Related papers: Transport-diffusion equations with irregular data …
Under natural spectral stability assumptions motivated by previous investigations of the associated spectral stability problem, we determine sharp $L^p$ estimates on the linearized solution operator about a multidimensional planar periodic…
Excitonic transport in static disordered one dimensional systems is studied in the presence of thermal fluctuations that are described by the Haken-Strobl-Reineker model. For short times, non-diffusive behavior is observed that can be…
Here, we study a discrete Coagulation-Fragmentation equation with a multiplicative coagulation kernel and a constant fragmentation kernel, which is critical. We apply the discrete Bernstein transform to the original…
Much effort has been spent in recent years on restoring uniqueness of McKean-Vlasov SDEs with non-smooth coefficients. As a typical instance, the velocity field is assumed to be bounded and measurable in its space variable and…
Transport events in turbulent tokamak plasmas often exhibit non-local or non-diffusive action at a distance features that so far have eluded a conclusive theoretical description. In this paper a theory of non-local transport is investigated…
We consider a rather general class of non-local in time Fokker-Planck equations and show by means of the entropy method that as $t\to \infty$ the solution converges in $L^1$ to the unique steady state. Important special cases are the…
In the paper, we consider a path-dependent Hamilton-Jacobi equation with coinvariant derivatives over the space of continuous functions. Such equations arise from optimal control problems and differential games for time-delay systems. We…
We study well-posedness for the stochastic transport equation with transport noise, as introduced by Flandoli, Gubinelli and Priola. We consider periodic solutions in $\rho \in L^{\infty}_{t} L_{x}^{p}$ for divergence-free drifts $u \in…
By means of an original approach, called "method of the moving frame", we establish existence, uniqueness and stability results for mild and weak solutions of stochastic partial differential equations (SPDEs) with path dependent…
In this paper, we show the non-uniqueness of the weak solution in the class $\rho\in L^{s}_tL^p_x$ for the transport equation driven by a divergence-free vector field $\boldsymbol{u}\in L^{\tilde{s}}_tW^{1,q}_x\cap L_t^{s'}L_x^{p'}$ happens…
The goal of this investigation was to derive strictly new properties of chaotic systems and their mutual relations. The generalized Fokker-Planck equation with a non stationary diffusion has been derived and used for chaos analysis. An…
We study a mathematical model describing the dynamics of dislocation densities in crystals. This model is expressed as a one-dimensional system of a parabolic equation and a first order Hamilton-Jacobi equation that are coupled together. We…
In this article we study (possibly degenerate) stochastic differential equations (SDE) with irregular (or discontiuous) coefficients, and prove that under certain conditions on the coefficients, there exists a unique almost everywhere…
We investigate the long time behavior of weakly dissipative semilinear Hamilton-Jacobi-Bellman (HJB) equations and the turnpike property for the corresponding stochastic control problems. To this aim, we develop a probabilistic approach…
Stability of the kink-like and soliton-like travelling wave solutions to the generalized convection-reaction-diffusion equation is studied by means of the qualitative methods and numerical simulation.
In this paper we study the stability of explicit finite difference discretizations of linear advection-diffusion equations (ADE) with arbitrary order of accuracy in the context of method of lines. The analysis first focuses on the stability…
In this work we rigorously establish a number of properties of "turbulent" solutions to the stochastic transport and the stochastic continuity equations constructed by Le Jan and Raimond in [Ann. Probab. 30(2): 826-873, 2002]. The advecting…
We deal with the vanishing viscosity scheme for the transport/continuity equation $\partial_t u + \text{div }(u\boldsymbol{b} ) = 0$ drifted by a divergence-free vector field $\boldsymbol{b}$. Under general Sobolev assumptions on…
Motivated by the viewpoint of integrable systems, we study commuting flows of 2-component quasilinear equations, reducing to investigate the solutions of the wave equation with non-constant speed. In this paper, we apply the reduction…
This paper examines the mathematical properties of the relativistic diffusion equation. The peculiar solution which Hiscock and Lindblom identified as an instability is shown to emerge from an ill-posed initial value problem. These do not…