Existence and Uniqueness Results

Abstract

Existence and uniqueness theorems play a very central part in the theory of partial differential equations, particularly in the context of mathematical physics. The well-posedness of a Cauchy or boundary value problem is of tantamount importance for the physical interpretation and/or practical application of the equation under consideration. For instance, numerical calculations become a touchy business in the absence of uniqueness or continuous dependence on the data, and the function spaces in which existence theorems can be proved usually contain intrinsic useful information about the solutions. Moreover, having in mind that the Boltzmann equation is a schematization of the reality (described at a more detailed level by the Newton laws) expected to be valid only in the asymptotic regime when a gas is extremely rarefied, a good existence theorem for the solutions of such an equation is, at least, the first check of the validity of the mathematical model under investigation.KeywordsCauchy ProblemBoltzmann EquationGlobal ExistenceUniqueness ResultMild SolutionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.