Free Boundary Problems via Da Prato–Grisvard Theory

An <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper L 1"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">L</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">\mathrm {L}_1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-maximal regularity theory for parabolic evolution equations inspired by the pioneering work of Da Prato and Grisvard (J. Math. Pures Appl. (9) <bold>54</bold> (1975), no. 3, 305–387) is developed. Besides of its own interest, the approach yields a framework allowing <italic>global-in-time</italic> control of the change of Eulerian to Lagrangian coordinates in various problems related to fluid mechanics. This property which is of course decisive for free boundary problems is, firstly, illustrated by the analysis of the free boundary value problem describing the motion of viscous, incompressible Newtonian fluids without surface tension and, secondly, the motion of compressible, pressureless gases. To this end, an endpoint maximal <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper L 1"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">L</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">\mathrm {L}_1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-regularity approach to the Stokes and Lamé systems is developed. It is applied then to establish global, strong well-posedness results for the free boundary problems described above in the case where the initial domain coincides with the half-space, and the initial velocity is small with respect to a suitable scaling invariant norm.