Strong solvability of the Stokes and Navier-Stokes equations in weak $L^{n}$ space (The structure of function spaces and its environment)
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(2) 106. then there exists. a. time. periodic. solution. u. of. u(t)=\displaystyle \int_{-\infty}^{t}e^{(t-s) $\Delta$}\mathb {P}f(s)ds-\int_{-\infty}^{t}e^{(t-s) $\Delta$}\mathb {P}u\cdot\nabla u(s)ds,. t\in \mathbb{R}. (IE). ,. period as f such thatu\in BC(\mathbb{R};L_{ $\sigma$}^{n,\infty}(\mathbb{R}^{n})) with \nabla u\in BC(\mathbb{R};L^{\frac{n}{2},\infty}(\mathbb{R}^{n}) Moreover, for \displaystyle \frac{n}{3}<p<\infty there exists $\epsilon$_{n_{\rangle}p}>0 with $\epsilon$_{n,p}\leq$\epsilon$_{n} such that if f addition‐ ally belongs to BC(\mathbb{R};L^{p,\infty}(\mathbb{R}^{n})) and satisfies. with the. same. .. ,. \displaystyle \sup_{t\in \mathb {R} \Vert f(t)\Vert_{\frac{n}{3},\infty}<$\epsilon$_{n} then the solution. u. of (IE). obtained. ,. r. and q. p. above, also satisfies. u\in BC(\mathbb{R};L_{ $\sigma$}^{r,\infty}(\mathbb{R}^{n}) where the exponents. ). and. \nabla u\in BC(\mathbb{R};L^{q,\infty}(\mathbb{R}^{n})). satisfy. \displaystyle \frac{n}{2}\leq q\leq\frac{np}{n-p}. \left\{ begin{ar ay}{l n\leqr\leq\frac{np}{n-2p}ifp<\frac\d{isnpl}ay{s2ty}le,\\frac {n}{2\}\{le qp,\end{ar ay}\right. \displaystyle\frac{n}{2}\leqq<\infty Let. n. 3. =. BC(\mathbb{R};L^{1}(\mathbb{R}^{3})). There exists $\epsilon$_{3}. .. if p<n,. if. if. n\leq r<\infty. (ii). ,. >. 0 with the. satisfies f(t)=f(t+ $\omega$) for. n\leq p.. following property. Suppose some period $\omega$>0 If. t\in \mathbb{R} with. that. f. \in. period. $\omega$. .. \displaystyle \sup_{t\in \mathb {R} \Vert f(t)\Vert_{1}<$\epsilon$_{3}, then there exists time. periodic function. u. in. such that. BC(\mathbb{R};L_{ $\sigma$}^{n,\infty}(\mathbb{R}^{3}). with the. u(t)=\displaystyle \int_{-\infty}^{t}\mathb {P}e^{(t-s) $\Delta$}f(s)ds-\int_{-\infty}^{t}\nabla\cdot e^{(t-s) $\Delta$}\mathb {P}(u\otimes u)(s)ds,. same. t\in \mathbb{R}. that. Moreover, for 1<p<\infty there exists $\epsilon$_{3,p}>0 with $\epsilon$_{3,p}\leq$\epsilon$_{3} such belongs to BC(\mathbb{R}; If (\mathbb{R}^{3}) and satisfies. if f additionally. \displaystyle \sup_{t\in \mathb {R} \Vert f(t)\Vert_{1}<$\epsilon$_{3,p} then the solution. u. of (IE^{*}) obtained above, satisfies (IE) ,. u\in BC(\mathbb{R};L_{ $\sigma$}^{r,\infty}(\mathbb{R}^{3}) where the exponents. r. and q. and. and also. ,. satisfy. \displaystyle \frac{3}{2}<q\leq\frac{3p}{3-p}. \left\{ begin{ar ay}{l 3\leqr\leq\frac{3p}{3-2p}\mathrm{\id}ifsp1lay<styple\f\rafc{r3a}{2c}\{l3eq}p{,2},\ \{ \end{ar ay}\right. \displaystyle\frac{3}{2}<q<\infty 3\leq r<\infty. satisfies. \nabla u\in BC(\mathbb{R};L^{q,\infty}(\mathbb{R}^{3})). if. (\mathrm{I}\mathrm{E}^{*}). if. 1<p<3,. if. 3\leq p..
(3) 107. [11]. periodic solution in L^{n,\infty}( $\Omega$) of (N‐S) with weak‐mild form. In [11], the regularity and strong solvability is discussed in terms of the topology of some sum space of the Sobolev spaces with negative differentiability. So we discuss the strong solvability of (N‐S) in the topology of L^{n,\infty}(\mathbb{R}^{n}) Since the Stokes (the heat) semigroup on L^{n,\infty}(\mathbb{R}^{n}) is not strongly continuous, we may not expect the strong solvability of the Stokes equations for each f So we introduce the restriction on the external forces, not on initial data, as follows: Here,. note that Yamazaki. we. is. obtained the time. firstly. .. .. \displaystyle \lim_{ $\epsilon$\rightar ow 0}\Vert e^{ $\epsilon \Delta$}f(t)-f(t)\Vert_{n,\infty}=0 Indeed with the condition. (A),. we. obtain the. for each t. following. (A). ,. theorem.. Suppose that f \in BC(\mathbb{R}_{\text{）} \cdot L^{n,\infty}(\mathbb{R}^{n}) and that u \in periodic solution of (IE) which satisfies u\in BC(\mathbb{R};L_{ $\sigma$}^{r}(\mathbb{R}^{n}) BC(\mathbb{R};L_{ $\sigma$}^{n,\infty}(\mathbb{R}^{n}) with some r > n and \nabla u \in BC(\mathbb{R};L^{q,\infty}(\mathbb{R}^{n})) with some q \geq \displaytle\frac{n}2 If \mathbb{P}f is Hölder continuous on \mathbb{R} with values in L^{n,\infty}(\mathbb{R}^{n}) and $\iota$ f\mathbb{P}f satisfies (A) then the periodic solution u satisfies the following properties,. ([9]).. Theorem 1.2. is. Let. a. \geq 3. n. .. time. .. ,. ,. (i) u\in BC(\mathbb{R}_{\text{）} L_{ $\sigma$}^{n,\infty}(\mathbb{R}^{n}) \cap C^{1}(\mathbb{R}; L^{n,\infty}(\mathbb{R}^{n}) (ii) u(t). \{u \in L_{ $\sigma$}^{n,\infty}(\mathbb{R}^{n});\partial_{j}\partial_{k}u \in L^{n,\infty}(\mathbb{R}^{n}), j, k = 1, . . . , n\} for. \in. $\Delta$ u\in C(\mathbb{R};L^{n,\infty}(\mathbb{R}^{n})). (iii). u. ,. all t \in \mathbb{R} and. ,. satisfies. \displaystyle \frac{du}{dt}(t)- $\Delta$ u(t)+\mathbb{P}[u\cdot\nabla u](t)=\mathbb{P}f(t). in. L_{ $\sigma$}^{n,\infty}(\mathbb{R}^{n}). ). t\in \mathbb{R}^{n}.. proof of Theorem 1.2, the local in time existence theorem plays an important a direction, Kozono‐Yamazaki [6] construct a local in time strong solution of in sum space the r>n On the other hand, we try to construct a (N‐S) L^{n,\infty}( $\Omega$)+L^{r}( $\Omega$) local solution which satisfies the differential equation of (N‐S) in the topology of L^{n,\infty}(\mathbb{R}^{n}) For the. role. For such. .. ,. .. 2. Result. Before stating our results, we introduce the following notations and some function spaces. Let denotes the set of all C^{\infty} ‐solenoidal vectors $\phi$ with compact support in \mathbb{R}^{n},. C_{0, $\sigma$}^{\infty}(\mathb {R}^{n}). in \mathbb{R}^{n}.. i.e., \mathrm{d}\mathrm{i}\mathrm{v} $\phi$=0 1<r<\infty. 1. \leq r \leq. \infty.. ) L^{r}(\mathbb{R}^{n}). L_{ $\sigma$}^{r}(\mathbb{R}^{n}). is the closure of. C_{0, $\sigma$}^{\infty}(\mathb {R}^{n}). with respect to the L^{r} ‐norm. \Vert\cdot\Vert_{r},. duality pairing between L^{r}(\mathbb{R}^{n}) and L^{r'}(\mathbb{R}^{n}) , where 1/r+1/r'=1, and W^{m,r}(\mathbb{R}^{n}) denote the usual (vector‐valued) L^{r} ‐Lebesgue space. is the. and L^{r} ‐Sobolev space over \mathbb{R}^{n} , respectively. Moreover, S(\mathbb{R}^{n}) denotes the set of all of the Schwartz functions. S'(\mathbb{R}^{n}) denotes the set of all tempered distributions. When X is a Banach space,. \Vert\cdot\Vert_{X}. denotes the. norm on. X. .. Moreover, C(I;X) BC(I;X) and L^{r}(I;X) ,. denote the X ‐valued continuous and bounded continuous functions and X ‐valued L^{r}. functions, respectively.. over. the interval I\subset \mathbb{R}.
(4) 108. Moreover, for 1 integrable functions. |E|. (quasi). with. norm. L^{p,q}(\mathbb{R}^{n}). let. \infty. \Vert f\Vert_{p,q}<\infty. ,. be the space of all. locally. where. \left{\begin{ar y}{l (\int_{0}^\infty}($\lambd$|\{xin\mathb{R}^n;|f(x)>$\lambd$\}|^{frac1}{p)^q}\frac{d$\lambd$}{ \lambd$})^{\frac1}{\mathr{q} &1\leq<\infty,\ sup_{$\lambd$>0} \lambd$|\{xin\mathb{R}^n;|f(x)>$\lambd$\}|^{frac1}{p,&q=\infty, \end{ar y}\ight.. \Vert f\Vert_{p,q}= where. and 1 \leq q \leq. < p < \infty. denotes the. Lebesgue measure of E\subset \mathbb{R}^{n} For the following norm: with any 1\leq r<p .. Banach space with the. case. q=\infty,. If^{\infty}(\mathbb{R}^{n}). is. a. \displaystyle \Vert f\Vert_{L^{p,\infty} =\sup_{0<|E|<\infty}|E|^{-\frac{1}{r}+\frac{1}{\mathrm{p} (\int_{E}|f(x)|^{r}dx)^{\frac{1}{r} Here,. we. note that. To construct. a. \Vert\cdot\Vert_{L^{n}. ). is. \infty. equivalent. local solution of. to. (N‐S),. \overline{L}_{ $\sigma$}^{n,\infty}(\mathb {R}^{n})=\overline{L_{ $\sigma$}^{n,\infty}(\mathb {R}^{n})\cap L^{\infty}(\mathb {R}^{n}) ^{\Vert\cdot\Vert_{n,\infty}. \Vert. \Vert_{n,\infty}.. we. and. introduce the. following function. spaces.. L_{0, $\sigma$}^{n,\infty}(\mathb {R}^{n})=\overline{\{ $\phi$\in C_{0}^{\infty}(\mathb {R}^{n});\mathrm{d}\mathrm{i}\mathrm{v} $\phi$=0\} ^{\Vert\cdot\Vert_{n,\infty}. See, Taniuchi [10] and Koba [5].. \overline{L}_{ $\sigma$}^{n,\infty}(\mathb {R}^{n}) and f \in BC([0, \infty);L^{n,\infty}(\mathbb{R}^{n})) Suppose f is Hölder [0, \infty ) with value in L^{n,\infty}(\mathbb{R}^{n}) and satisfies (A) There are T > 0 and a function u\in BC((0, T);L_{ $\sigma$}^{n,\infty}(\mathbb{R}^{n})) with \nabla t^{1/2}u\in BC((0, T);L^{n,\infty}(\mathbb{R}^{n})) which satisfies Theorem 2.1. Let a\in continuous. .. on. .. (i) u\in BC((0, T);L_{ $\sigma$}^{n,\infty}(\mathbb{R}^{n}))\cap C^{1}((0, T);L_{ $\sigma$}^{n,\infty}(\mathbb{R}^{n})). (ii) u(t). \{u \in L_{ $\sigma$}^{n,\infty}(\mathbb{R}^{n});\partial_{j}\partial_{k}u \in L^{n,\infty}(\mathbb{R}^{n}), j, k = 1, . . . , n\} for. \in. $\Delta$ u\in C((0, T);L^{n,\infty}(\mathbb{R}^{n})). (iii). u. ,. (0, T). all t \in. and. ,. satisfies. \left\{ begin{ar y}{l \frac{du}{dt}()-$\Delta$u(t)+\mathb {P}[u\cdot\nabl u](t)=\mathb {P}f(t)inL_{$\sigma$}^{n,\infty}(\mathb {R}^{n}),t\in(0,T),\ u(t)-aweakly*inL_{$\sigma$}^{n,\infty}(\mathb {R}^{n})ast\searow0. \end{ar y}\right. Moreover, if expressed as. with. some. a\in. L_{ $\sigma$}^{n,\infty}(\mathbb{R}^{n})\cap L^{r}(\mathbb{R}^{n}) for. some r. > n,. then the existence time T. >. 0 is. T\displaystyle\geq\min\{1,(\frac{$\eta$_{*}{\Verta\Vert_{r}+\sup_{0<s \infty}\Vert\mathb {P}f(s)\Vert_{n,\infty})^{\frac{2r}{ -n}\},. absolute constant. Remark 2.1.. satisfy. (i). if. a. $\eta$_{*}>0.. L_{ $\sigma$}^{n,\infty}(\mathbb{R}^{n}). \in. and. f. \in. L^{1}(0, \infty {}_{\text{）}}L^{n,\infty}(\mathbb{R}^{n}))\cap BC([0, \infty);L^{n,\infty}(\mathbb{R}^{n})). \displaystyle \Vert a\Vert_{n,\infty}+\Vert \mathbb{P}f\Vert_{L^{1}(0,\infty;L^{n,\infty})}+\sup_{s\in \mathbb{R} s\Vert \mathbb{P}f(s)\Vert_{n,\infty}\l 1, then. we can. take T=\infty.. (ii) Along to Koba [5], if. a\in L_{0, $\sigma$}^{n_{\text{）} \infty}(\mathbb{R}^{n}). is small. if. a\in L_{0, $\sigma$}^{n,\infty}(\mathbb{R}^{n}\rangle we also see that \displaystyle \lim_{t\rightar ow 0}\Vert u(t)-a\Vert_{n,\infty}=0. enough. and. f\equiv 0. then. t\rightar ow\infty \mathrm{h}\mathrm{m}\Vert u(t)\Vert_{n,\infty}=0.. .. Moreover.
(5) 109. Key lemma. 3. In this. subsection,. reconstruct. we. group which is not. strongly. strongly. on. For. continuous. a. bounded. a. theory of abstract evolution equations with the semi‐ 0 Indeed, the Stokes semigroup is not. continuous at t. L_{ $\sigma$}^{r,\infty}(\mathbb{R}^{n}). =. .. .. while, let A be a general closed operator and analytic on X with the estimates. on. Banach space X and. a. \displaystyle \sup_{0<t<\infty}\Vert e^{tA}\Vert_{\mathcal{L}(X)}\leq N, \Vert Ae^{tA}\Vert_{\mathcal{L}(X)} \leq\frac{M}{t}, t>0 where norm.. \{e^{tA}\}. \mathrm{a}. (3.1). ,. is the space of all bounded linear operators on X equipped with the operator Especially, we note that e^{tA} is strongly continuous in X for t\neq 0.. \mathcal{L}(X). Definition 3.1. Let $\theta$\in in X with the order. (0,1].. $\theta$_{f} if for. We call. f. is the Hölder continuous. every T>0 there exists. K_{T}>0. on. [0, \infty). with value. such that. \Vert f(t)-f(s)\Vert_{X}\leq K_{T}|t-s|^{ $\theta$}, 0\leq t\leq T, 0\leq s\leq T. Assumption.. Let. f. [0, \infty). :. \rightarrow X. .. We. assume. for. every t>0. \displaystyle \lim_{ $\epsilon$\sear ow 0}\Vert e^{ $\epsilon$ A}f(t)-f(t)\Vert_{X}=0 Lemma 3.1. Let a\in. X and let. f. \in. with value in X with order $\theta$>0 and. C([0, \infty);X). (A). .. be the Hölder continous. satisfy Assumption.. on. Then. [0, \infty ). u(t)=e^{tA}a+\displaystyle \int_{0}^{t}e^{(t-s)A}f(s)ds satisfies. \displaystyle \frac{d}{dt}u=Au+f. Remark 3.1. We note that initial data. a. condition. If X^{*} , then. we. The. local in. need. a. restriction. only. Lemma 3.1 does not focus. have. information of the. we recover. some. f. not. on. A^{*} and of the dual space a suitable sense.. proof. 2.1 is fulfilled. by the standard iteration method developed by Fujita Miyakawa [3] and Giga [2]. The difficulty to construct [1], [4], Giga time mild solution comes from the lack of the density of C_{0, $\sigma$}^{\infty}(\mathb {R}^{n}) in L_{ $\sigma$}^{n,\infty}(\mathbb{R}^{n}) Kato. and. .. \overline{L}_{ $\sigma$}^{n,\infty}(\mathb {R}^{n}). a. the external force. the verification of the initial. adjoint operator. For this reason, we restrict initial data within in time solution in a suitable function spaces, Lemma 3.1 to. on. on. the verification of the initial condition with. proof of Theorem. and Kato. we. t>0.. Moreover,. Outline of. 4. a. .. in X. .. Then. once we. obtain. a. local. guarantees the mild solution is. strong solution, i.e., satisfies the differential equations of (N‐S), since it is not difficult see that the nonlinear term satisfies the assumption (A) by the regularity of the mild. solution..
(6) 110. Application. 5. As is mentioned in the previous section, our motivation is to prove the strong solvability of the time periodic problem of (N‐S), see [9]. For this purpose, to construct a local strong solution and the uniqueness theorem of the mild solution of (N‐S) are essential. So introduce the uniqueness theorem in weak L^{n} space proved by Kozono and Yamazaki Theorem 5.1. ([7]).. Let n<r<\infty. .. Then there exists. a. constant. following property. Let a\in L_{ $\sigma$}^{n,\infty}(\mathbb{R}^{n})\cap L_{ $\sigma$}^{r}(\mathbb{R}^{n}) Suppose v is [0, T) of (N‐S) obtained by Theorem 2.1. Suppose w is also a mild (N‐S) which satisfies t^{\frac{1}{2}-\frac{n}{2r}}w\in BC((0, T);L^{r}(\mathbb{R}^{n})) If. the. .. $\kappa$= $\kappa$(n, r). >0 with. the mild solution solution. on. we. [7]. on. [0, T ) of. .. \displaystyle \lim_{t\rightar ow}\sup_{0}t^{\frac{1}{2}-\frac{n}{2r} \Vert w(t)\Vert_{r}\leq $\kap a$ then. v\equiv w. on. (0, T). (5.1). .. only give the sketch of proof of Theorem 1.2. Firstly, we construct a mild periodic solution of (N‐S) with suitable regularity. Then solve the initial value problem of (N‐S) where the initial state is the point on the periodic orbit. Finally, by the uniqueness theorem, we may conclude the time periodic mild solution Then. we. solution of the time. satisfies the differential equation of. (N‐S).. References [1]. H.. Fujita and T. Kato, On the Navier‐Stokes initial value problem. I) Arch. Rational Anal., 16 (1964), 269‐315.. Mech.. [2]. Y.. Giga, Solutions for semilinear parabohc equations in IP and regularity of weak of the Navier‐Stokes system, J. Differential Equations, 62 (1986), 186‐212.. solutions. [3] Y.Giga. and T.. Miyakawa, Solutions in L_{r} of the Navier‐Stokes initial Anal., 89 (1985), 267‐281.. value. problem,. Arch. Rational Mech.. [4]. T.. Kato, Strong Ii ‐solutions of the Navier‐Stokes equation solutions, Math. Z., 187 (1984), 471‐480.. in \mathrm{R}^{m} , with. applications. to weak. [5]. [6]. H.. Koba, On L^{3,\infty} ‐stability of. ferential Equations, 262 H. Kozono and M. exterior. problem. the Navier‐Stokes system in exterior 2618‐2683.. domains, J. Dif‐. (2017),. Yamazaki, Local and global unique solvability of the Navier‐Stokes Cauchy data in the space L^{n,\infty} Houston J. Math., 21 (1995),. with. ,. 755‐799.. [7]. H. Kozono and M.. Yamazaki, On a larger class of stable solutions to the Navier‐ domains, Math. Z., 228 (1998), 751‐785.. Stokes equations in exterior.
(7) 111. [8]. Y.. Meyer, Wavelets, paraproducts, and Navier‐Stokes equations, In;Current develop‐ mathematics, 1996 (Cambridge, MA), 105‐212, Int. Press, Boston, MA,. ments in. 1997.. [9]. T. Okabe and Y.. Tsutsui,. [10]. Y.. periodic strong solutions to the incompressible non‐divergence external force, preprint.. Time. Navier‐Stokes equations with. Taniuchi, On the uniqueness of time‐periodic solutions to the Navier‐Stokes domains, Math. Z., 261 (2009), 597‐615.. equa‐. tions in unbounded. [11]. M.. Yamazaki, The Navier‐Stokes equations in the weak-L^{n} force, Math. Ann., 317 (2000), 635‐675.. external. space with. time‐dependent.
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