On the local solvability of a class of degenerate second order operators with complex coefficients

We study the local solvability of a class of operators with multiple characteristics. The class considered here complements and extends the one previously studied in Federico and Parmeggiani (CPDEs 2016, Vol. 41), in that in this paper we consider some cases of operators with complex coefficients that were not present in Federico and Parmeggiani. The class of operators considered here ideally encompasses classes of degenerate parabolic and Schrodinger type operators. We will give local solvability theorems. In general, one has L-2 local solvability, but also cases of local solvability with better Sobolev regularity will be presented

On a mixed problem for the parabolic Lam'e type operator

We consider a boundary value problem for the parabolic Lam\'e type operator being a linearization of the Navier-Stokes' equations for compressible flow of Newtonian fluids. It consists of recovering a vector-function, satisfying the parabolic Lam\'e type system in a cylindrical domain, via its values and the values of the boundary stress tensor on a given part of the lateral surface of the cylinder. We prove that the problem is ill-posed in the natural spaces of smooth functions and in the corresponding H\"older spaces; besides, additional initial data do not turn the problem to a well-posed one. Using the Integral Representation's Method we obtain the Uniqueness Theorem and solvability conditions for the problem

Nonlinear theory for coalescing characteristics in multiphase Whitham modulation theory

The multiphase Whitham modulation equations with $N$ phases have $2N$ characteristics which may be of hyperbolic or elliptic type. In this paper a nonlinear theory is developed for coalescence, where two characteristics change from hyperbolic to elliptic via collision. Firstly, a linear theory develops the structure of colliding characteristics involving the topological sign of characteristics and multiple Jordan chains, and secondly a nonlinear modulation theory is developed for transitions. The nonlinear theory shows that coalescing characteristics morph the Whitham equations into an asymptotically valid geometric form of the two-way Boussinesq equation. That is, coalescing characteristics generate dispersion, nonlinearity and complex wave fields. For illustration, the theory is applied to coalescing characteristics associated with the modulation of two-phase travelling-wave solutions of coupled nonlinear Schr\"odinger equations, highlighting how collisions can be identified and the relevant dispersive dynamics constructed.Comment: 40 pages, 2 figure

On the Volterra property of a boundary problem with integral gluing condition for mixed parabolic-hyperbolic equation

In the present work we consider a boundary value problem with gluing conditions of integral form for parabolic-hyperbolic type equation. We prove that the considered problem has the Volterra property. The main tools used in the work are related to the method of the integral equations and functional analysis.Comment: 18 page