Related papers: Non-uniform dependence for the Novikov equation in…

In this work we consider the initial value problem (IVP) associated to the Ostrovsky equations $$\left. \begin{array}{rl} u_t+\partial_x^3 u\pm \partial_x^{-1}u +u \partial_x u &\hspace{-2mm}=0,\qquad\qquad x\in\mathbb R,\; t\in\mathbb R,\\…

We show that the so called Robinson-Trautman-Maxwell equations do not constitute a well posed initial value problem. That is, the dependence of the solution on the initial data is not continuous in any norm built out from the initial data…

We consider three special cases of the initial value problem of the first Painlev\'e equation (PI). Our approach is based on the method of uniform asymptotics introduced by Bassom, Clarkson, Law and McLeod. A rigorous proof of a property of…

We introduce a decoupling method on the Wiener space to define a wide class of an\-iso\-tro\-pic Besov spaces. The decoupling method is based on a general distributional approach and not restricted to the Wiener space. The class of Besov…

For the famous Camassa-Holm equation, the well-posedness in $B^{1+\frac{1}{p}}_{p,1}(\mathbb{R})$ with $1\leq p\leq2$ and the ill-posedness in $B^{1+\frac{1}{p}}_{p,r}(\mathbb{R})$ with $1\leq p\leq+\infty,\ 1<r\leq+\infty$ had been studied…

In this paper, we obtain the global well-posedness and scattering for the radial solution to the defocusing conformal invariant nonlinear wave equation with initial data in the critical Besov space…

In this paper, we prove the local well-posedness of 3-D density-dependent liquid crystal flows with initial data in the critical Besov spaces, without assumptions of small density variation. Furthermore, if the initial density is close…

We consider a one-parameter family of beam equations with Hamiltonian non-linearity in one space dimension under periodic boundary conditions. In a unified functional framework we study the long time evolution of initial data in two…

In this work, we investigate the Cauchy problem for the spatially inhomogeneous non-cutoff Kac equation. If the initial datum belongs to the spatially critical Besov space, we can prove the well-posedness of weak solution under a…

We give a construction of a divergence-free vector field $u_0 \in H^s \cap B^{-1}_{\infty,\infty}$, for all $s<1/2$, such that any Leray-Hopf solution to the Navier-Stokes equation starting from $u_0$ is discontinuous at $t=0$ in the metric…

This paper investigates instances of Sobolev embeddings characterized by local compactness at every point within their domain, except for a single point. We obtain the sharp conditions that distinguish compactness from non-compactness and…

For the cubic ab-family of equations with $a\neq0$, it is proved that there exists an initial data in the Sobolev space $H^s$, $s<3/2$, with non-unique solutions on the circle.

There are known trace and extension theorems relating functions in a weighted Sobolev space in a domain U to functions in a Besov space on the boundary bU. We extend these theorems to the case where the Sobolev exponent p is less than one…

For $0 < s < 1 < q < \infty$, we characterize the homeomorphisms $\varphi : \real^n \to \real^n$ for which the composition operator $f \mapsto f \circ \varphi$ is bounded on the homogeneous, scaling invariant Besov space…

Analytic smooth solutions of a general, strongly parabolic semi-linear Cauchy problem of $2m$-th order in $\mathbb{R}^N\times (0,T)$ with analytic coefficients (in space and time variables) and analytic initial data (in space variables) are…

We consider a perturbation of the Benjamin Ono equation with periodic boundary conditions on a segment. We consider the case where the perturbation is Hamiltonian and the corresponding Hamiltonian vector field is analytic as a map form…

Let $0<p<\infty$, $0<q\leq\infty$, and $s\in\mathbb{R}$. We introduce a new type of generalized Besov-type spaces $B_{p,q}^{s,\varphi}(\mathbb{R}^d)$ and generalized Triebel-Lizorkin-type spaces $F_{p,q}^{s,\varphi}(\mathbb{R}^d)$, where…

We consider the Cauchy problem for the nonlinear Schr\"odinger equation $iu_t+ \Delta u+ \lambda |u|^\alpha u=0$ in $\R^N $, in the $H^s$-subcritical and critical cases $0<\alpha \le 4/(N-2s)$, where $0<s<N/2$. Local existence of solutions…

We prove a result on the convex dependence of solutions of ordinary differential equations on an ordered finite-dimensional real vector space with respect to the initial data.

A H\"older type inequality in Besov spaces is established and applied to show that every strong solution $u(t,x)$ on (0,T) of the Navier-Stokes equations can be continued beyond $t>T$ provided that the vorticity $\omega(t,x)\in L^{\frac…