The study of a class of analytical solutions for six fluid dynamical systems

Abstract

The systems of gas dynamics (Euler, Euler-Poisson, Navier-Stokes, Navier-Stokes-Poisson equations) and of shallow water (Camassa-Holm and Degasperis-Procesi equations) are important basic models in fluid mechanics and astrophysics. Constructing analytical or exact solutions for the partial differential equations is a vital part in nonlinear sciences. Indeed, scientists and mathematicians are eager to seek analytical solutions for better understanding of the evolution of these kinds of systems. In this PhD thesis, I consider the construction of analytical solutions for the above systems. As these systems share similar mathematical structures in some aspects, I will exhibit some common features among them, including certain blowup and stability phenomena. In detail, I attempt to employ the well-known separation method to its fullest extent and introduce a novel pertubational method to seek analytical solutions with free boundaries. The main idea is to reduce the nonlinear partial differential systems into several ordinary or functional differential equations, or to simpler partial differential equations under some suitable assumptions on the functional structures of the solutions. After proving the existence of solutions of the corresponding simpler differential equations, the analytical solutions for the original nonlinear systems are constructed. One of the applications of such analytical solutions is to test numerical methods designed for these systems. Another application is to provide samples of concrete solutions so as to affirm or support theoretical hypotheses or conjectures about these complicated systems. A substantial percentage of the results presented in this thesis have appeared in print. In total, sixteen published papers (not counting preprints; see the lists in the next three pages) are the direct outcome of work done during my PhD study. The fact that these results are well-received by referees and editors attests to the great interest of others in these analytical solutions. The most significant contributions of this thesis are as follows: - I am the first to reduce the compressible density-dependent Navier-Stokes equations in RN to new 1 + N differential functional equations, which lead to solutions with elliptical symmetry and drift phenomena. - I am the first to obtain self-similar solutions in explicit form for the 2-component shallow water systems. - We construct the first rotational solutions in explicit form for the 2-dimensional Euler- Poisson equations and demonstrate the principle that rotation can prevent blowup. The thesis is organized as follows: - A brief introduction of the above six models is provided. - The separation method is applied to construct solutions with free boundaries for the systems of gas dynamics and shallow water. - In addition, some solutions with rotation are constructed for the 2-dimensional Euler-Poisson and 3-dimensional Euler equations. - Based on the separation method, a novel pertubational method is used to obtain more general classes of analytical solutions for the 1-dimensional Euler and Camassa-Holm equations. - Finally a summary is provided to conclude the works done and other related works in the PhD studies, together with some future research insights for the further development of this thesis is included.