Related papers: Stochastic continuity equation with non-smooth vel…
In the present article, solvability in Sobolev spaces is investigated for a class of degenerate stochastic integro-differential equations of parabolic type. Existence and uniqueness is obtained, and estimates are given for the solution.
We consider a degenerate stochastic differential equation that has a sticky point in the Markov process sense. We prove that weak existence and weak uniqueness hold, but that pathwise uniqueness does not hold nor does a strong solution…
This paper presents an overview of physical ideas and mathematical methods for implementing non-smooth and discontinuous substitutions in dynamical systems. General purpose of such substitutions is to bring the differential equations of…
Whether singularities can form in fluids remains a foundational unanswered question in mathematics. This phenomenon occurs when solutions to governing equations, such as the 3D Euler equations, develop infinite gradients from smooth initial…
We establish partial regularity for vector-valued solutions to inhomogeneous elliptic systems in divergence form where the coefficients are possibly discontinuous with respect to $x$. More precisely, we assume a VMO-condition with respect…
In our previous papers we proposed a continuum model for the dynamics of the systems of self-propelling particles with conservative kinematic constraints on the velocities. We have determined a class of stationary solutions of this…
We discuss the stability theory and numerical analysis of the Helmholtz equation with variable and possibly non-smooth or oscillatory coefficients. Using the unique continuation principle and the Fredholm alternative, we first give an…
We prove the existence of local stable, unstable, and center manifolds for stochastic semiflows induced by rough differential equations driven by rough paths valued stochastic processes around random fixed points of the equation. Examples…
In this paper, the stability behaviors of stochastic differential equations (SDEs) driven by time-changed Brownian motions are discussed. Based on the generalized Lyapunov method and stochastic analysis, necessary conditions are provided…
We investigate the sedimentation properties of quasi-neutrally buoyant inertial particles carried by incompressible zero-mean fluid flows. We obtain generic formulae for the terminal velocity in generic space-and-time periodic (or steady)…
We consider the following stochastic partial differential equation, \begin{align*} &dY_t=L^\ast Y_tdt+A^\ast Y_t\cdot dB_t\\ &Y_0=\psi, \end{align*} associated with a stochastic flow $\{X(t,x)\}$, for $t \geq 0$, $x \in \mathbb{R}^d$, as in…
In this article the question on uniqueness of weak solution of the incompressible Navier-Stokes Equations in the 3-dimensional case is studied. Here the investigation is carried out with use of another approach. The uniqueness of velocity…
In this paper, we study the existence and uniqueness of periodic solutions of the differential equation of the form . Here, we obtain some sufficient conditions which guarantee the existence of periodic solutions. This equation is a quite…
We review recent progress in the study of infinite-dimensional stochastic differential equations with symmetry. This paper contains examples arising from random matrix theory.
In this paper we discuss the existence of stationary incompressible fluids with splash singularities. Specifically, we show that there are stationary solutions to the Euler equations with two fluids whose interfaces are arbitrarily close to…
The solutions of stochastic differential equations without an external drift are stochastically invariant under time reversal. This singles out the "anti-Ito" integral.
This paper is devoted to study the asymptotic stability of wave equations with constant coefficients coupled by velocities. By using Riesz basis approach, multiplier method and frequency domain approach respectively, we find the sufficient…
We obtain $T_2(C)$ for stochastic differential equations with Dini continuous drift and $T_1(C)$ stochastic differential equations with singular coefficients.
The concept of square-mean almost automorphy for stochastic processes is introduced. The existence and uniqueness of square-mean almost automorphic solutions to some linear and non-linear stochastic differential equations are established…
This article aims to investigate sufficient conditions for the stability of stochastic differential equations with a random structure, particularly in contexts involving the presence of concentration points. The proof of asymptotic…