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We study constrained 2-dimensional Navier-Stokes Equations driven by a multiplicative Gaussian noise in the Stratonovich form. In the deterministic case [4] we showed the existence of global solutions only on a two dimensional torus and…

The classical system of shallow-water (Saint--Venant) equations describes long surface waves in an inviscid incompressible fluid of a variable depth. Although shock waves are expected in this quasilinear hyperbolic system for a wide class…

This paper studies the global well-posedness of classical solutions to the isentropic compressible Navier-Stokes equations in 3D domains D under non-slip boundary conditions. D will separate into the inner and boundary parts along a free…

Direct algebraic method of obtaining exact solutions to nonlinear PDE's is applied to certain set of nonlinear nonlocal evolutionary equations, including nonlinear telegraph equation, hyperbolic generalization of Burgers equation and some…

We obtain general weak existence and stability results for stochastic convolution equations with jumps under mild regularity assumptions, allowing for non-Lipschitz coefficients and singular kernels. Our approach relies on weak convergence…

This paper is concerned with the study of a class of nonlinear nonlocal functional evolution problems defined in an abstract Banach algebra. We introduce an abstract functional setting that encompasses a wide range of structured population…

We study the well solvability of nonlinear backward stochastic evolutionary equations driven by a space-time white noise. We first establish a novel a priori estimate for solution of linear backward stochastic evolutionary equations, and…

In this paper we prove the global existence and uniqueness (regularity) of strong solutions to the three-dimensional viscous primitive equations, which model large scale ocean and atmosphere dynamics.

In this overview paper, we show existence of smooth solitary-wave solutions to the nonlinear, dispersive evolution equations of the form \begin{equation*} \partial_t u + \partial_x(\Lambda^s u + u\Lambda^r u^2) = 0, \end{equation*} where…

In this paper we consider an alternative formulation of a class of stochastic wave and master equations with scalar noise that are used in quantum optics for modelling open systems and continuously monitored systems. The reformulation is…

We establish the existence of martingale solutions to a class of stochastic conservation equations. The underlying models correspond to random perturbations of kinetic models for collective motion such as the Cucker-Smale and Motsch-Tadmor…

This article considers the variational wave equation with viscosity and transport noise as a system of three coupled nonlinear stochastic partial differential equations. We prove pathwise global existence, uniqueness, and temporal…

This paper is devoted to reviewing several recent developments concerning certain class of geophysical models, including the primitive equations (PEs) of atmospheric and oceanic dynamics and a tropical atmosphere model. The PEs for…

In this paper, we consider the initial-boundary value problem of the viscous 3D primitive equations for oceanic and atmospheric dynamics with only vertical diffusion in the temperature equation. Local and global well-posedness of strong…

We study the existence of weak martingale solutions to a stochastic moving boundary problem arising from the interaction between an isentropic compressible fluid and a viscoelastic structure. In the model, we consider a three-dimensional…

In this paper we will develop linear and nonlinear filtering methods for a large class of nonlinear wave equations that arise in applications such as quantum dynamics and laser generation and propagation in a unified framework. We consider…

We study the Navier-Stokes equations governing the motion of isentropic compressible fluid in three dimensions driven by a multiplicative stochastic forcing. In particular, we consider a stochastic perturbation of the system as a function…

We establish the global existence and uniqueness of classical solutions to the three-dimensional full compressible Navier-Stokes system with smooth initial data which are of small energy but possibly large oscillations where the initial…

Localized patterns and nonlinear oscillation formation on the bounded free surface of an ideal incompressible liquid are analytically investigated . Cnoidal modes, solitons and compactons, as traveling non-axially symmetric shapes are…

WE PRESENT THE RANDOM REPRESENTATIONS FOR THE NAVIER-STOKES VORTICITY EQUATIONS FOR AN INCOMPRESSIBLE FLUID IN A SMOOTH MANIFOLD WITH BOUNDARY AND REFLECTING BOUNDARY CONDITIONS FOR THE VORTICITY. WE SPECIALIZE OUR CONSTRUCTIONS TO…