Zakoni održanja i njihova stohastiÄka aproksimacija - PhDData

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Zakoni održanja i njihova stohastiÄka aproksimacija

The thesis was published by Marković Branko, in June 2022, University of Novi Sad.

Abstract:

Mathematical models that use partial differential equations today make an important approach to modeling various phenomena from physics, biology,mechanics, etc. In the past, only deterministic quantities appeared as variablesin the model, while today they are often used together with the stochasticprocesses. Generally, models are incomplete descriptions of some phenomena,and stochastic processes may stand for some effects that are present in thephenomenon itself but missing in the model. And it has been observed that byadding adequate stochastic quantities to a deterministic problem, wesometimes get a solution with better features comparing to the solutionwithout stochastic quantities. And there are cases where deterministic modelshave no solutions, while their stochastic approximations have them.In this dissertation we deal with 2×2 systems of conservation laws and theirstochastic approximations. We get acquainted with the various conservationlaws from the theory of elasticity, chromatography, electrophoresis, isentropicand the Born-Infeld system. For each of the previous systems, some importantroperties are determined, the Riemann problems are solved, and their elementary solutions are found. Also, we get reminded of some basic concepts concerning stochastic processes and make a stochastic perturbation of the system of conservation law. The stochastic perturbation itself was made byusing an adequate stochastic source, additional modifications of the initial conditions and symmetrization of the system. As a result a stochastic Cauchy problem is obtained. We solve it by applying the method of vanishing viscosity to a sequence of new approximate problems, and the viscosity term goes to zero by using the Skorohod’s theorem. That is how we get some solution on a new stochastic basis. Returning to the original stochastic basis is done by showing pathwise uniqueness and applying the Yamada-Watanabe theorem. Since the stochastic solution is given on a random interval, its local and global estimates of existence are given in probability. Finally, for a concrete system from the theory of elasticity, whose stochastic approximation we obtain without using symmetrization, we show that the stochastic solution converges in some adequate sense to the deterministic solution, in the zero-noise limit.



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