Related papers: Transport-diffusion equations with irregular data …
We study stability of travelling wave solutions to Korteweg--de Vries type equations which has the fractional dispersion and integer-indices double power nonlinearities. It may depend on parity combinations of the two indices and the…
This article studies a Fokker-Planck type equation of fractional diffusion with conservative drift $\partial$f/$\partial$t = $\Delta$^($\alpha$/2) f + div(Ef), where $\Delta$^($\alpha$/2) denotes the fractional Laplacian and E is a…
In this paper, we give a uniqueness result to a transport equation fulfilled by probability measure on a infinite dimensional Hilbert space. Main arguments are based on projective aspects and a probabilistic representation of the solutions.…
The method of potential solutions of Fokker-Planck equations is used to develop a transport equation for the joint probability of N stochastic variables with Lochner's generalized Dirichlet distribution (R.H. Lochner, A Generalized…
Pathwise uniqueness for stochastic PDEs with drift in differential form is a main open problem in the recent literature on regularisation by noise. This paper establishes a self-contained theory in the framework of stochastic evolution…
We consider reaction-diffusion equations that are stochastically forced by a small multiplicative noise term. We show that spectrally stable travelling wave solutions to the deterministic system retain their orbital stability if the…
In this article we are concerned with the instability and stability properties of traveling wave solutions of the double dispersion equation $~u_{tt} -u_{xx}+a u_{xxxx}-bu_{xxtt} = - (|u|^{p-1}u)_{xx}~$ for $~p>1$, $~a\geq b>0$. The main…
The singular set of a viscosity solution to a Hamilton-Jacobi equation is known to propagate, from any noncritical singular point, along singular characteristics which are curves satisfying certain differential inclusions. In the…
This paper is devoted to $\phi$-entropies applied to Fokker-Planck and kinetic Fokker-Planck equations in the whole space, with confinement. The so-called $\phi$-entropies are Lyapunov functionals which typically interpolate between Gibbs…
This article addresses the local boundedness and H\"older continuity of weak solutions to kinetic Fokker-Planck equations with general transport operators and rough coefficients. These results are due to the mixing effect of diffusion and…
We consider the road-field reaction-diffusion model introduced by Berestycki, Roquejoffre, and Rossi. By performing a "thin-front limit," we are able to deduce a Hamilton-Jacobi equation with a suitable effective Hamiltonian on the road…
We study self-regulating processes modeling biological transportation networks. Firstly, we write the formal $L^2$-gradient flow for the symmetric tensor valued diffusivity $D$ of a broad class of entropy dissipations associated with a…
The classical result by It\^o on the existence of strong solutions of stochastic differential equations (SDEs) with Lipschitz coefficients can be extended to the case where the drift is only measurable and bounded. These generalizations are…
We discuss a connection between a generative model, called the diffusion model, and nonequilibrium thermodynamics for the Fokker-Planck equation, called stochastic thermodynamics. Using techniques from stochastic thermodynamics, we derive…
We investigate the differentiability issue of the drift-diffusion equation with nonlocal L\'evy-type diffusion at either supercritical or critical type cases. Under the suitable conditions on the drift velocity and the forcing term in terms…
Here we study the nonnegative solutions of the viscous Hamilton-Jacobi problem \[ \left\{\begin{array} [c]{c}% u_{t}-\nu\Delta u+|\nabla u|^{q}=0, u(0)=u_{0}, \end{array} \right. \] in $Q_{\Omega,T}=\Omega\times\left(0,T\right) ,$ where…
We study the unique continuation property for the neutron transport equation and for a simplified model of the Fokker-Planck equation in a bounded domain with absorbing boundary condition. An observation estimate is derived. It depends on…
We study the relaxation to equilibrium for a class linear one-dimensional Fokker-Planck equations characterized by a particular subcritical confinement potential. An interesting feature of this class of Fokker-Planck equations is that, for…
This is a survey paper on the quantitative analysis of the propagation of singularities for the viscosity solutions to Hamilton-Jacobi equations in the past decades. We also review further applications of the theory to various fields such…
We consider the numerical approximation of the ill-posed data assimilation problem for stationary convection-diffusion equations and extend our previous analysis in [Numer. Math. 144, 451--477, 2020] to the convection-dominated regime.…