Related papers: H\"{o}lder continuous solutions to stochastic 3D E…

We construct global-in-time weak solutions to the pressureless Euler alignment system posed on the whole line and supplemented with initial conditions, where an initial density is an arbitrary, nonnegative, bounded, and integrable function…

We establish short-time existence of solutions to the surface quasi-geostrophic equation in both the H\"{o}lder spaces $C^r(\mathbb{R}^2)$ for $r>1$ and the uniformly local Sobolev spaces $H^s_{ul}(\mathbb{R}^2)$ for $s\geq 3$. Using…

We prove a novel stability estimate in $L^\infty _t (L^p _x)$ between the regular Lagrangian flow of a Sobolev vector field and a piecewise affine approximation of such flow. This approximation of the flow is obtained by a (sort of)…

We present a new class of numerical methods for solving stochastic differential equations with additive noise on general Riemannian manifolds with high weak order of accuracy. In opposition to the popular approach with projection methods,…

We develop a McKean-Vlasov interpretation of Navier-Stokes equations with external force field in the whole space, by associating with local mild $L^p$-solutions of the 3d-vortex equation a generalized nonlinear diffusion with random…

In this paper we investigate the long time stability of the implicit Euler scheme for the Cahn-Hilliard equation with polynomial nonlinearity. The uniform estimates in $H^{-1}$ and $\mathcal H^s_\alpha$ ($s=1,2,3$) spaces independent of the…

In this article we give necessary and sufficient conditions providing regularity of solutions to stochastic Volterra equations with infinite delay on a $d$-dimensional torus. The harmonic analysis techniques and stochastic integration in…

We examine the two-dimensional Euler equations including the local energy (in)equality as a differential inclusion and show that the associated relaxation essentially reduces to the known relaxation for the Euler equations considered…

We consider the $\alpha$-Euler equations on a bounded three-dimensional domain with frictionless Navier boundary conditions. Our main result is the existence of a strong solution on a positive time interval, uniform in $\alpha$, for…

We consider solutions to the two-dimensional incompressible Euler system with only integrable vorticity, thus with possibly locally infinite energy. With such regularity, we use the recently developed theory of Lagrangian flows associated…

We characterize, using commuting zero-flux homologies, those volume-preserving vector fields on a $3$-manifold that are steady solutions of the Euler equations for some Riemannian metric. This result extends Sullivan's homological…

We examine the blow-up claims of the incompressible Euler equations for several specific flow-fields, (1) the columnar eddies in the vicinity of stagnation; (2) a quasi-three-dimensional structure for illustrating oscillations and…

We consider an abstract functional-differential equation derived from the pressure-less Euler system with variable coefficients that includes several systems of partial differential equations arising in the fluid mechanics. Using the method…

We derive a priori estimates for the compressible free-boundary Euler equations with surface tension in three spatial dimensions in the case of a liquid. These are estimates for local existence in Lagrangian coordinates when the initial…

This paper is concerned with the numerical approximation of stochastic ordinary differential equations, which satisfy a global monotonicity condition. This condition includes several equations with super-linearly growing drift and diffusion…

Incompressible Euler flows in narrow domains, in which the horizontal length scale is much larger than other scales, play an important role in applications, and their leading-order behavior can be described by the hydrostatic Euler…

It is well known that the incompressible Euler equations can be formulated in a very geometric language. The geometric structures provide very valuable insights into the properties of the solutions. Analogies with the finite-dimensional…

The spatial discretization of the magnetic vector potential formulation of magnetoquasistatic field problems results in an infinitely stiff differential-algebraic equation system. It is transformed into a finitely stiff ordinary…

Online and stochastic gradient methods have emerged as potent tools in large scale optimization with both smooth convex and nonsmooth convex problems from the classes $C^{1,1}(\reals^p)$ and $C^{1,0}(\reals^p)$ respectively. However to our…

We construct Gaussian invariant measures for the two-dimensional Euler equation on the plane. We show the existence of solution with initial conditions in the support of the measures, namely $H^\beta_{loc}(\R^2)$ with $\beta<-1$. Uniqueness…