相关论文: Non-renormalized solutions to the continuity equat…
We consider the Cauchy problem for the continuity equation with a bounded nearly incompressible vector field $b\colon (0,T) \times \mathbb R^d \to \mathbb R^d$, $T>0$. This class of vector fields arises in the context of hyperbolic…
We consider the stochastic transport equation with a possibly unbounded H\"older continuous vector field. Well-posedness is proved, namely, we show existence, uniqueness and strong stability of W^{1,p}-weak solutions.
We consider the problem of proving uniqueness of the solution of the continuity equation with a vector field $u \in [L^1 (0,T; W^{1,p}(\mathbb{T}^d)) \cap L^\infty ((0,T) \times \mathbb{T}^d)]^d$ with $\operatorname{div}(u) ^- \in L^1 (0,T;…
We consider the linear transport equations driven by an incompressible flow in dimensions $d\geq 3$. For divergence-free vector fields $u \in L^1_t W^{1,q}$, the celebrated DiPerna-Lions theory of the renormalized solutions established the…
In this note, we construct solutions to the 2D Euler equations which belong to the Yudovich class but lose $W^{1,p}$ regularity continuously with time.
The aim of this note is to prove sharp regularity estimates for solutions of the continuity equation associated to vector fields of class $W^{1,p}$ with $p>1$. The regularity is of "logarithmic order" and is measured by means of suitable…
We construct a large class of examples of non-uniqueness for the linear transport equation and the transport-diffusion equation with divergence-free vector fields in Sobolev spaces $W^{1,p}$.
An evidence of temporal dis-continuity of the solution in $F^s_{1, \infty}(\mathbb{R}^d)$ is presented, which implies the ill-posedness of the Cauchy problem for the Euler equations. Continuity and weak-type continuity of the solutions in…
We construct a large class of incompressible vector fields with Sobolev regularity, in dimension $d \geq 3$, for which the chain rule problem has a negative answer. In particular, for any renormalization map $\beta$ (satisfying suitable…
We consider the Cauchy problem for the continuity equation in space dimension ${d \geq 2}$. We construct a divergence-free velocity field uniformly bounded in all Sobolev spaces $W^{1,p}$, for $1 \leq p<\infty$, and a smooth compactly…
We construct a set of bounded vector fields dense in $L^p((0,2);W^{s,p}_{loc}(\mathbb{R}^2;\mathbb{R}^2))$ for $1\leq p <+\infty$ and $0\leq s<1$ with $p<1/s$ for which smooth regularisation of the vector field does not give a selection…
We study the incompressible Euler equations in the $\alpha$-modulation $M^{s,\alpha}_{p,q}$ and H\"older $C^{1+\sigma}$ spaces on the plane. We show that for these spaces the associated data-to-solution map is not continuous on bounded…
We prove the non-uniform continuity of the data-to-solution map of the incompressible Euler equations in Besov spaces $B_{p,q}^{s}$, where the parameters $p, q$ and $s$ considered here are such that the local existence and uniqueness result…
It has been known for some time that the (1,3) perturbations of the (2k+1,2) Virasoro minimal models have conserved currents which are also singular vectors of the Virasoro algebra. This also turns out to hold for the (1,2) perturbation of…
For every odd p \geq 3, we investigate representation theory of the vertex algebra WW_{2,p} associated to (2,p) minimal models for the Virasoro algebras. We demonstrate that vertex algebras WW_{2,p} are C_2--cofinite and irrational.…
A linear stochastic continuity equation with non-regular coefficients is considered. We prove existence and uniqueness of strong solution, in the probabilistic sense, to the Cauchy problem when the vector field has low regularity, in which…
These notes address two problems. First, we investigate the question of ``how many'' are (in Baire sense) vector fields in $L^1_t L^q_x$, $q \in [1, \infty)$, for which existence and/or uniqueness of local, distributional solutions to the…
In this paper, we extend our earlier unique continuation results \cite{PZ2} for the Schr\"odinger-type inequality $ |\bar\partial u| \le V|u|$ on a domain in $\mathbb C^n$ by removing the smoothness assumption on solutions $u = (u_1,…
A class of singular 3D-velocity vector fields is constructed which satisfy the incompressible 3D-Euler equation. It is shown that such a solution scheme does not exist in dimension 2. The solutions constructed are bounded and smooth up to…
We investigate the unique solvability of second order parabolic equations in non-divergence form in $W_p^{1,2}((0,T) \times \bR^d)$, $p \ge 2$. The leading coefficients are only measurable in either one spatial variable or time and one…