L p approach to free boundary problems of the Navier-Stokes equation

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1 L p approach to free boundary problems of the Navier-Stokes equation address: yshibata@waseda.jp address: ssshimi@ipc.shizuoka.ac.jp Ω R n (n 2) v Ω. Ω,,,, perturbed infinite layer, perturbed half-space. t > Ω t, Navier-Stokes t v + (v )v Div S(v, θ) = f(x, t) in Ω t, t >, div v = in Ω t, t >, S(v, θ)ν t = σhν t p ν t on Γ t, t >, v = on Γ b, t >, v t= = v in Ω, (1) v(x, t) = t (v 1,..., v n ) θ(x, t) (x Ω t ) Ω t = Γ t Γ b. ν t is the unit outward normal to Γ t, S(v, θ) = µd(v) θi is the stress tensor, D(v) = (D(v)) ij = v i / x j + v j / x i is the deformention tensor, H is the mean curvature: Hν t = Γ(t) x, µ > is the vicsous coefficient, σ > is the coefficient of surface tension, p > is the atmospheric pressure

2 (1). Ω Γ b =, (1) drop. Ω x n perturbed infinite layer f(x, t) = g a x n (g a > ), (1) ocean. If Ω x n perturbed half-space f(x, t) = g a x n, Γ b =, (1) ocean. (1). drop, Solonnikov [11, 13, 14, 15] (1) W 2+l,1+l/2 2 (1/2 < l < 1, n = 3). Moglilevskiĭ-Solonnikov [3], Solonnikov [15] Hölder. σ =,, Solonnikov [12], Mucha and W. Zaj aczkowski [4, 5] Wq 2,1 (n < q < ), - [7, 8] Wq,p 2,1 (2 < p <, n < q < ). ocean, Allain [1] 2, [16], W 2+l,1+l/2 2 (1/2 < l < 1, n = 3). ocean, Prüss-Simonett[6] height function η η Wp 2,1 (p > n + 2)., drop, ocean, ocean (1),, W 2,1 q,p ( 2 < p <, n < q < ). Ω = Ω, Γ = Γ., u(ξ, t) Lagrange Euler Lagrange x = ξ + u(ξ, τ) dτ = X u (ξ, t) (2) f(x, t) = g a x n (1) Lagrange : t u Div S(u, π) = Div Q(u) + R(π) in Ω, t >, div u = E(u) = divẽ(u) in Ω, t >, (S(u, π) + Q(u))ν tu σhν tu = on Γ, t >, u = on Γ b, t >, u t= = u (ξ) in Ω. (3) θ(x u (ξ, t), t) = π(ξ, t), u (ξ) = v (x). ν tu = t A 1 ν / t A 1 ν A (2)

3 Jacobi a jk = x j u j = δ jk + dτ ξ k ξ k. Q(u), R(π), E(u) Ẽ(u) V j() = (j = 1, 2, 3, 4) : Q(u) = µv 1 ( E(u) = V 3 (. u dτ) u, R(π) = V 2 ( u dτ) u, Ẽ(u) = V 4 ( u dτ) π, u dτ)u. Theorem 1. Ω R n,,,, perturbed infinite layer, perturbed half-space. Γ Wq 3, Γ b Wq 2. 2 < p <, n < q <, 2(1 1/p) > 1 + 1/q. : div u = in Ω, D(u ) (D(u )ν, ν )ν = on Γ, u = on Γ b u Bq,p 2(1 1/p) (Ω) n = [L q (Ω), Wq 2 (Ω)] n 1 1/p,p, T > (3) u L p ((, T ), W 2 q (Ω)) n W 1 p ((, T ), L q (Ω)) n, π L p ((, T ), Ŵ 1 q (Ω)). π Γ = π Γ. π Hp 1/2 ((, T ), L q (Ω)) L p ((, T ), Wq 1 (Ω)) Remark 2. p, q. 2 < p < W 1 q ((, T ), L q (Ω)) L p ((, T ), W 2 q (Ω)) BUC ( (, T ), [L q (Ω), W 2 q (Ω)] 1 1/p,p ) (e.g. Amann [2]). [L q (Ω), Wq 2 (Ω)] 1 1/p,p = Bq,p 2(1 1/p) (Ω). 2 < p < Bq,p 2(1 1/p) (Ω) Wq 1 (Ω) W 1 q ((, T ), L q (Ω)) L p ((, T ), W 2 q (Ω)) BUC ( (, T ), W 1 q (Ω) )..

4 n < q < W 1 q (Ω) L (Ω). 2(1 1/p) > 1 + 1/q D(u ) (D(u )ν, ν )ν Γ. Γ η, (3) : t u µ u + π = f x Ω, t >, div u = f d = div f d x Ω, t >, t η ν u = d x Γ, t >, S(u, π)ν σ(m Γ )η ν = h x Γ, t >, u = x Γ b, t >, u t= =, η t= =. (4) m Γ. Theorem 1, (4) L p -L q. Theorem 3. Ω R n,,,, perturbed infinite layer, perturbed half-space. Γ W 3 q, Γ b W 2 q. 1 < p <, n 1 < q < (n 3 ), 2 q < (n = 2 ). : f d t= =, h t= =, f L p ((, T ), L q (Ω)) n, f d L p ((, T ), Wq 1 (Ω)), fd Wp 1 ((, T ), L q (Ω)) n, d L p ((, T ), Wq 2 (1/q) (Γ)), h Hp 1/2 ((, T ), L q (Ω)) n L p ((, T ), Wq 1 (Ω)) n (4) (u, π, η) u W 1 q ((, T ), L q (Ω)) n L p ((, T ), W 2 q (Ω)) n, π L p ((, T ), Ŵ 1 q (Ω)),. π Γ = π Γ. η W 1 p ((, T ), W 2 1/q q (Γ)) L p ((, T ), Wq 3 1/q (Γ)). π Hp 1/2 ((, T ), L q (Ω)) L p ((, T ), Wq 1 (Ω)). [1] G. Allain, Small-time existence for the Navier-Stokes equations with a free surface, Appl. Math. Optim., 16 (1987) 37 5.

5 [2] H. Amann, Linear and Quasilinear Parabolic Problems Vol. I Abstract Linear Theory, Monographs in Math., Vol 89, 1995, Birkhäuser Verlag, Basel Boston Berlin [3] I. Sh. Mogilevskiĭ and V. A. Solonnikov, On the solvability of a free boundary problem for the Navier-Stokes equations in the Hölder spaces of functions, Nonlinear Analysis. A Tribute in Honour of Giovanni Prodi, Quaderni, Pisa (1991) [4] P. B. Mucha and W. Zaj aczkowski, On the existence for the Cauchy-Neumann problem for the Stokes system in the L p -framework, Studia Mathematica, 143 (2) [5] P. B. Mucha and W. Zaj aczkowski, On local existence of solutions of the free boundary problem for an incompressible viscous self-gravitating fluid motion, Applicationes Mathematicae, 27 (2) [6] J. Prüss and G. Simonett, On the two-phase Navier-Stokes equations with surface tension, preprint. [7] Y. Shibata and S. Shimizu, On some free boundary problem for the Navier-Stokes equations, Differential Integral Equations, 2 (27), [8] Y. Shibata and S. Shimizu, A On the L p -L q maximal regularity of the Neumann problem for the Stokes equations in a bounded domain, J. Reine Angew. Math. (Crelles Journal) 615 (28), [9] V. A. Solonnikov, On the solvability of the second initial-boundary value problem for the linear nonstationary Navier-Stokes system, Zap. Nauchn. Sem. (LOMI), 69 (1977), (in Russian); English transl.: J. Soviet Math., 1 (1978) [1] V. A. Solonnikov, Solvability of a problem on the motion of a viscous incompressible fluid bounded by a free surface, Izv. Akad. Nauk SSSR Ser. Mat., 41 (1977), (in Russian); English transl.: Math. USSR Izv., 11 (1977) [11] V. A. Solonnikov, Solvability of the evolution problem for an isolated mass of a viscous incompressible capillary liquid, Zap. Nauchn. Sem. (LOMI), 14 (1984) (in Russian); English transl.: J. Soviet Math., 32 (1986) [12] V. A. Solonnikov, On the transient motion of an isolated volume of viscous incompressible fluid, Izv. Akad. Nauk SSSR Ser. Mat., 51 (1987) (in Russian); English transl.: Math. USSR Izv., 31 (1988) [13] V. A. Solonnikov, On nonstationary motion of a finite isolated mass of self-gravitating fluid, Algebra i Analiz, 1 (1989) (in Russian); English transl.: Leningrad Math. J., 1 (199) [14] V. A. Solonnikov, Solvability of the problem of evolution of a viscous incompressible fluid bounded by a free surface on a finite time interval, Algebra i Analiz, 3 (1991) (in Russian); English transl.: St. Petersburg Math. J., 3 (1992) [15] V. A. Solonnikov, Lectures on evolution free boundary problems: classical solutions, L. Ambrosio et al.: LNM 1812, P.Colli and J.F.Rodrigues (Eds.), (23) , Springer-

6 Verlag, Berlin, Heidelberg. [16] A. Tani, Small-time existence for the three-dimensional incompressible Navier-Stokes equations with a free surface, Arch. Rat. Mech. Anal., 133 (1996)