Related papers: Parabolic equations and SDEs with time-inhomogeneo…

We consider a large class of nonlinear FPKEs with coefficients of Nemytskii-type depending explicitly on time and space, for which it is known that there exists a sufficiently Sobolev-regular distributional solution u in L^1 and L^\infty.…

We study functional stochastic differential equations with a locally unbounded, functional drift focusing on well-posedness, stability and the strong Feller property. Following the non-functional case, we only consider integrability…

We consider regularity properties of stochastic kinetic equations with multiplicative noise and drift term which belongs to a space of mixed regularity ($L^p$-regularity in the velocity-variable and Sobolev regularity in the…

In this paper we show that the rotational Smagorinsky model for turbulent flows, can be put, for a wide range of parameters in the setting of Bochner pseudo-monotone evolution equations. This allows to prove existence of weak solutions a)…

Following Smale, we study simple symmetric mechanical systems of $n$ point particles in the plane. In particular, we address the question of the linear and spectral stability properties of relative equilibria, which are special solutions of…

We consider multidimensional SDEs with singular drift $b$ and Sobolev diffusion coefficients $\sigma$, satisfying Krylov--R\"ockner type assumptions. We prove several stability estimates, comparing solutions driven by different…

We carry on the investigation started in [2] about the regularity of weak solutions to the strongly degenerate parabolic equation \[ u_{t}-\mathrm{div}\left[(\vert Du\vert-1)_{+}^{p-1}\frac{Du}{\vert…

In this paper we develop a new approach to stochastic evolution equations with an unbounded drift $A$ which is dependent on time and the underlying probability space in an adapted way. It is well-known that the semigroup approach to…

The Dirichlet problem for a class of stochastic partial differential equations is studied in Sobolev spaces. The existence and uniqueness result is proved under certain compatibility conditions that ensure the finiteness of…

This paper constructs a solvability theory for a system of stochastic partial differential equations. On account of the Kolmogorov continuity theorem, solutions are looked for in certain H\"older-type classes in which a random field is…

We prove universality of a macroscopic behavior of solutions of a large class of semi-linear parabolic SPDEs on $\mathbb{R}_+\times\mathbb{T}$ with fractional Laplacian $(-\Delta)^{\sigma/2}$, additive noise and polynomial non-linearity,…

We study a time-inhomogeneous SDE in $\R^d$ driven by a cylindrical L\'evy process with independent coordinates which may have different scaling properties. Such a structure of the driving noise makes it strongly spatially inhomogeneous and…

We analytically derive novel explicit integral representations for the solution of nonhomogeneous initial-boundary-value problems for a large category of evolution partial differential equations of Sobolev-Galpern type with generic…

In this paper, we develop a universal, conceptually simple and systematic method to prove well-posedness to Cauchy problems for weak solutions of parabolic equations with non-smooth, time-dependent, elliptic part having a variational…

We are concerned in this paper with the degenerate fractional diffusion advection equations posed in bounded domains. Due to a suitable formulation, we show the existence of weak entropy solutions for measurable and bounded initial and…

One proves the well-posedness in the Sobolev space H^{-1} of nonlinear Fokker-Planck equations with singular drifts.Applications to existence of strong solutions to McKean-Vlasov equations are given.

We show uniqueness in law for a general class of stochastic differential equations in $\mathbb{R}^d$, $d\ge 2$, with possibly degenerate and/or fully discontinuous locally bounded coefficients among all weak solutions that spend zero time…

In this paper parabolic random partial differential equations and parabolic stochastic partial differential equations driven by a Wiener process are considered. A deterministic, tensorized evolution equation for the second moment and the…

Motivated by applications to gas filtration problems, we study the regularity of weak solutions to the strongly degenerate parabolic PDE $u_{t}-\mathrm{div}\left((\vert Du\vert-\nu)_{+}^{p-1}\frac{Du}{\vert Du\vert}\right)=f$ in…

We study nonlinear time-inhomogeneous Markov processes in the sense of McKean's seminal work [32]. These are given as families of laws $\mathbb{P}_{s,\zeta}$, $s\geq 0$, on path space, where $\zeta$ runs through a set of admissible initial…