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On a free boundary problem for the Navier-Stokes equations. (English) Zbl 1212.35353

Summary: We consider a free boundary problem for the Navier-Stokes equation in \(\mathbb R^n\) (\(n\geq 2\)). We prove a local in time unique existence theorem for any initial data, and a global in time unique existence theorem for some small initial data. The problem we consider in this paper was already treated by V. A. Solonnikov [Math. USSR, Izv. 31, No. 2, 381–405 (1988); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 51, No. 5, 1065–1087 (1987; Zbl 0850.76180)]. But, recently in [J. Reine Angew. Math. 615, 157–209 (2008; Zbl 1145.35053)] we proved an \(L_p\)-\(L_q\) maximal regularity theorem for the Stokes equation with Neumann boundary condition which is a linearized version of the free boundary problem for the Navier-Stokes equation treated in this paper. Our proof is based on this theorem. Therefore our solution is obtained in the space \(W^{2,1}_{q,p}\) (\(2<p<\infty \) and \(n<q<\infty \)) while a solution in the above paper of Solonnikov is in \(W^{2,1}_q=W^{2,1}_{q,q}\) \((n<q<\infty)\). Moreover, our proof of the global in time existence theorem is much simpler than the above paper of Solonnikov, because in our paper cited above we established a maximal regularity theorem on the whole time interval \((0,\infty )\) with exponential stability. The results obtained in this paper were already announced in Y. Shibata and S. Shimizu [Proc. Japan Acad., Ser. A 81, No. 9, 151–155 (2005; Zbl 1188.35139)].

MSC:

35Q30 Navier-Stokes equations
76D07 Stokes and related (Oseen, etc.) flows
35R35 Free boundary problems for PDEs
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