Resumen de Uniqueness and existence of solutions in the BVt (Q) space to a doubly nonlinear parabolic problem

Juan Francisco Padial , Jesús Ildefonso Díaz Díaz

• In this paper we present some results on the uniqueness and existence of a class of weak solutions (the so called BV solutions) of the Cauchy-Dirichlet problem associated to the doubly nonlinear diffusion equation $$ b(u)_t-\operatorname{div}(|\nabla u-k(b(u))\boldkey{e}|^{p-2}(\nabla u-k(b(u))\boldkey{e})) +g(x,u) = f(t,x).

  $$ This problem arises in the study of some turbulent regimes: flows of incompressible turbulent fluids through porous media, gases flowing in pipelines, etc. The solvability of this problem is established in the $BV_t(Q)$ space. We prove some comparison properties (implying uniqueness) when the set of jumping points of the BV solution has $N$-dimensional null measure and suitable additional conditions as, for instance, $b^{-1}$ locally Lipschitz. The existence of this type of weak solution is based on suitable uniform estimates of the BV norm of an approximated solution.