Mixed Type Partial Differential Equations With Initial and Boundary Values in Fluid Mechanics

This paper includes various parts of the theory of mixed type partial differential equations with initial and boundary conditions in fluid mechanics ,such as: The classical dynamical equation of mixed type due to Chaplygin (1904), regularity of solutions in the sense of Tricomi (1923) and in brief his fundamental idea leading to singular integral equations, and the new mixed type boundary value problems due to Gellerstedt (1935), Frankl (1945), Bitsadze and Lavrent’ev (1950), and Protter (1950-2007 ). Besides this work contains the classical energy integral method and quasi-regular solutions and weak solutions , as well as the well-posedness of the Tricomi , Frankl, and Bitsadze Lavrent’ev problems in the sense that: “There is at most one quasi-regular solution and a weak solution exists”. Furthermore Rassias ( Ph.D. dissertation, U. C. , Berkeley, (1977) ) generalized the Tricomi and Frankl problems in n dimensions based on Protter’s proposal. This author generalizes even further the results obtained through the said thesis. Also this paper provides a maximum principle for the Cauchy problem of hyperbolic equations in multi-dimensional space-time regions, the formulation and solution of the TricomiProtter problem, a selection of several uniqueness and existence theorems and recent open problems suggested by Rassias in the theory of mixed type partial differential equations and systems with applications in fluid mechanics. 2000 Mathematics Subject Classification. 35K15, 76H05, 35M05