On the regularity and uniqueness of weak solutions for the Navier-Stokes equations
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概要
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Let $ \Omega $ be a bounded domain in 3-dimensional Euclidean space $ R^3 $, or $ R^n $ ($ n \geqq 2 $). Assume that a rigid vessel $ \Omega $ having a sufficiently smooth boundary $ \partial{ \Omega } $ is fixed in the space and filled with some incompressible fluid of viscosity $ \nu > 0 $ and of density 1. Furthermore it is assumed that the fluid is adhered at the boundary at all times and some external force $ f(x,t) $ is exerted on the fluid. When an initial velocity $ a(x) $ is given to the fluid, what will be the future velocity $ u(x,t) $ of it ? In accordance with the fundamental laws of fluid dynamics this physical problem is mathematically formulated in seeking the unknown velocity $ u = u(x,t) $ and the unknown pressure $ p = p(x,t) $ satisfying the following equations and conditions: (1.1) $ u_t - v \Delta u + u \cdot \nabla u = f - \nabla p $, $ x \in \Omega $, $ t>0 $ (1.2) $ div u = 0 $, $ x \in \Omega $, $ t>0 $ (1.3) $ u \mid_{ \partial{ \Omega } } = 0 $, $ t>0 $ (1.4) $ u \mid_{t=0} = a(x) $, $ x \in \Omega $ where the vector-valued functions $ u = (u^1(x,t), u^2(x,t), \cdots , u^n(x,t)) $, $ a = ( a^1(x), a^2(x), \cdote , a^n(x)) $, $ f = (f^1(x,t), f^2(x,t), \cdots , f^n(x,t)) $ and the scalar function $ p = p(x,t) $ are considered. We use the notations: $ u \cdot \nabla v = \sum_{i=1}^{n}{u^i} \fraq{ \partial{v} }{ x_i }} $, $ div u = \sum_{i=1}^{n} \fraq{ \partial{ u^i }}{ \partial{ x_i }}$ for vector-valued functions $ u $, $ v $. For this problem Hopf [10] succeeded in showing that there exists a global weak solution and that for an arbitrary domain in $ n $-dimensional Euclidean space $ R^n $ ($ n \geqq 2 $), but he left the investigation on the uniqueness and the smoothness of the solution for later works. To establish the uniqueness theorem together with the existence theorem, various researches on strong solutions have been made by many authors. It is well known that if the dimension $ n $ is two Hopf weak solution is unique and regular (see Ladyzhenskaia [16], Lions [18], Lions and Prodi [19], Serrin [25], Temam [29]). However, for $ n \geqq 3 $ the uniquencess and the regularity of Hopf weak solution are still open problems. Up to the present the unique existence of local regular solutions has been proved and the global existence of a unique regular solution has been shown only when the data $ a(x) $ and $ f(x,t) $ are sufficiently small (see Fujita and T. Kato [4], Giga and Miyakawa [9], Ito [11], Kiselev and Ladyzhenskaia [15], see also Ladyzhenskaia [16], Serrin [25], Temam [29], Wahl [30], and papers cited there). In order to solve the open problems mentioned above, various investigations have been done as approaches to them. In the present paper, firstly we shall show regularity criteria for weak solutions of (1.1)-(1.4). To this end we discuss Hausdorff dimension of the set E of possible time singularity of a weak solution $ u(x,t) $ and we shall show the following results (see Theorem 2.1 and 2.2). Let $ A $ be the Stokes operator in the Hilbert space $ H_{ \sigma } $. Our result for the space dimension $ n = 3 $ reads: 1. If $ u $ is in $ L_2 ((0, T); D(A^{ \gamma })) $ for $\gamma = \freq{3}{4} - \barepsilpn $, then $ 2 \barepsilon $-dimensional Hausdorff measure of the set $ E $ of $ u $ is zero and $ u $ is in $ C^{\infty} (\bar{ \Omega } \times ((0,T) - E)) $. ($ 0 < \barepsilon < \fraq{1}{2} $). 2. If $ u \in L_2 ((0, T); D(A^{\frac{3}{4}})) $, the set $ E $ is empty. More generally, let the space dimension be $ n $ and $ A_p $ be the Stokes operator in the Banach space $ L_{p, \sigmma} $. Then our result reads: 1. If u belongs to the space $ L_q((O,T);D(A_p^{ \gamma })) $ ($ 0 \leqq {\gamma}_0 \equiv \frac{n}{2p} - \frac{1}{2} < \gamma < 1 $) with $ k \equiv 1 - q( \gamma - {\gamma}_0) > 0 $, then $ k $-dimensional Haudorff measure of the set $ E $ of $ u $ is zero and $ u $ is in $ C^{\infty} (\bar{ \Omega } \times ((0,T) - E) $). 2. If $ u $ belongs to the space $ L_q((O,T); D(A_p^{ \gamma })) $ ($ 0 \leqq { \gamma }_0 < \gamma < 1 $) with $ k \leqq 0 $, then the set $ E $ is empty, i.e. $ u $ is regular in $ \bar{ \Omega } \times (0,T) $. ($ D(A^{ \gamma }) $ means the domain of the fractional power of the operator $ A $.) Recently, Giga [8] has proved the regularity criteria: if a weak solution $ u $ is in $ L_q((O,T); L_{p, \sigma }) $ with $ k = 2 - q + \frac{nq}{p} > 0 $ and $ p > n $, for some $ p,q \geqq 1 $, then there is a closed set $ E $ of $ (0,T) $ whose $ \frac{k}{2} $-dimensional Hausdorff measure vanishes and such that $ u $ is in $ C^{\infty}( \bar{ \Omega } \times ((0,T) - E)) $. If $ u $ is in $ L_q((O,T); L_{p,\sigma}) $ with $ k \leqq 0 $, $ P > n $, then $ u $ is in $ C^{\infty}(\bar{ \Omega } \^times (0,T)) $. Here, $ L_{p, \sigma } $ means the closure in $ L_p(\Omega) $ of all divergencefree vector fields with compact support in $ \Omega $. It seems that our result is included in Giga [8] for $\gamma < \frac{n}{2p} $ by the imbedding theorem, but for $ \gamma \geqq \frac{n}{2p} $ it is not included in Giga's theorem. We notice that many investigations have been made on the regularity of weak solutions (see Caffarelli, Kohn and Nirenberg [1], Scheffer [22], [23], [24], Sohr and Wahl [28]). Besides we note that the regularity problem of the weak solutions belonging to the space $ L_{\infty} $($ (0,T); D(A^{{\gamma}_0}) $) (${ \gamma }_0 \equiv \frac{n}{4} - \frac{1}{2} $) or $ L_{\infty}((0,T); L_n( \Omega )) $ is an important open problem. Secondly we shall show the following result on the uniqueness of the weak solutions (see Theorem 3.1 and 3.2): Let $ { \gamma }_0 \equiv \frac{n}{4} - \frac{1}{2} $, and $ 0< { \gamma }_0 < 1 $. 1. the weak solution $ u $ of (1.1)-(1.4) in the class $ L_{\infty}((O,T); D(A^{ \gamma_0})) $ has the properties: $ u(t) \in D(A^{ \gamma_0 }) $ for all $ t \in [0,T] $, and $ u(t) $ is right continuous for $ t $ as a $ D(A^{ \gamma_0 }) $-valued function, 2. the weak solution belonging to the space $ L_{\infty}((0,T); D(A^{ \gamma_0 })) $ is the only weak solution of the problem (1.1)-(1.4). In this way, the assumption of the right continuity in the uniqueness theorem of Masuda [20] will be removed if we consider the space $ L_{\infty}((O,T); D(A^{ {\gamma}_0} )) $ instead of $ L_{\infty}((O,T); L_n(\Omega)) $. It seems that we give a remark to the proof of Wahl [30] on the uniqueness. Concerning the uniqueness, Serrin [25] has proved a uniqueness theorem of weak solutions $ L_q((O,T); L_p( \Omega )) $ for some $ p $ and $ q $ satisfying $ \frac{n}{p} + \frac{2}{q} \leqq 1 $, $ p > n $ and for a general domain $ Omega $ of $ R^n $ ($ n = 2, 3, 4 $). Foias [3] has proved a similar theorem for $ \Omega = R^n $. Masuda [20] has generalized the Foias-Serrin uniqueness theorem and has shown that if there is a weak solution $ u $ in $ L_{\infty}((0,T); L_n(\Omega)) $ which is right continuous for t as an $ L_n $-valued function, then $ u $ is the only weak solution. Wahl [30] also has shown that the space $ L_{\infty}((O,T); L_n(\Omega)) $ is a uniqueness class for weak solutions. As an approach to the global existence of a unique regular solution of the Navier-Stokes equations Ladyzhenskaia [16], [17] has introduced modified Navier-Stokes equations, and has discussed the equation for the initial value problem or limit states problems. Finally we shall prove the existence of a strong periodic solution of the modified Navier-Stokes equations under no smallness condition on $ f(x,t) $. The problem we consider is as follows (see Theorem 4.1): (1.5) $ u_t - \Delta{u} - \mu {\paralle \nabla{u} \parallel}^2 \Delta{u} + u \cdot \nabla{u} = f - \nabla{p} $, $ x \in \Omega $, $ t \in R^1 $, (1.6) $ u(x, t+\omega) = u(x,t) $, $ x \in \Omega $, $ t \in R^1 $, with the conditions (1.7) $ div u = 0 $, $ x \in \Omega $, $ t \in R^1 $, (1.8) $ u \mid_{\partial{ \Omega }} = 0 $, $ t \in R^1 $, where $ f(x,t) $ is an $ \omega $-periodic function in $ t $. The constant $ \mu $ is an arbitrary positive number. We note that the Navier-Stokes equation (1.1) is approximated by (1.5) with a parameter $ \mu $. To the problem we will apply the method which we have considered on the periodicity problem for a semilinear parabolic equation and a semilinear wave equation (see H. Kato and Nakao [14], Nakao and H. Kato [21]). Concerning the periodicity problem of the Navier-Stokes equations Shinbrot [26] has prove a reproductive property when the external force $ f $ is sufficiently small. while we assume no smallness condition on $ f $. In Section 2 we discuss the regularity weak solutions. In Section 3 we study the uniqueness of weak solutions. In Section 4 we consider modified Navier-Stokes equations. Through this paper $ L_2(\Omega) $ is a Hilbert space equipped with the inner product $ (u,v) = \sum_{i=1}^{n} \int_{\Omega} {u^i(x)v^i(x)}dx $. We denote $ L_2( \Omega ) $-norm by $ \parallel \cdot \parallel $. $ H^m(\Omega) $ and $ W_p^m(\Omega) $ are the Sobolev spaces of vector-valued functions which are in $ L_2(\Omega) $ and $ L_p(\Omega) $ together with their derivatives up to the order $ m $, respectively.
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