Related papers: Low regularity well-posedness for the generalized …

We investigate the well-posedness of $\alpha$-SQG equations in the half-plane, where $\alpha=0$ and $\alpha=1$ correspond to the 2D Euler and SQG equations respectively. For $0<\alpha \le 1/2$, we prove local well-posedness in certain…

We consider the modified Zakharov-Kuznetsov (mZK) equation in two space dimensions in both focusing and defocusing cases. Using the $I$-method, we prove the global well-posedness of the $H^s$ solutions for $s>\frac{3}{4}$ for any data in…

We regard the Cauchy problem for a particular Whitham-Boussinesq system modelling surface waves of an inviscid incompressible fluid layer. The system can be seen as a weak nonlocal dispersive perturbation of the shallow water system. The…

We consider the cubic non-linear Schr\"odinger equation on general closed (compact without boundary) Riemannian surfaces. The problem is known to be locally well-posed in $H^s(M)$ for $s>1/2$. Global well-posedness for $s\geq 1$ follows…

In this paper, we consider a family of piecewise constant solutions of the quasi-geostrophic shallow-water (QGSW) equation. We derive the contour dynamics equation of the QGSW front, which is a nonlinear, nonlocal dispersive equation, and…

We prove almost global well-posedness for quasilinear strongly coupled wave-Klein-Gordon systems with small and localized data in two space dimensions. We assume only mild decay on the data at infinity as well as minimal regularity. We…

We explore probabilistic approaches to the deterministic energy equality for the forced Surface Quasi-Geostrophic (SQG) equation on a torus. First, we prove the zero-noise dynamical large deviations for a corresponding stochastic SQG…

We consider the defocusing periodic fractional nonlinear Schr\"odinger equation $$ i \partial_t u +\left(-\Delta\right)^{\alpha}u=-\lvert u \rvert ^2 u, $$ where $\frac{1}{2}< \alpha < 1$ and the operator $(-\Delta)^\alpha$ is the…

In this article we consider the following generalized quasi-geostrophic equation \partial_t\theta + u\cdot\nabla \theta + \nu \Lambda^\beta \theta =0, \quad u= \Lambda^\alpha \mathcal{R}^\bot\theta, \quad x\in\mathbb{R}^2, where $\nu>0$,…

This paper studies a family of generalized surface quasi-geostrophic (SQG) equations for an active scalar $\theta$ on the whole plane whose velocities have been mildly regularized, for instance, logarithmically. The well-posedness of these…

We prove the local well-posedness for the generalized Korteweg-de Vries equation in $H^s(\mathbb{R})$, $s>1/2$, under general assumptions on the nonlinearity $f(x)$, on the background of an $L^\infty_{t,x}$-function $\Psi(t,x)$, with…

This paper considers a family of active scalar equations which modify the generalized surface quasi-geostrophic (gSQG) equations through its constitutive law and a dissipative perturbation. These modifications are characteristically mild in…

We study the stochastic complex Ginzburg-Landau equation (SCGL) with an additive space-time white noise forcing on the two-dimensional torus. This equation is singular and thus we need to renormalize the nonlinearity in order to give proper…

In this paper we prove global well-posedness of the critical surface quasigeostrophic equation on the two dimensional sphere building on some earlier work of the authors. The proof relies on an improving of the previously known pointwise…

We establish global well-posedness and scattering results for the logarithmically energy-supercritical nonlinear wave equation, under the assumption that the initial data satisfies a partial symmetry condition. These results generalize and…

We show that the generalized SQG equation with $\alpha\in(0,\frac 14]$ is locally well-posed on the half-plane in spaces of bounded integrable solutions that are natural for its dynamic on domains with boundaries, and allow for some power…

We consider fractional wave equations with exponential or arbitrary polynomial nonlinearities. We prove the global well-posedness on the support of the corresponding Gibbs measures. We provide ill-posedness constructions showing that the…

In this note, we prove the local well-posedness in the energy space of the $k$-generalized Zakharov-Kuznetsov equation posed on $ \R\times \T $ for any power non-linearity $ k\ge 2$. Moreover, we obtain global solutions under a precise…

In this paper we construct non-trivial solutions to the stationary dissipative surface quasi-geostrophic equation on the two dimensional torus which lie strictly below the critical regularity threshold of $\dot{H}^{-1/2}(\mathbb{T}^2)$.…

We obtain improved local well-posedness results for the Lorentzian timelike minimal surface equation. In dimension $d=3$, for a surface of arbitrary co-dimension, we show a gain of $1/3$ derivative regularity compared to a generic equation…