Pavel Holba
Complete Classification of Local Conservation Laws for Generalized Cahn--Hilliard--Kuramoto--Sivashinsky Equation
In this talk we consider nonlinear multidimensional Cahn-Hilliard and Kuramoto-Sivashinsky equations that have many important applications in physics, chemistry, and biology, and a certain natural generalization of these equations.
Namely, for an arbitrary number $n$ of spatial independent variables we present a complete list of cases when the following PDE in $n+1$ independent variables $t, x_1\ldots, x_n$ and one dependent variable $u$
$$
u_t= a \Delta^2u+b(u)\Delta u+f(u)|\nabla u|^2+g(u),
$$
to which we refer as to the generalized Cahn-Hilliard-Kuramoto-Sivashinsky equation, admits nontrivial local conservation laws of any order, and for each of those cases we give an explicit form of all the local conservation laws of all orders modulo trivial ones admitted by the equation under study. Here $b,f,g$ are arbitrary smooth functions of the dependent variable $u$, $a$ is a nonzero constant, $\Delta=\sum_{i=1}^n\partial^2/\partial x_i^2$ is the Laplace operator and $|\nabla u|^2=\sum_{i=1}^n(\partial u/\partial x_i)^2$.
In particular, we show that the original Kuramoto-Sivashinsky equation admit no nontrivial local conservation laws and list all of the nontrivial local conservation laws for original Cahn-Hilliard equation.
For more details see P. Holba, Complete classification of local conservation laws for generalized Cahn-Hilliard-Kuramoto-Sivashinsky equation. Stud. Appl. Math., to appear, https://doi.org/10.1111/sapm.12576.
