Maximal \(L^1\) regularity for solutions to inhomogeneous incompressible Navier-Stokes equations. (English) Zbl 1495.35143
Summary: This paper is devoted to the maximal \(L^1\) 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 \(L^1\)-in-time estimate for the Lipschitz seminorm of the velocity field without any smallness assumption on the initial density fluctuation. In the derivation of this estimate, we study the maximal \(L^1\) 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. Our method can be used to study some other issues arising from incompressible or compressible viscous fluids.
MSC:
| 35Q30 | Navier-Stokes equations |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 76D03 | Existence, uniqueness, and regularity theory for incompressible viscous fluids |
| 76D05 | Navier-Stokes equations for incompressible viscous fluids |
| 76D07 | Stokes and related (Oseen, etc.) flows |
Keywords:
inhomogeneous Navier-Stokes equations; maximal \(L^1\) regularity; Lagrangian coordinates; Stokes system; elliptic gradient estimatesReferences:
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