Related papers: Non-uniqueness of Weak Solutions to the Wave Map P…
Consider the focusing energy-critical wave equation in space dimension 3, 4 or 5. In a previous paper, we proved that any solution which is bounded in the energy space converges, along a sequence of times and in some weak sense, to a…
This paper aims to establish the local and global well-posedness theory in $L^1$, inspired by the approach of Keel and Tao [Internat. Math. Res. Notices, 1998], for the forced wave map equation in the ``external'' formalism. In this…
We consider the Cauchy problem for the weakly dissipative wave equation $$ \square u+\frac\mu{1+t} u_t=0 $$ with parameter $\mu\ge2$. Based on the explicit representations of solutions provided in [Math. Meth. Appl. Sci. 2004; {\bf…
This paper concerns the McKean-Vlasov stochastic differential equation (SDE) with common noise. An appropriate definition of a weak solution to such an equation is developed. The importance of the notion of compatibility in this definition…
The existence of proper weak solutions of the Dirichlet-Cauchy problem constituted by the Navier-Stokes-Fourier system which characterizes the incompressible homogeneous Newtonian fluids under thermal effects is studied. We call proper weak…
We prove weak-strong uniqueness in the class of admissible measure-valued solutions for the isentropic Euler equations in any space dimension and for the Savage-Hutter model of granular flows in one and two space dimensions. For the latter…
Given a Hilbert space, we investigate the well-posedness of the Cauchy problem for the wave equation for operators with discrete non-negative spectrum acting on it. We consider the cases when the time-dependent propagation speed is regular,…
We prove existence and uniqueness of distributional, bounded, nonnegative solutions to a fractional filtration equation in ${\mathbb R}^d$. With regards to uniqueness, it was shown even for more general equations in [19] that if two bounded…
So far existence of dissipative weak solutions for the compressible Navier-Stokes equations (i.e. weak solutions satisfying the relative energy inequality) is known only in the case of boundary conditions with non zero inflow/outflow (i.e.,…
In this paper we mainly investigate the Cauchy problem of a three-component Camassa-Holm system. By using the method of approximation of smooth solutions, a regularization technique and the special structure of the system, we prove the…
This work is on the Cauchy problem for critical wave maps coupled to Einstein's equations of general relativity. The main result of this work is the proof that the energy of the Einstein-equivariant wave map system does not concentrate…
Let $M$ be a compact Riemannian homogeneous space (e.g. a Euclidean sphere). We prove existence of a global weak solution of the stochastic wave equation \mathbf D_t\partial_tu=\sum_{k=1}^d\mathbf…
We show that 1-D rarefaction wave solutions are unique in the class of bounded entropy solutions to the multidimensional compressible Euler system. Such a result may be viewed as a counterpart of the recent examples of non-uniqueness of the…
In this paper, we study the wave equation on infinite graphs. On one hand, in contrast to the wave equation on manifolds, we construct an example for the non-uniqueness for the Cauchy problem of the wave equation on graphs. On the other…
In this paper, we first establish regularity of the heat flow of biharmonic maps into the unit sphere $S^L\subset\mathbb R^{L+1}$ under a smallness condition of renormalized total energy. For the class of such solutions to the heat flow of…
This paper is concerned with decay and symmetry properties of solitary wave solutions to a nonlocal shallow water wave model. It is shown that all supercritical solitary wave solutions are symmetric and monotone on either side of the crest.…
We follow the idea of Wang \cite{W} to show the existence of global weak solutions to the Cauchy problems of Landau-Lifshtiz type equations and related heat flows from a $n$-dimensional Euclidean domain $\Om$ or a $n$-dimensional closed…
In this paper we consider stochastic differential equations with discontinuous diffusion coefficient of varying sign, for which weak existence and uniqueness holds but strong uniqueness fails. We introduce the notion of $\varphi $-strong…
This paper deals with the existence of global weak solutions for 3D MHD equations when the initial data belong to the weighted spaces $L^2_{w_\gamma}$, with $w_\gamma(x)=(1+| x|)^{-\gamma}$ and $0 \leq \gamma \leq 2$. Moreover, we prove the…
We consider the stochastic Navier--Stokes equations in three dimensions and prove that the law of analytically weak solutions is not unique. In particular, we focus on three examples of a stochastic perturbation: an additive, a linear…