Asymptotic approximation of fluid flows from the compressible Navier-Stokes equations
Welter, Roland Kuha
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In this thesis a method for studying the asymptotic behavior of solutions to dissipative partial differential equations is developed, motivated by the study of the compressible Navier-Stokes equations in the past works of Hoff and Zumbrun,1995, Hoff and Zumbrun, 1997. In its most basic form, this method allows one to compute n^th order approximations in terms of Hermite functions of solutions of the heat equation having n^th order moments. The main advantage is that these approximations can be efficiently computed, and are often given explicitly in terms of elementary functions. It is shown how this method can be extended to increasingly complicated systems, leading the way toward the asymptotic analysis of the compressible Navier-Stokes equations. A number of challenges must be overcome to apply this method to the compressible Navier-Stokes system. For technical reasons, the analysis is carried out on the divergence and curl of the velocity field, and hence a means of recovering the velocity field from these quantities is established first. The linear part of the evolution is then studied, and an extended version of the artificial viscosity decomposition previously developed (Kawashima, Hoff and Zumbrun1995) is introduced. This decomposition is in terms of the heat and combined heat-wave operators, and hence general estimates on their evolution in weighted L^p spaces are obtained. A modified compressible Navier-Stokes system is then introduced which captures the dominant behavior of the linear evolution and possesses similar nonlinear terms. Solutions to this modified system are proven to exist in weighted spaces, showing that solutions initially having a certain number of moments possess this same number of moments for all time. An analysis of the asymptotic behavior of the modified compressible Navier-Stokes system is then carried out, and it is shown that the method developed herein extends and unifies the approach of Hoff and Zumbrun with that of Gallay and Wayne, 2002a, Gallay and Wayne, 2002b, where it was originally developed to study the behavior of the incompressible Navier-Stokes equations. The thesis is concluded with a discussion of how the results obtained for the modified compressible Navier-Stokes system pave the way for an analysis of the true compressible Navier-Stokes system, the generalization of this asymptotic analysis to arbitrary order, and with a comparison of this asymptotic analysis to that found in the recent work of Kagei and Okita, 2017.
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