| Abstract: |
| This talk presents the maximal $L^p$ regularity for the heat equation subject to nonhomogeneous Neumann boundary conditions and a divergence-form external source. While maximal regularity is well-established for homogeneous cases, nonhomogeneous boundary data often introduce technical complexities, particularly in fluid dynamics models. Our work establishes a robust maximal $L^p$ estimate for the generalized solution to this problem. As a key application, we demonstrate how this estimate improves the weak solution framework for the compressible Navier-Stokes equations with inflow-outflow boundary conditions. Specifically, we show that our regularity results allow for the removal of the artificial terms previously required in approximate systems (e.g., [Chang, Jin, and Novotny]), thereby providing a more physical and streamlined analysis of the Navier-Stokes equations under nonhomogeneous boundary settings. |
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