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- W2934944186 abstract "The aim of this thesis is to develop methods for proving the existence and uniqueness of solutionsof local and nonlocal stochastic evolution equations in phase transition problems. In chapter 1, we studyan initial value problem for a nonlocal stochastic reaction-diffusion equation with homogeneous Neumannboundary conditions in an open bounded set of ℝn, with a smooth boundary. We consider the case of ageneral nonlinear elliptic operator and we suppose that the noise is additive and induced by a Q-Wiener process.The deterministic problem with a linear diffusion term is used to model phase separation in a binarymixture. The proof of existence for the stochastic problem is based on a change of function which involvesthe solution of the stochastic heat equation with a nonlinear diffusion term. We obtain a problem withoutthe noise term. This simplifies the application of the monotonicity method, which we use to identify thelimit of the nonlinear terms. In chapter 2, we prove the existence and uniqueness of the solution for a phasefield problem with multiplicative noises induced by Q-Wiener processes. This problem models for instancethe process of melting and solidification. To that purpose we apply the Galerkin method and derive a prioriestimates for the approximate solutions. The last step is to identify the limit of the nonlinear terms whichwe do by the so-called stochastic monotonicity method. Finally, in chapter 3, we prove the existence anduniqueness of a pathwise solution in space dimension up to 6 for the stochastic nonlocal Allen-Cahn equationwith a multiplicative noise induced by a Q-Wiener process. The usual compactness method for deterministicproblems cannot be applied in a stochastic context because of the additional probability variable. Therefore,we apply the stochastic compactness method." @default.
- W2934944186 created "2019-04-11" @default.
- W2934944186 creator A5076589431 @default.
- W2934944186 date "2018-11-27" @default.
- W2934944186 modified "2023-10-16" @default.
- W2934944186 title "Local and Nonlocal Stochastic Evolution Equations in Phase Transition Problems." @default.
- W2934944186 hasPublicationYear "2018" @default.
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