The Cauchy Problem in Kinetic TheorySIAM, 01.01.1996 - 254 Seiten This volume studies the basic equations of kinetic theory in all of space. It contains up-to-date, state-of-the-art treatments of initial-value problems for the major kinetic equations, including the Boltzmann equation (from rarefied gas dynamics) and the Vlasov-Poisson/Vlasov-Maxwell systems (from plasma physics). This is the only existing book to treat Boltzmann-type problems and Vlasov-type problems together. Although these equations describe very different phenomena, they share the same streaming term. The author proves that solutions starting from a given configuration at an initial time exist for all future times by imposing appropriate hypotheses on the initial values in several important cases. He emphasizes those questions that a mathematician would ask first: Is there a solution to this problem? Is it unique? Can it be numerically approximated? The topics treated include the study of the Boltzmann collision operator, the study of the initial-value problem for the Boltzmann equation with "small" and "near equilibrium" data, global smooth solvability of the initial-value problem for the Vlasov-Poisson system with smooth initial data of unrestricted size, conditions under which the initial-value problem for the Vlasov-Maxwell system has global-in-time solutions (in both the smooth and weak senses), and more. |
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Anal Appl apply approximation argument Assume Boltzmann Equation bounded called Cauchy Problem chapter characteristics claim classical collision Collisionless Comm compact completes compute condition conservation consider constant continuous convergence define definition denote density depending derivatives desired dimensions energy errors estimate expression field first follows function given Global Existence Global Solutions Hence holds inequality initial data integral Kinetic Theory known Lemma limit linear Lions Math Maxwell Meth methods nonlinear norm Note obtain operator particle Phys Physics plasma problem Proof proves Recall relativistic remains representation result satisfies satisfy side similar Similarly smooth solution space Stat sufficiently symmetric term Theorem unique value problem variables velocity Vlasov Vlasov-Maxwell System Weak Solutions write