Viscosity Solutions of Fully Nonlinear Parabolic Path Dependent PDEs: Part I

The main objective of this paper and the accompanying one \cite{ETZ2} is to provide a notion of viscosity solutions for fully nonlinear parabolic path-dependent PDEs. Our definition extends our previous work \cite{EKTZ}, focused on the semilinear case, and is crucially based on the nonlinear optimal stopping problem analyzed in \cite{ETZ0}. We prove that our notion of viscosity solutions is consistent with the corresponding notion of classical solutions, and satisfies a stability property and a partial comparison result. The latter is a key step for the wellposedness results established in \cite{ETZ2}. We also show that the value processes of path-dependent stochastic control problems are viscosity solutions of the corresponding path dependent dynamic programming equation.Comment: 42 page

On a vector-valued generalisation of viscosity solutions for general PDE systems

We propose a theory of non-differentiable solutions which applies to fully nonlinear PDE systems and extends the theory of viscosity solutions of Crandall-Ishii-Lions to the vectorial case. Our key ingredient is the discovery of a notion of extremum for maps which extends min-max and allows "nonlinear passage of derivatives" to test maps. This new PDE approach supports certain stability and convergence results, preserving some basic features of the scalar viscosity counterpart. In this first part of our two-part work we introduce and study the rudiments of this theory, leaving applications for the second part.Comment: 34 pages, 6 figure

Extended equation for description of nonlinear waves in liquid with gas bubbles

Nonlinear waves in a liquid with gas bubbles are studied. Higher order terms with respect to the small parameter are taken into account in the derivation of the equation for nonlinear waves. A nonlinear differential equation is derived for long weakly nonlinear waves taking into consideration liquid viscosity, inter--phase heat transfer and surface tension. Additional conditions for the parameters of the equation are determined for integrability of the mathematical model. The transformation for linearization of the nonlinear equation is presented too. Some exact solutions of the nonlinear equation are found for integrable and non--integrable cases. The nonlinear waves described by the nonlinear equation are numerically investigated