Maximal L1 Regularity for a Class of Parabolic Systems and Applications to Navier-Stokes Equations
Type of DegreePhD Dissertation
Mathematics and Statistics
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This dissertation is devoted to the maximal L1-in-time regularity for a class of linear parabolic systems with variable coefficients. This theory can be applied to investigate the global-in-time well-posedness and stability issues for density-dependent viscous fluids, even if the initial fluctuation of the density is large. The results in Chapter 3 and most of the results in Chapter 4 have been addressed in the author's papers  and , respectively. The main result in Chapter 3 concerns the maximal L1 regularity and asymptotic behavior for solutions to the inhomogeneous incompressible Navier-Stokes equations under a scaling-invariant smallness assumption on the initial velocity. We obtain a new global L1-in-time estimate for the Lipschitz seminorm of the velocity field without any smallness assumption on the fluctuation of the initial density. In the derivation of this estimate, we study the maximal L1 regularity for a linear Stokes system with variable coefficients. The analysis tools are a use of the semigroup generated by a generalized Stokes operator to characterize some Besov norms and a new gradient estimate for a class of second-order elliptic equations of divergence form. In Chapter 4, we generalize the concrete maximal L1 regularity result obtained in Chapter 3 and establish an abstract one for a class of Cauchy problems associated with composite operators. Then we apply this abstract theory to study maximal L1 regularity for the Lame system with rough variable coefficients. To lower the regularity of the coefficients, we work in the Lp (in space) framework. For this, we use a classical method to establish Gaussian bounds of the fundamental matrix of a generalized parabolic Lame system with only bounded and measurable coefficients. As applications, we use a Lagrangian approach to study the global-in-time well-posedness of systems of compressible flows.