global existence of solutions for the case of regular potentials of arbitrary…