Regularity for the fully nonlinear parabolic thin obstacle problem

We prove [Formula: see text] regularity (in the parabolic sense) for the viscosity solution of a boundary obstacle problem with a fully nonlinear parabolic equation in the interior. Following the method which was first introduced for the harmonic case by L. Caffarelli in 1979, we extend the results of I. Athanasopoulos (1982) who studied the linear parabolic case and the results of E. Milakis and L. Silvestre (2008) who treated the fully nonlinear elliptic case.

Convergence of time-splitting approximations for degenerate convection-diffusion equations with a random source

In this paper we propose a time-splitting method for degenerate convection-diffusion equations perturbed stochastically by white noise. This work generalizes previous results on splitting operator techniques for stochastic hyperbolic conservation laws for the degenerate parabolic case. The convergence in $\begin{array}{} \displaystyle L^p_{loc} \end{array}$ of the time-splitting operator scheme to the unique weak entropy solution is proven. Moreover, we analyze the performance of the splitting approximation by computing its convergence rate and showing numerical simulations for some benchmark examples, including a fluid flow application in porous media.

External optimal control of fractional parabolic PDEs

In [Antil et al. Inverse Probl. 35 (2019) 084003.] we introduced a new notion of optimal control and source identification (inverse) problems where we allow the control/source to be outside the domain where the fractional elliptic PDE is fulfilled. The current work extends this previous work to the parabolic case. Several new mathematical tools have been developed to handle the parabolic problem. We tackle the Dirichlet, Neumann and Robin cases. The need for these novel optimal control concepts stems from the fact that the classical PDE models only allow placing the control/source either on the boundary or in the interior where the PDE is satisfied. However, the nonlocal behavior of the fractional operator now allows placing the control/source in the exterior. We introduce the notions of weak and very-weak solutions to the fractional parabolic Dirichlet problem. We present an approach on how to approximate the fractional parabolic Dirichlet solutions by the fractional parabolic Robin solutions (with convergence rates). A complete analysis for the Dirichlet and Robin optimal control problems has been discussed. The numerical examples confirm our theoretical findings and further illustrate the potential benefits of nonlocal models over the local ones.

On the plane into plane mappings of hydrodynamic type. Parabolic case

Singularities of plane into plane mappings described by parabolic two-component systems of quasi-linear partial differential equations of the first order are studied. Impediments arising in the application of the original Whitney’s approach to such a case are discussed. Hierarchy of singularities is analyzed by the double-scaling expansion method for the simplest [Formula: see text]-component Jordan system. It is shown that flex is the lowest singularity while higher singularities are given by [Formula: see text] curves which are of cusp type for [Formula: see text], [Formula: see text] Regularization of these singularities by deformation of plane into plane mappings into surface [Formula: see text] to plane is discussed. Applicability of the proposed approach to other parabolic type mappings is noted. We finally compare the results obtained for the parabolic case with non-generic gradient catastrophes for hyperbolic systems.

Slow motion for a hyperbolic variation of Allen–Cahn equation in one space dimension

The aim of this paper is to prove that, for specific initial data [Formula: see text] and with homogeneous Neumann boundary conditions, the solution of the IBVP for a hyperbolic variation of Allen–Cahn equation on the interval [Formula: see text] shares the well-known dynamical metastability valid for the classical parabolic case. In particular, using the “energy approach” proposed by Bronsard and Kohn [On the slowness of phase boundary motion in one space dimension, Comm. Pure Appl. Math. 43 (1990) 983–997], if [Formula: see text] is the diffusion coefficient, we show that in a time scale of order [Formula: see text] nothing happens and the solution maintains the same number of transitions of its initial datum [Formula: see text]. The novelty consists mainly in the role of the initial velocity [Formula: see text], which may create or eliminate transitions in later times. Numerical experiments are also provided in the particular case of the Allen–Cahn equation with relaxation.

On generalisation of H-measures

In some applications it is useful to consider variants of H-measures different from those introduced in the classical or the parabolic case. We introduce the notion of admissible manifold and define variant H-measures on Rd x P for any admissible manifold P. In the sequel we study one special variant, fractional H-measures with orthogonality property, where the corresponding manifold and projection curves are orthogonal, as it was the case with classical or parabolic H-measures, and prove the localisation principle. Finally, we present a simple application of the localisation principle.

Analytical and numerical investigation of traveling waves for the Allen–Cahn model with relaxation

A modification of the parabolic Allen–Cahn equation, determined by the substitution of Fick’s diffusion law with a relaxation relation of Cattaneo–Maxwell type, is considered. The analysis concentrates on traveling fronts connecting the two stable states of the model, investigating both the aspects of existence and stability. The main contribution is the proof of the nonlinear stability of the wave, as a consequence of detailed spectral and linearized analyses. In addition, numerical studies are performed in order to determine the propagation speed, to compare it to the speed for the parabolic case, and to explore the dynamics of large perturbations of the front.