Related papers: Time decay for solutions to the Stokes equations w…
We study the smoothness of the upper and lower value functions of stochastic differential games in the framework of time-homogeneous (possibly degenerate) diffusion processes in a domain, under the assumption that the diffusion, drift and…
We consider linear, time-dependent and skew-adjoint perturbations of periodic transport equations on the one-dimensional torus. We describe the long-time behavior of solutions for all non-degenerate perturbations in resonant regime, proving…
This paper shows how the theory of dynamic risk measures provides viscosity solutions to a family of second-order parabolic partial differential equations, even in the degenerate case. First, motivated by the martingale problem approach of…
Relying on the analysis of characteristics, we prove the uniqueness of conservative solutions to the variational wave equation $u_{tt}-c(u) (c(u)u_x)_x=0$. Given a solution $u(t,x)$, even if the wave speed $c(u)$ is only H\"older continuous…
This paper investigates the Cauchy problem of the time-space fractional Keller-Segel-Navier- Stokes model, which can describe both memory effect and L\'evy process of the system. The local existence and global existence in Lebesgue space…
We consider the hypodissipative Navier-Stokes equations on $[0,T]\times\mathbb{T}^{d}$ and seek to construct non-unique, H\"older-continuous solutions with epochs of regularity (smooth almost everywhere outside a small singular set in…
We consider the stationary (time-independent) Navier-Stokes equations in the whole threedimensional space, under the action of a source term and with the fractional Laplacian operator (--$\Delta$) $\alpha$/2 in the diffusion term. In the…
Consider the Cauchy problem of incompressible Navier-Stokes equations in $\mathbb{R}^3$ with uniformly locally square integrable initial data. If the square integral of the initial datum on a ball vanishes as the ball goes to infinity, the…
We consider the discrete time stopping problem \[ V(t,x) = \sup_{\tau}E_{(t,x)}[g(\tau, X_\tau)],\] where $X$ is a random walk. It is well known that the value function $V$ is in general not smooth on the boundary of the continuation set…
We study the strong approximation of stochastic differential equations with discontinuous drift coefficients and (possibly) degenerate diffusion coefficients. To account for the discontinuity of the drift coefficient we construct an…
In this paper, we examine the averaging effect of a highly oscillating external force on the solutions of the Navier-Stokes equations. We show that, as long as the force time-average decays over time, if the frequency and amplitude of the…
We study the Stokes problem in a bounded planar domain $\Omega$ with a friction type boundary condition that switches between a slip and no-slip stage. Unlike our previous work [6], in the present paper the threshold value may depend on the…
We study the motion of a particle in a random time-dependent vector field defined by the 2D Navier-Stokes system with a noise. Under suitable non-degeneracy hypotheses we prove that the empirical measures of the trajectories of the pair…
We study the Cauchy problem for a system of cubic nonlinear Klein-Gordon equations in one space dimension. Under a suitable structural condition on the nonlinearity, we will show that the solution exists globally and decays of the order…
This work explores Navier-Stokes equation with no gravitational forces. In short, it shows that any smooth solution that decays quickly must take the form $$ \textbf{u}(x,t)- \dfrac{1}{4\pi}\textbf{Curl}\Biggl(…
We consider solutions of the Cauchy problem for semilinear equations with (possibly) different L\'evy operators. We provide various results on their convergence under the assumption that symbols of the involved operators converge to the…
Stochastic Stokes' drift and hypersensitive transport driven by dichotomous noise are theoretically investigated. Explicit mathematical expressions for the asymptotic probability density and drift velocity are derived including the…
In this paper, the Cauchy problem for the three-dimensional (3-D) isentropic compressible Navier-Stokes equations is considered. When viscosity coefficients are given as a constant multiple of the density's power ($\rho^\delta$ with…
This paper introduces a randomized tamed Euler scheme tailored for L\'evy-driven stochastic differential equations (SDEs) with superlinear random coefficients and Carath\'eodory-type drift. Under assumptions that allow for time-irregular…
The Degasperis-Procesi (DP) equation \begin{align} &u_t-u_{txx}+3\kappa u_x+4uu_x=3u_x u_{xx}+uu_{xxx}, \nonumber \end{align} serving as a model delineating the propagation of shallow water waves, stands as a completely integrable system…