Related papers: Development of singularities in the relativistic E…

Existence of finite-time blow ups in the classical one-dimensional nonlinear Schr\"odinger equation (NLS) (1) i \partial_t u + u_{x x} + |u|^{2r} u = 0, u(x,0) = u_0(x) has been one of the central problems in the studies of the singularity…

In this paper we provide a complete local well-posedness theory for the free boundary relativistic Euler equations with a physical vacuum boundary on a Minkowski background. Specifically, we establish the following results: (i) local…

This paper deals with the longstanding quest of the possible existence of finite-time singularities in the equations governing the dynamics of inviscid fluids, namely, Euler equations. Here, two contributions are brought for the case of…

We are concerned with the formation of singularity and breakdown of strong solutions to the Cauchy problem of the three-dimensional full compressible magnetohydrodynamic equations with zero heat conduction. It is proved that for the initial…

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…

Let $u(t,x)$ be the solution to the Cauchy problem of a scalar conservation law in one space dimension. It is well known that even for smooth initial data the solution can become discontinuous in finite time and global entropy weak solution…

This paper concerns the global well-posedness and large time asymptotic behavior of strong and classical solutions to the Cauchy problem of the Navier-Stokes equations for viscous compressible barotropic flows in two or three spatial…

R. Glassey and W. Strauss have proved in [Arch. Rational Mech. Anal. 92 (1986), 59--90] that classical solutions to the relativistic Vlasov-Maxwell system in three space dimensions do not develop singularities as long as the support of the…

Measure-valued solutions to fluid equations arise naturally, for instance as vanishing viscosity limits, yet exhibit non-uniqueness to a vast extent. In this paper, we show that some measurevalued solutions to the two-dimensional isentropic…

In this paper, we present strong numerical evidences that the incompressible axisymmetric Euler equations with degenerate viscosity coefficients and smooth initial data of finite energy develop a potential finite-time locally self-similar…

We consider the Cauchy problem for the 3D incompressible axisymmetric swirl-free Euler equations. The convex integration method developed by De Lellis and Sz\'ekelyhidi rules out the possibility that the Euler equations admit unique…

We consider existence and uniqueness issues for the initial value problem of parabolic equations $\partial_{t} u = {\rm div} A \nabla u$ on the upper half space, with initial data in $L^p$ spaces. The coefficient matrix $A$ is assumed to be…

In this manuscript, we highlight a new phenomenon of complex algebraic singularity formation for solutions of a large class of genuinely nonlinear partial differential equations (PDEs). We start from a unique Cauchy datum, which is…

We consider the Cauchy problem for doubly non-linear degenerate parabolic equations on Riemannian manifolds of infinite volume, or in $\R^N$. The equation contains a weight function as a capacitary coefficient which we assume to decay at…

We prove a localized big bang formation result, which does not require proximity of the initial data to any background solution. Suppose that we are given initial data for the Einstein--nonlinear scalar field equations on an open set $U…

We consider the two dimensional unsteady Prandtl system. For a special class of outer Euler flows and solutions of the Prandtl system, the trace of the tangential derivative of the tangential velocity along the transversal axis solves a…

We consider solutions to the hyperbolic system of equations of ideal granular hydrodynamics with conserved mass, total energy and finite momentum of inertia and prove that these solutions generically lose the initial smoothness within a…

For the Cauchy problem of nonlinear elastic wave equations of three dimensional isotropic, homogeneous and hyperelastic materials satisfying the null condition, global existence of classical solutions with small initial data was proved in…

For periodic initial data with initial density, we establish the global existence and uniqueness of strong and classical solutions for the two-dimensional compressible Navier-Stokes equations with no restrictions on the size of initial data…

This paper is devoted to the study of the global existence of smooth solutions for the 3+1 dimensional Einstein-Klein-Gordon systems with a $U(1) \times \mathbb{R}$ isometry group for a class of regular Cauchy data. In our first paper…