Doubly nonlinear evolution equations

In the first part of the thesis we consider doubly nonlinear history-dependent problems of the form $\partial_t[k \ast (b(v) - b(v_0))] - \text{div}~ a(x, \nabla v) = f$. The kernel $k$ satisfies certain assumptions which are, in particular, satisfied by $k(t)= \frac{t^{-\alpha}}{\Gamma(1-\alpha)}$, i.e., the case of fractional derivatives of order $\alpha \in (0,1)$ is included. We show existence of entropy solutions in the case of a nondecreasing $b$. An existence result in the case of a strictly increasing $b$ is used to get entropy solutions of approximate problems. Kruzhkov's method of doubling variables, a comparison principle and the diagonal principle are used to obtain a.e. convergence for approximate solutions. A uniqueness result has been shown in a previous work. The second part of the thesis deals with a doubly nonlinear evolution equation with multiplicative noise and we show existence and pathwise uniqueness of a strong solution. Using a semi-implicit time discretization we get approximate solutions with monotonicity arguments. We establish a-priori estimates for the approximate solutions and show tightness of the sequence of pushforward measures induced by the sequence of approximate solutions. As a consequence of the theorems of Prokhorov and Skorokhod we get a.s. convergence of a subsequence on a new probability space which allows to show the existence of martingale solutions. Pathwise uniqueness is obtained by an $L^1$-method. Using this result, we are able to show existence and uniqueness of strong solutions. Finally, in the third part of the thesis we consider a $p$-Laplace evolution problem with stochastic forcing on a bounded domain $D\subset\mathbb{R}^d$ with homogeneous Dirichlet boundary conditions for $1<p<\infty$. The additive noise term is given by a stochastic integral in the sense of Itô. The technical difficulties arise from the merely integrable random initial data $u_0$ under consideration. Due to the poor regularity of the initial data, estimates in $W^{1,p}_0(D)$ are available with respect to truncations of the solution only and therefore well-posedness results have to be formulated in the sense of generalized solutions. We extend the notion of renormalized solution for this type of SPDEs, show well-posedness in this setting and study the Markov properties of solutions.


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