Related papers: Growth of solutions for QG and 2D Euler equations

For two-dimensional Euler equation on the torus, we prove that the uniform norm of the gradient can grow superlinearly for some infinitely smooth initial data. We also show the exponential growth of the gradient for the finite time.

This article is concerned with pointwise growth and spreading speeds in systems of parabolic partial differential equations. Several criteria exist for quantifying pointwise growth rates. These include the location in the complex plane of…

We prove the global existence of solutions with small and smooth initial data of a nonlinear dispersive equation for the motion of generalized surface quasi-geostrophic (GSQG) fronts in a parameter regime $1<\alpha<2$, where $\alpha=1$…

We consider a nonlinear, spatially-nonlocal initial value problem in one space dimension on $\mathbb{R}$ that describes the motion of surface quasi-geostrophic (SQG) fronts. We prove that the initial value problem has a unique local smooth…

In this paper, we perform a careful numerical study of nearly singular solutions of the 3D incompressible Euler equations with smooth initial data. We consider the interaction of two perturbed antiparallel vortex tubes which was previously…

We prove strong ill-posedness in $L^{\infty}$ for linear perturbations of the 2d Euler equations of the form: \[\partial_t \omega + u\cdot\nabla\omega = R(\omega),\] where $R$ is any non-trivial second order Riesz transform. Namely, we…

In this announcement, we report results on the existence of families of large-amplitude internal hydrodynamic bores. These are traveling front solutions of the full two-phase incompressible Euler equation in two dimensions. The fluids are…

This work gathers new results concerning the semi-geostrophic equations: existence and stability of measure valued solutions, existence and uniqueness of solutions under certain continuity conditions for the density, convergence to the…

We consider the classical compressible Euler's Equations in three space dimensions with an arbitrary equation of state, and whose initial data corresponds to a constant state outside a sphere. Under suitable restriction on the size of the…

Whether the 3D incompressible Euler equations can develop a singularity in finite time from smooth initial data is one of the most challenging problems in mathematical fluid dynamics. This work attempts to provide an affirmative answer to…

The process of breaking of inviscid incompressible flows along a rigid body with slipping boundary conditions is studied. Such slipping flows are compressible, which is the main reason for the formation of a singularity for the gradient of…

Coboundary and cosystolic expansion are notions of expansion that generalize the Cheeger constant or edge expansion of a graph to higher dimensions. The classical Cheeger inequality implies that for graphs edge expansion is equivalent to…

In this paper we investigate the integrability of two-dimensional partial difference equations using the newly developed techniques of study of the degree of the iterates. We show that while for generic, nonintegrable equations, the degree…

For the 2d Euler dynamics of patches, we investigate the convergence to the singular stationary solutions in the presence of a regular strain. It is proved that the rate of merging can be made double exponential for all time.

In this paper for either the sharp front Surface Quasi-Geostrophic equation or the Muskat problem we rule out the "splash singularity" blow-up scenario; in other words we prove that the contours evolving from either of these systems can not…

For continuous boundary data, including data of polynomial growth, modified Poisson integrals are used to write solutions to the half space Dirichlet and Neumann problems in $\mathbb{R}^{n}$. Pointwise growth estimates for these integrals…

We prove the existence of multiple solutions for a quasilinear elliptic equation containing a term with natural growth, under assumptions that are invariant by diffeomorphism. To this purpose we develop an adaptation of degree theory.

Using probabilistic methods, we establish a-priori estimates for two classes of quasilinear parabolic systems of partial differential equations (PDEs). We treat in particular the case of a nonlinearity which has quadratic growth in the…

Growth occurs in a wide range of systems ranging from biological tissue to additive manufacturing. This work considers surface growth, in which mass is added to the boundary of a continuum body from the ambient medium or from within the…

We consider reaction-diffusion equations $\partial_tu=\Delta u+f(u)$ in the whole space $\mathbb{R}^N$ and we are interested in the large-time dynamics of solutions ranging in the interval $[0,1]$, with general unbounded initial support.…