相关论文: Local regularity for the modified SQG patch equati…
The formalism of exact 1D quantization is reviewed in detail and applied to the spectral study of three concrete Schr\"odinger Hamiltonians $[-\d^2/\d q^2 + V(q)]^\pm$ on the half-line $\{q>0\}$, with a Dirichlet (-) or Neumann (+)…
The motivation of the present study is to discuss the global (in time) existence of small data solutions to the following semi-linear structurally damped $\sigma$-evolution models: \begin{equation*}…
We consider the maximal regularity problem for non-autonomous evolution equations \begin{equation} \left\{ \begin{array}{rcl} u'(t) + A(t)\,u(t) &=& f(t), \ t \in (0, \tau] u(0)&=&u_0. \end{array} \right. \end{equation} Each operator $A(t)$…
We consider the maximal regularity problem for non-autonomous evolution equations of the form $u(t) + A(t) u(t) = f(t)$ with initial data $u(0) = u\_0$ . Each operator $A(t)$ is associated with a sesquilinear form $a(t; *, *)$ on a Hilbert…
In this paper we consider the generalized surface quasi-geostrophic $\alpha$-SQG equations, in the "sublinear regime" $\alpha \in (0, 1)$ and we study the stability of vortex patches close to vortex discs. We shall prove that for regular,…
We consider solutions to linear parabolic SPDEs of the form \[ \mathrm{d} u(t) + A u(t)\, \mathrm{d} t = g(t)\, \mathrm{d} \beta, \qquad u(0)=0, \] where $A$ is a positive, invertible, and self-adjoint operator on a Hilbert space $X$,…
This paper studies the regularity of weak solutions to a class of parabolic perturbed fractional $1$-Laplace equations. Our analysis combines finite difference quotients, energy estimates, and iterative arguments, with a key step being the…
In this paper we address the existence of time periodic solutions for the generalized inviscid SQG equation in the unit disc with homogeneous Dirichlet boundary condition when $\alpha\in (0,1)$. We show the existence of a countable family…
Scattering problems with locally perturbed periodic surfaces have been studied both theoretically and numerically in recent years. In this paper, we will discuss the regularity results of the Bloch transform of the total fields. The idea is…
We study partial regularity for degenerate elliptic systems of double-phase type, where the growth function is given by $H(x,t)=t^p+a(x)t^q$ with $1<p\leq q$ and $a(x)$ a nonnegative $C^{0,\alpha}$-continuous function. Our main result…
We construct a one parameter family of finite time blow ups to the co-rotational wave maps problem from $S^2\times \RR$ to $S^2,$ parameterized by $\nu\in(1/2,1].$ The longitudinal function $u(t,\alpha)$ which is the main object of study…
We study the stability of quantum motion of classically regular systems in presence of small perturbations. Onthe base of a uniform semiclassical theory we derive the fidelity decay which displays a quite complexbehaviour, from Gaussian to…
We prove $C^{1,\nu}$ regularity for local minimizers of the \oh{multi-phase} energy: \begin{flalign*} w \mapsto \int_{\Omega}\snr{Dw}^{p}+a(x)\snr{Dw}^{q}+b(x)\snr{Dw}^{s} \ dx, \end{flalign*} under sharp assumptions relating the couples…
We consider the Surface Quasi-Geostrophic equation (SQG) driven by space-time white noise and show the existence of a local in time solution by applying the theory of regularity structures. A main difficulty is the presence of…
It is well known that the Euler vortex patch in $\mathbb{R}^{2}$ will remain regular if it is regular enough initially. In bounded domains, the regularity theory for patch solutions is less complete. We study here the Euler vortex patch in…
We consider homogeneous (stationary self-similar) solutions to the generalized surface quasi-geostrophic (gSQG) equations parametrized by the constant $0<s<1$, representing the 2D Euler equations ($s=1$), the SQG equations $(s=1/2)$, and…
This paper studies the regularity and energy conservation problems for the 2D supercritical quasi-geostrophic (SQG) equation. We apply an approach of splitting the dissipation wavenumber to obtain a new regularity condition which is weaker…
We study existence, uniqueness, norm estimates and asymptotic time behaviour (in some cases can be claimed to be sharp) for the solution of a general evolutionary integral (differential) equation of scalar type on a locally compact…
The issue of so-called maximal regularity is discussed within a Hilbert space framework for a class of evolutionary equations. Viewing evolutionary equations as a sums of two unbounded operators, showing maximal regularity amounts to…
In this article we study the regularity of stationary points of the knot energies $E^\alpha$ introduced by O'Hara in the range $\alpha \in (2,3)$. In a first step we prove that $E^\alpha$ is $C^1$ on the set of all regular embedded closed…