This paper presents a study to achieve discrete entropy consistency using artificial and physical diffusion mechanisms. The study begins with the one-dimensional viscous Burgers equation, specifically looking at the shock results of entropy-conserved fluxes combined with a few choices of artificial and physical viscous diffusions. The approach is then repeated for the Navier–Stokes equations. O...

In this paper we construct numerical schemes to approximate linear transport equations with slab geometry by one-dimensional diffusion equations. We treat both the case of pure diffusive scaling and the case where kinetic and diffusive scalings coexist. The diffusion equations and their data are derived from asymptotic and layer analysis which allows general scattering kernels and general data....

Space and time scales are not independent in diffusion. In fact, numerical simulations show that different patterns are obtained when space and time steps (∆x and ∆t) are varied independently. On the other hand, anisotropy effects due to the symmetries of the discretization lattice prevent the quantitative calibration of models. We introduce a new class of explicit difference methods for numeri...

Unbounded stochastic control problems may lead to Hamilton-Jacobi-Bellman equations whose Hamiltonians are not always defined, especially when the diffusion term is unbounded with respect to the control. We obtain existence and uniqueness of viscosity solutions growing at most like o(1+ |x|p) at infinity for such HJB equations and more generally for degenerate parabolic equations with a superli...

In a series of papers [1]–[4] we considered parabolic equations for measures on R. Our motivation was a study of the Kolmogorov equations for transition probabilities of diffusion processes. Here we are concerned with similar problems in infinite dimensions. Applications are given to stochastic partial differential equations such as stochastic equations of Navier– Stokes and reaction-diffusion ...

In some cases, solutions to nonlinear PDEs happen to be asymptotically (for large x and/or t) invariant under a group G which is not a symmetry of the equation. After recalling the geometrical meaning of symmetries of differential equations – and solution-preserving maps – we provide a precise definition of asymptotic symmetries of PDEs; we deal in particular, for ease of discussion and physica...

We prove existence, uniqueness, and stability of transition fronts (generalized traveling waves) for reaction-diffusion equations in cylindrical domains with general inhomogeneous ignition reactions. We also show uniform convergence of solutions with exponentially decaying initial data to time translates of the front. In the case of stationary ergodic reactions the fronts are proved to propagat...

We report on recent progress in the study of evolution processes involving degenerate parabolic equations which may exhibit free boundaries. The equations we have selected follow two recent trends in diffusion theory: considering anomalous diffusion with long-range effects, which leads to fractional operators or other operators involving kernels with large tails; and the combination of diffusio...

We devise a new geometric approach to study the propagation of disturbance compactly supported data in reaction-diffusion equations. The method builds a bridge between the propagation of disturbance and of almost planar solutions. It applies to very general reaction-diffusion equations. The main consequences we derive in this paper are: a new proof of the classical Freidlin-Gärtner formula for ...