Related papers: Generalized surface quasi-geostrophic equations wi…

We show that weak solutions to the 3D quasi-geostrophic system in the class $C^\zeta_{t,x}$ for $\zeta<\frac{1}{5}$ are not unique and may achieve any smooth, non-negative energy profile. Our proof follows a convex integration scheme which…

We look for solutions to the Schr\"odinger equation \[ -\Delta u + \lambda u = g(u) \quad \text{in } \mathbb{R}^N \] coupled with the mass constraint $\int_{\mathbb{R}^N}|u|^2\,dx = \rho^2$, with $N\ge2$. The behaviour of $g$ at the origin…

We consider boundary value problems for quasilinear first-order one-dimensional hyperbolic systems in a strip. The boundary conditions are supposed to be of a smoothing type, in the sense that the $L^2$-generalized solutions to the…

In this talk we extend a family of geodesically complete $G_{2}$ stiff fluid cosmological models to the case in which the velocity of the fluid is not orthogonal to the gradient of the transitivity surface element.

We consider the quasilinear parabolic-parabolic Keller-Segel system $$ u_t=\nabla \cdot (D(u)\nabla u) - \nabla \cdot (S(u)\nabla v), \qquad x\in\Omega, \ t>0, v_t=\Delta v -v + u, x\in\Omega, \ t>0, $$ under homogeneous Neumann boundary…

In this paper, we study the local behaviors of nonnegative local solutions of fractional order semi-linear equations $(-\Delta)^\sigma u=u^{\frac{n+2\sigma}{n-2\sigma}}$ with an isolated singularity, where $\sigma\in (0,1)$. We prove that…

In this paper, we consider the generalized Navier-Stokes equations with fritional dissipation $(-\Delta)^{\beta}$ with $\beta>\frac{1}{2}$. When $\beta\in(1,2)$, We prove that smooth solutions of the generalized Navier-Stokes equations are…

We study the nodal set of solutions to equations of the form $$ (-\Delta)^s u = \lambda_+ (u_+)^{q-1} - \lambda_- (u_-)^{q-1}\quad\text{in $B_1$}, $$ where $\lambda_+,\lambda_->0, q \in [1,2)$, and $u_+$ and $u_-$ are respectively the…

We consider general relativity with cosmological constant minimally coupled to electromagnetic field and assume that four-dimensional space-time manifold is the warped product of two surfaces with Lorentzian and Euclidean signature metrics.…

We establish the existence of quasi-periodic traveling wave solutions for the $\beta$-plane equation on $\mathbb{T}^2$ with a large quasi-periodic traveling wave external force. These solutions exhibit large sizes, which depend on the…

We consider the parabolic-elliptic Keller-Segel system \[ \left\{ \begin{aligned} u_t &= \Delta u - \chi \nabla \cdot (u \nabla v), \\ 0 &= \Delta v - v + u \end{aligned} \right. \tag{$\star$} \] in a smooth bounded domain $\Omega \subseteq…

In this paper we present a new family of solutions to the singularly perturbed Allen-Cahn equation $\alpha^2 \Delta u + u(1-u^2)=0, \quad \hbox{in }\Omega\subset \R^N $ where $N=3$, $\Omega$ is a smooth bounded domain and $\A>0$ is a small…

We consider the initial value problem for the fractionally dissipative quasi-geostrophic equation \[ \partial_t \theta + \mathcal{R}^\perp \theta \cdot \nabla \theta + \Lambda^\gamma \theta = 0, \qquad \theta(\cdot,0) =\theta_0 \] on…

This paper is concerned with the global existence and uniqueness of classical solutions to the barotropic compressible Navier-Stokes equations with degenerate viscosity coefficients in three-dimensional bounded domains or in the whole space…

For an abelian surface $A$, we explicitly construct two new families of stable vector bundles on the generalized Kummer variety $K_n(A)$ for $n\geqslant 2$. The first is the family of tautological bundles associated to stable bundles on…

In this article, we investigate the quantitative unique continuation properties of complex-valued solutions to drift equations in the plane. We consider equations of the form $\Delta u + W \cdot \nabla u = 0$ in $\mathbb{R}^2$, where $W =…

We study the inequality $$ {\rm div}\big(|x|^{-\alpha}|\nabla u|^{m-2}\nabla u\big)\geq (I_\beta\ast u^p)u^q \quad\mbox{ in } B_1\setminus\{0\}\subset {\mathbb R}^N, $$ where $\alpha>0$, $N\geq 1$, $m>1$, $p, q>m-1$ and $I_\beta$ denotes…

This article studies the vortex-wave system for the Surface Quasi-Geostrophic equation with parameter 0 < s < 1. We obtained local existence of classical solutions in H^4 under the standard ''plateau hypothesis'', H^2-stability of the…

The existence of positive solutions is considered for the Dirichlet problem \[ \left\{ \begin{array} [c]{rcll}% -\Delta_{p}u & = & \lambda\omega_{1}(x)\left\vert u\right\vert ^{q-2}% u+\beta\omega_{2}(x)\left\vert u\right\vert…

We establish new non-uniqueness results for the forced inviscid surface quasi-geostrophic equation, via an alternating formulation of convex integration techniques. Our results imply non-uniquenesss in the class of weak solutions with…