Related papers: Solutions at vacuum and rarefaction waves in press…
We consider several modifications of the Euler system of fluid dynamics including its pressureless variant driven by non-local interaction repulsive-attractive and alignment forces in the space dimension $N=2,3$. These models arise in the…
In this paper, we study the large time asymptotic behavior toward rarefaction waves for solutions to the 1-dimensional compressible Navier-Stokes equations with density-dependent viscosities for general initial data whose far fields are…
Within the class of nonlinear hyperbolic balance laws posed on a curved spacetime (endowed with a volume form), we identify a hyperbolic balance law that enjoys the same Lorentz invariance property as the one satisfied by the Euler…
A class of differentiable solutions is proved for the isentropic Euler equations in two and three space dimensions. The solutions are explicitly given in terms of solutions to inviscid Burgers equations, and several directions of…
We study a pressureless Euler system with a nonlinear density-dependent alignment term, originating in the Cucker-Smale swarming models. The alignment term is dissipative in the sense that it tends to equilibrate the velocities. Its density…
We introduce a new wave formulation for the relativistic Euler equations with vacuum boundary conditions that consists of a system of non-linear wave equations in divergence form with a combination of acoustic and Dirichlet boundary…
We consider admissible weak solutions to the compressible Euler system with source terms, which include rotating shallow water system and the Euler system with damping as special examples. In the case of anti-symmetric sources such as…
We are interested in the life span and the asymptotic behaviour of the solutions to a system governing the motion of a pressureless gas, submitted to a strong, inhomogeneous magnetic field $ \e^{-1} B(x)$, of variable amplitude but fixed…
We consider the (barotropic) Euler system describing the motion of a compressible inviscid fluid driven by a stochastic forcing. Adapting the method of convex integration we show that the initial value problem is ill-posed in the class of…
In this paper, we study the asymptotic stability of rarefaction waves for the compressible isentropic Navier-Stokes equations with density-dependent viscosity. First, a weak solution around a rarefaction wave to the Cauchy problem is…
The global existence of smooth solutions to the vacuum free boundary problem with physical singularity of compressible Euler equations with damping and gravity is proved in space dimensions $n=1, 2, 3$, for the initial data being small…
We consider the Cauchy problem for the isentropic compressible Euler equations in a three-dimensional periodic domain under general pressure laws. For any smooth initial density away from the vacuum, we construct infinitely many entropy…
We consider the isentropic Euler equations of gas dynamics in the whole two-dimensional space and we prove the existence of a $C^\infty$ initial datum which admits infinitely many bounded admissible weak solutions. Taking advantage of the…
We consider an abstract functional-differential equation derived from the pressure-less Euler system with variable coefficients that includes several systems of partial differential equations arising in the fluid mechanics. Using the method…
In this paper, we consider the existence of global weak solutions to a one dimensional fluid-particles interaction model: inviscid Burgers-Vlasov equations with fluid velocity in $L^\infty$ and particles' probability density in $L^1$. Our…
We investigate the long-time behavior of solutions to the isothermal Euler, Korteweg or quantum Navier Stokes equations, as well as generalizations of these equations where the convex pressure law is asymptotically linear near vacuum. By…
Global stability of the spherically symmetric nonisentropic compressible Euler equations with positive density around global-in-time background affine solutions is shown in the presence of free vacuum boundaries. Vacuum is achieved despite…
We investigate the large time behavior of solutions to the two-dimensional viscous Burgers equation $u_t+uu_x+uu_y=\Delta u$, toward a non-self-similar rarefaction wave of inviscid Burgers equation with two initial constant states,…
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 develop a global wellposedness theory for weak solutions to the 1D Euler-alignment system with measure-valued density, bounded velocity, and locally integrable communication protocol. A satisfactory understanding of the low-regularity…