Abstract
This paper is concerned with the uniqueness of bounded continuous \(L^{3,\infty }\)-solutions on the whole time axis \(\mathbb R\) or the half-line \((-\infty ,T)\) to the Navier–Stokes equations in 3-dimensional unbounded domains. When \(\Omega \) is an unbounded domain, it is known that a small solution in \(BC(\mathbb R;L^{3,\infty })\) is unique within the class of solutions which have sufficiently small \(L^{\infty }(\mathbb R; L^{3,\infty })\)-norm; i.e., if two solutions u and v exist for the same force f, both u and v are small, then the two solutions coincide. There is another type of uniqueness theorem. Farwig et al. (Commun Partial Differ Equ 40:1884–1904, 2015) showed that if two solutions u and v exist for the same force f, u is small and if v has a precompact range \({\mathscr {R}}(v):=\{v(t); -\infty<t<T\}\) in \(L^{3,\infty },\) then the two solutions coincide. However, there exist many solutions which do not have precompact range. In this paper, instead of the precompact range condition, by assuming some decay property of v(x, t) with respect to the spatial variable x near \(t=-\infty ,\) we show a modified version of the above-mentioned uniqueness theorem. As a by-product, in the half-space \(\mathbb R^3_+,\) we obtain a non-existence result of backward self-similar \(L^{3,\infty }\)-solutions sufficiently close to some homogeneous function Q(x/|x|)/|x| in a certain sense.
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1 Introduction
The motion of a viscous incompressible fluid in 3-dimensional domains \(\Omega \) is governed by the Navier–Stokes equations:
where \(u=(u^1(x,t), u^2(x,t), u^3(x,t))\) and \(p= p(x, t)\) denote the velocity vector and the pressure, respectively, of the fluid at the point \((x, t) \in \Omega \times \mathbb R.\) Here f is a given external force. In this paper we consider the uniqueness of bounded mild \(L^{3}_{w}\)-solutions to (N-S) on the whole time axis \((-\infty ,\infty )\) or the half-line \((-\infty ,T)\) in unbounded domains \(\Omega .\) Typical examples of such solutions are stationary, periodic-in-time and almost periodic-in-time solutions.
In case where \(\Omega \subset \mathbb R^3\) is bounded, the existence and uniqueness of time-periodic solutions were considered by several authors; see e.g. [12] and references therein. Maremonti [38, 39] was the first to prove the existence of unique time-periodic regular solutions to (N-S) in unbounded domains, namely for \(\Omega =\mathbb R^3\) and \(\Omega =\mathbb R^3_+.\) In the case of more general unbounded domains, the existence of time-periodic solutions was proven by e.g. Kozono–Nakao [30], Maremonti–Padula [40], Salvi [47], Yamazaki [55], Galdi–Sohr [21], Kubo [35], Crispo–Maremonti [6], Kang–Miura–Tsai [28], Geissert–Hieber–Nguyen [22], Okabe–Tsutsui [46], Galdi–Kyed [20], Galdi [19], Eiter–Kyed [8] and Eiter–Kyed–Shibata [9]. Some of them constructed solutions to (N-S) on the whole time axis without any time-periodic condition on the external force. In particular, Kozono–Nakao [30] introduced a new approach to the study of time-periodic solutions by using the following integral equation:
where the definitions of P and A will be mentioned later. In [30], they constructed global \(L^3\)-solutions on the whole time axis \((-\infty ,\infty )\) to (1.1), which are called mild solutions to (N-S) on \((-\infty ,\infty ).\) More precisely, Kozono–Nakao [30] showed that if the domain \(\Omega \) is \(\mathbb R^n,\mathbb R^n_+\) \((n\ge 3)\) or an n-dimensional exterior domain \((n\ge 4)\) and if the external force f is small in some function space, then there exists a small \(L^n\)-solution to (1.1) in the class
where \(\delta \) is a small number. They also showed that this small solution is unique within the class (1.2). Furthermore, they proved that this small \(L^n\)-solution is a strong solution to (N-S) and that if f is a small time-periodic function in some function space, then their small solution u is also time-periodic. In the case where \(\Omega \subset \mathbb R^n(n\ge 3)\) is a perturbed half space or an aperture domain with \(\partial \Omega \in C^{\infty },\) Kubo [35] proved the same uniqueness and existence theorem as Kozono–Nakao [30]. When \(\Omega =\mathbb R^3, \mathbb R^3_+,\) a perturbed half space or an aperture domain with \(\partial \Omega \in C^{\infty },\) Farwig, Nakatsuka and the present author [11] showed that if f is small and has a precompact range in some function space, then the small solution u also has a precompact range in \(L^3.\)
In the case where \(\Omega \) is a 3-dimensional exterior domain with \(\partial \Omega \in C^{\infty },\) Yamazaki [55] succeeded in proving the existence of small mild \(L^{3,\infty }\)-solutions on the whole time axis, if the external force f is sufficiently small in some sense. More precisely, he showed the existence of small mild \(L^{n,\infty }\)-solutions in the case \(\Omega \) is an n-dimensional exterior domain for \(n\ge 3.\) Here \(L^{p,q}\) denotes the Lorentz space and \(L^{p,\infty }\) is equivalent to the weak-\(L^p\) space \((L^p_w).\) Yamazaki [55] also proved uniqueness of small solutions within the class
where \(\delta =\delta (\Omega ,n)\) is a small number and \(L^{n,\infty }_{\sigma }\) will be defined in the next section. See Lemma 8 below. Furthermore, if \(n=3,\) Kang–Miura–Tsai [28] showed the existence of mild solutions u on the whole time axis with the spatial uniform decay:
for some \(\alpha >1,\) \(L>0\) and some function V(x) with \(\sup _{|x|>L}|x||V(x)|<\infty ,\) if f satisfies adequate conditions. Note that \(V\in L^{3,\infty }(\Omega )\) and \(\sup _{t}\Vert v(t)-V\Vert _{ L^{r,\infty }(\Omega )}<\infty \) for some \(1<r<3.\) They also dealt with the inhomogeneous boundary value problem.
Concerning the uniqueness of solutions on the whole time axis, roughly speaking, it was shown in [30, 35, 55] that a small solution in some function spaces (e.g. \(BC(\mathbb R;L^{3,\infty }(\Omega ))\)) is unique within the class of solutions which are sufficiently small; i.e., if u and v are solutions for the same force f and if both of them are small, then \(u=v.\) It is notable that, concerning time-periodic solutions, Galdi–Sohr [21] show that a small time-periodic solution is unique within the larger class of all periodic weak solutions v with \(\nabla v\in L^2(0,T_{per};L^2),\) satisfying the energy inequality \(\int _0^{T_{per}}\Vert \nabla v\Vert _{L^2}^2\,d\tau \le -\int _0^{T_{per}}(F, \nabla v)\,d\tau \) and mild integrability conditions on the corresponding pressure; here \(T_{per}\) is a period of F and \(f=\nabla \cdot F.\)
Another type of uniqueness theorem for solutions on the whole time axis or the half-line \((-\infty ,T)\) was proven by Farwig, Nakatsuka and the present author [10] without time-periodic condition, where it was proven that if u and v are solutions for the same force f, u is small and if v satisfies the precompact range condition (PRC):
then \(u=v\) on \((-\infty ,T).\) Note that the smallness condition is assumed only on one of solutions. See also [16, 17, 43, 44, 51].
Since almost periodic-in-time solutions satisfy (PRC), this uniqueness theorem is applicable to almost periodic-in-time solutions. In [10], without (PRC), it was also shown that a similar uniqueness theorem holds under the smallness condition of u, (1.3) and the condition: \(u,v\in BC(-\infty ,T;L^{3,\infty }\cap L^q)\) for some \(q>3.\) On the other hand, there are many mild \(L^{3,\infty }\)-solutions to (N-S) that satisfy neither (PRC) nor (1.3). For example, traveling solutions \(u(x,t)=u(x-a\cdot t, 0)\) \((a\not =0, u\not \equiv 0)\) satisfy neither (PRC) nor (1.3), when \(\Omega =\mathbb R^3.\) Other examples are backward self-similar solutions \(u(x,t)=\frac{1}{\sqrt{-t}} U(\frac{x}{\sqrt{-t}})\) in \(\mathbb R^3\times (-\infty ,-1),\) when \(U\not \equiv 0\) and \(U\in L^3.\)
Very recently the present author showed a similar uniqueness theorem without (PRC) or (1.3) in [52, Theorem 3], where the \(L^r(r<3)\) and \(L^3\) integrabilities of solutions are assumed instead of (PRC). These integrabilities, however, are too restrictive for solutions to the 3D exterior problem. Indeed, if \(\Omega \) is a 3D exterior domain and u is a stationary weak solution to (N-S), then u cannot belong to \(L^3,\) excepting in the case where the net force exerted on \(\partial \Omega \) is equal to zero. See e.g. [32]. In the present paper, we will improve the uniqueness theorems given in [10, 52] in order to deal with \(L^{3,\infty }\)-solutions to the 3D exterior problem.
Throughout this paper we impose the following assumption on the domain.
Assumption 1
\(\Omega \subset \mathbb R^3\) is an exterior domain, the half-space \(\mathbb R^3_+,\) the whole space \(\mathbb R^3,\) a perturbed half-space, or an aperture domain with \(\partial \Omega \in C^{2+\nu },\) \(0<\nu <1.\)
Here, the assumption \(\partial \Omega \in C^{2+\nu }\) means that for each \(x\in \partial \Omega \) there are an open ball \(B_x\) centered at x and a function \(g_x\in C^{2+\nu }(G)\) for some domain \(G\subset \mathbb R^2\) such that after a rotation of the Cartesian coordinates, if necessary,
\(y_3>g_x(y_1,y_2)\) for all \(y=(y_1,y_2,y_3)\in \Omega \cap B_x ,\) \(y_3<g_x(y_1,y_2)\) for all \(y\in (\mathbb R^3\backslash \bar{\Omega })\cap B_x\) and \(y_3=g_x(y_1,y_2)\) for all \(y\in (\partial \Omega )\cap B_x.\)
The definitions of a perturbed half-space and an aperture domain are as follows (See e.g. [13,14,15]). Let \(\mathbb R^3_+:=\{x\in \mathbb R^3; x_3>0\}\) and \(\Omega \subset \mathbb R^3\) be a domain. If there exists an open ball \(B\subset \mathbb R^3\) such that \(\Omega \cup B=\mathbb R^3_{+}\cup B,\) then \(\Omega \) is called a perturbed half-space. If there exists an open ball \(B\subset \mathbb R^3\) such that \(\Omega \cup B=\mathbb R^3_{+}\cup \mathbb R^3_-\cup B\) where \(d>0\) and \(\mathbb R^3_-:=\{x\in \mathbb R^3; x_3<-d\},\) then \(\Omega \) is called an aperture domain. Since the aperture domain \(\Omega \) should be connected, there are some apertures and one can take two disjoint subdomains \(\Omega _{\pm }\) and a smooth 2-dimensional manifold M such that \(\Omega =\Omega _-\cup M\cup \Omega _+,\) \(\Omega _{\pm }\backslash B=\mathbb R^3_{\pm }\backslash B\) and \(M\cup \partial M=\partial \Omega _-\cap \partial \Omega _+.\) We do not need to assume the connectedness of \(\mathbb R^3\backslash \bar{\Omega }.\)
Let BC(I; X) denote the set of all bounded continuous functions on an interval I with values in a Banach space X. The open ball with center x and radius \(R>0\) will be denoted by \(B_R(x).\) Let \(\mu (A)\) be the 3-dimensional Lebesgue measure of \(A\subset \mathbb R^3.\) The definition of mild \(L^{3,\infty }\)-solutions will be written in the next section. Let
where the definition of \(L^{3,\infty }_{\sigma }(\Omega )\) will be written in the next section. Now our main results on uniqueness of mild \(L^{3,\infty }\)-solutions reads as follows.
Theorem 1
Let \(\Omega \) satisfy Assumption 1. There exists constants \(\delta (\Omega ),c_*(\Omega )>0\) with the following property: Let \(T\le \infty ,\) u and v be mild \(L^{3,\infty }\)-solutions to (N-S) on \((-\infty ,T)\) for the same force f,
and
Assume that there exists a function \(V=V(x) \in L^{3,\infty }(\Omega )\) such that
Then \(u\equiv v\) on \((-\infty ,T).\) Here \(\{|v(t)-V|\le \gamma _0|t|^{-1/2} \}:=\{x\in \Omega \ ; \ |v(x,t)-V(x)|\le \gamma _0|t|^{-1/2}\}.\)
Corollary 2
Let \(\Omega \) satisfy Assumption 1 and \(1<r<3.\) There exist a constant \(c_{**}(\Omega ,r)>0\) with the following property : Let \(T\le \infty ,\) u and v be mild \(L^{3,\infty }\)-solutions to (N-S) on \((-\infty ,T)\) for the same force f. Assume that (1.4) and (1.5) hold. Furthermore assume that there exists a function \(V\in L^{3,\infty }(\Omega )\) such that
Then \(u\equiv v\) on \((-\infty ,T).\)
By the interpolation inequality:
we see that (1.8) implies (1.6) if \(c_{**}\) is sufficiently small. Hence, Corollary 2 is a direct consequence of Theorem 1.
Remark 1
-
(i)
When \(\Omega \) is an aperture domain, all mild \(L^{3,\infty }\)-solutions are assumed to belong to \(L^{3,\infty }_{\sigma }\) and hence satisfy the vanishing flux condition \(\phi (u(t))=0\) for all \(t\in (-\infty ,T),\) see the next section.
-
(ii)
Condition (1.8) can be replaced by the following simpler condition:
$$\begin{aligned} \limsup _{t\rightarrow -\infty }\Vert v(t)-V\Vert _{L^{r,\infty }(\Omega )}<\infty \end{aligned}$$(1.9)for some \(1<r<3\) and for some \(V\in L^{3,\infty },\) since (1.9) implies that the L.H.S. of (1.8) vanishes. We emphasis that (1.9) does not need any smallness conditions on v and V themselves. Furthermore, when \(\Omega =\mathbb R^3,\mathbb R^3_+,\) 3-dimensional perturbed half-spaces or aperture domains, letting \(V\equiv 0,\) we see that Corollary 2 is applicable to mild \(L^3\)-solutions on \((-\infty ,\infty )\) in the class (1.2) given by Kozono–Nakao [30] and Kubo [35], since \(L^3_{\sigma }(\Omega )\subset \tilde{L}^{3,\infty }_{\sigma }(\Omega ).\)
-
(iii)
For \(d(t):=|t|^{1/2}/\log |t|,\)
$$\begin{aligned} \Vert v(t)-V\Vert _{L^{3,\infty }(\{|v(t)-V|\le \gamma _0|t|^{-1/2}\}\cap B_{d(t)}(0))}\le C\{\mu (B_{d(t)}(0))\}^{1/3}\gamma _0|t|^{-1/2}\rightarrow 0 \end{aligned}$$as \(t\rightarrow -\infty .\) Then Condition (1.6) is equivalent to
$$\begin{aligned} \limsup _{t\rightarrow -\infty }\Vert v(t)-V\Vert _{L^{3,\infty }(\{|v(t)-V|\le \gamma _0|t|^{-1/2} \}\cap \{|x|\ge d(t)\})}<\delta . \end{aligned}$$Hence, roughly speaking, (1.6) requires only that v is close to some function \(V\in L^{3,\infty }\) in the \(L^{\infty }L^{3,\infty }\)-topology in the area near \(|x|=\infty \) and \(t=-\infty .\) In other words, v is assumed to behave like V(x) as \(|x|\rightarrow \infty \) near \(t=-\infty .\)
-
(iv)
In the case \(\Omega \) is a 3D exterior domain, Yamazaki [55] proved the existence of bounded continuous mild \(L^{3,\infty }\)-solutions u on the whole time axis, if f can be written in the form \(f=\nabla \cdot F,\) \(F\in BUC(\mathbb R; L^{3/2,\infty })\) and F is sufficiently small. We note that, in addition to this smallness condition on F, if we assume \(f\in BC(\mathbb R;L^{3,\infty }),\) then standard arguments easily prove that Yamazaki’s small solution u belongs to \(L^{\infty }(\mathbb R;L^9)\cap BC(\mathbb R;L^{3,\infty }_{\sigma }) \subset BC(\mathbb R; \tilde{L}_{\sigma }^{3,\infty });\) see [16, Remark 2]. Moreover, the existence of small mild solutions with (1.3) was also proven by Kang–Miura–Tsai [28] if f satisfies some conditions. Since (1.3) implies (1.9) and hence (1.8), Theorem 1 is applicable to their solutions. We also note that Corollary 2 is an improvement of our previous uniqueness theorem given in [52, Theorem 3], where the conditions \(u,v \in BC((-\infty ,T); L^3_{\sigma }),\) (1.5) and \(\limsup _{t\rightarrow -\infty }\Vert v(t)\Vert _{L^{r}}<\infty \) \((r<3)\) were assumed.
In Theorem 1, the function \(V=V(x)\) is assumed to be a function of x and independent of time-variable t. When V is assumed to be a function of (x, t), we have:
Theorem 3
Let \(\Omega \) satisfy Assumption 1. There exists a constant \(\delta (\Omega )>0\) with the following property : Let \(T\le \infty \) and let u and v be mild \(L^{3,\infty }\)-solutions to (N-S) on \((-\infty ,T)\) for the same force f. Assume that (1.4) and (1.5) hold. Furthermore assume that there exists a function \(V \in BC((-\infty ,T);L^{3,\infty }(\Omega ))\) such that
and the range \({\mathscr {R}}(V):=\{V(t) \in L^{3,\infty }(\Omega ); \ t\in (-\infty ,T)\}\) can be covered by finitely many open balls of radius \(\delta >0,\) i.e., there are finitely many functions \(\{V_l\}_{l=1}^N\subset L^{3,\infty }(\Omega )\) satisfying
Then \(u\equiv v\) on \((-\infty ,T).\) Here \(\{|v(t)-V(t)|\le \eta \}:=\left\{ x\in \Omega ; \ |v(x,t)-V(x,t)|\le \eta \right\} .\)
Remark 2
-
(i)
Condition (1.10) is more restricted compared to Condition (1.6) in some sense, since the constant \(\eta \) does not decay as \(t\rightarrow -\infty .\)
-
(ii)
If \(\sup _{t\le t_0}\Vert v(t)-V(t)\Vert _{L^{r,\infty }}(=:A)<\infty \) for some \(1<r<3\) and \(t_0<T,\) then
$$\begin{aligned} \sup _{t\le t_0}\Vert v(t)-V(t)\Vert _{L^{3,\infty }(\{|v(t)-V(t)|\le \eta \}}\le C\eta ^{1-r/3} A^{r/3}<\delta \end{aligned}$$for all \(\eta \in \big (0,\big (\delta /(CA^{r/3})\big )^{3/(3-r)}\big ) .\) Hence, Condition (1.10) can be replaced by
$$\begin{aligned} \limsup _{t\rightarrow -\infty }\Vert v(t)-V(t)\Vert _{L^{r,\infty }(\Omega )}<\infty \quad \text{ for } \text{ some } 1<r<3. \end{aligned}$$ -
(iii)
If v has a precompact range in \(L^{3,\infty },\) then by setting \(V=v\) we can see that (1.10) and (1.11) hold. Hence Theorem 3 is an improvement of the uniqueness theorem given in [10] with the precompact range condition.
Remark 3
In Theorems 1 and 3 and Corollary 2, Condition (1.4) can be replaced by the condition:
where \(I=(-\infty ,T)\) and \(\kappa \) is a small constant depending only on \(\Omega .\) See [10, Remark 1(iv)].
In the celebrated papers Nečas–Růžička–Šverák [45] and Tsai [53], the non-existence theorems of backward self-similar solutions in \(L^{3}(\mathbb R^3)\) and \(L^q(\mathbb R^3)\) for \(q\in (3,\infty ]\) were proven, respectively. More precisely, Nečas–Růžička–Šverák [45] proved that if \(\frac{1}{\sqrt{a(T-t)}}v(\frac{x}{\sqrt{a(T-t)}}),\) \(a>0,\) is a backward self-similar solution to (N-S) for \(f=0\) and if \(v\in L^3(\mathbb R^3),\) then \(v=0\) in \(\mathbb R^3.\) Tsai [53] proved that if \(v\in L^q(\mathbb R^3)\) for some \(3<q\le \infty ,\) then v is constant in \(\mathbb R^3\) and hence \(v=0\) if \(q<\infty .\) Moreover, Tsai [53] also proved the non-existence theorem of backward self-similar solutions satisfying local energy estimate. Their results were proven for the case where the domain \(\Omega \) is the 3D whole space \(\mathbb R^3.\) In the case where \(\Omega \) is the half-space \(\mathbb R^3_+,\) as a by-product of Corollary 2 we have the following non-existence result.
Corollary 4
Let \(\Omega =\mathbb R^3_+,\) \(a>0,\) \(1<r<3\) and functions \(v,V,R \in L^{3,\infty }(\mathbb R^3_+)\) and \(Q\in C(S^2)\) satisfy
Assume \(\frac{1}{\sqrt{-at}}v(\frac{x}{\sqrt{-at}})\) is a mild solution of (N-S) on \((-\infty , 0)\) in \(\mathbb R^3_+\) for \(f=0.\) Then \(v\equiv 0.\) Here \(c_{**}=c_{**}(\Omega ,r)\) is the constant given in Corollary 2.
We note that if functions v, V, R, Q satisfy (1.12)–(1.14), then \(\frac{1}{\sqrt{-at}}v(\frac{x}{\sqrt{-at}})\in BC((-\infty ,0);\tilde{L}^{3,\infty }(\mathbb R^3_+)),\) see Appendix. Since \(\frac{1}{\sqrt{-at}}V(\frac{x}{\sqrt{-at}})=V(x),\) we see that \(V(x)-\frac{1}{\sqrt{-at}}v(\frac{x}{\sqrt{-at}})(=\frac{1}{\sqrt{-at}}R(\frac{x}{\sqrt{-at}}))\) satisfies Condition (1.8). Thus, by letting \(u=0,\) we see that Corollary 2 directly yields Corollary 4.
Remark 4
-
(i)
For any given \(V(x)=Q(x/|x|)/|x|\in L^{3,\infty }_{\sigma }(\mathbb R^3_+)\) with \(Q\in C^1(S^2)\) and \(1<r<3,\) we can construct functions \(v\in \tilde{L}^{3,\infty }_{\sigma }(\mathbb R^3_+)\cap W^{2,q}(\mathbb R^3_+)\cap W^{1,q}_0(\mathbb R^3_+)\) \((q>3)\) and \(R\in L^{3,\infty }(\mathbb R^3_+)\cap L^{r,\infty }(\mathbb R^3_+)\) satisfying (1.12)–(1.15), if we do not assume that \(\frac{1}{\sqrt{-at}}v(\frac{x}{\sqrt{-at}})\) is a solution to (N-S) for \(f=0.\)
-
(ii)
Corollary 4 also holds for \(\Omega =\mathbb R^3.\) Compared with the results in [45, 53], Condition (1.15) is very restrictive, since (1.15) means that v is sufficiently close to the homogeneous function V.
2 Preliminaries
In this section, we introduce some notations, function spaces and key lemmata. Let \(C^{\infty }_{0,\sigma }(\Omega )=C^{\infty }_{0,\sigma }\) denote the set of all \(C^{\infty }\)-real vector fields \(\phi =(\phi ^1,\ldots ,\phi ^n)\) with compact support in \(\Omega \) such that \(\text{ div } \phi =0.\) Then \(L^r_{\sigma }(\Omega )=L^r_{\sigma },\) \(1<r<\infty ,\) is the closure of \(C^{\infty }_{0,\sigma }\) with respect to the \(L^r\)-norm \(\Vert \cdot \Vert _r.\) Concerning Sobolev spaces we use the notations \(W^{k,p}(\Omega )\) and \(W^{k,p}_0(\Omega ),\) \(k\in {\mathbb {N}},\) \(1\le p\le \infty .\) Note that very often we will simply write \(L^r\) and \(W^{k,p}\) instead of \(L^r(\Omega )\) and \(W^{k,p}(\Omega ),\) respectively. Let \(L^{p,q}(\Omega ),\) \(1< p< \infty ,1\le q\le \infty ,\) denote the Lorentz spaces and \(\Vert \cdot \Vert _{p,q}\) the norm (not quasi-norm) of \(L^{p,q}(\Omega );\) for the definition and properties of \(L^{p,q}(\Omega ),\) see e.g. [1]. The symbol \((\cdot ,\cdot )\) denotes the \(L^2\)-inner product and the duality pairing between \(L^{p,q}\) and \(L^{p',q'},\) where \(1/p+1/p'=1\) and \(1/q+1/q'=1.\) We note that \(L^{p,\infty }=L^p_{w}(\text{ weak- }L^p \text{ space})\) and \(L^{p,p} =L^p\) with equivalent norms. Moreover, when \(1<p<\infty \) and \(1\le q<\infty ,\) then the dual space of \(L^{p,q}\) is isometrically isomorphic to \(L^{p',q'}.\)
In this paper, we denote by C various constants. In particular, \(C=C(*,\ldots ,*)\) denotes a constant depending on the quantities appearing in the parentheses.
Let us recall the Helmholtz decomposition: \(L^r(\Omega ) = L^r_{\sigma }\oplus G_r\) \((1<r<\infty ),\) where \(G_r=\{\nabla p\in L^r; p\in L^r_{loc}(\overline{\Omega })\},\) see Miyakawa [42], Simader–Sohr [50], Borchers–Miyakawa [2], and Farwig–Sohr [13, 15]; \(P_r\) denotes the projection operator from \(L^r\) onto \(L^r_{\sigma }\) along \(G_r.\) The Stokes operator \(A_r\) on \(L^r_{\sigma }\) is defined by \(A_r=-P_r\Delta \) with domain \(D(A_r)=W^{2,r}\cap W^{1,r}_0\cap L^r_{\sigma }.\) It is known that \((L^r_{\sigma })^*\)(the dual space of \(L^r_{\sigma })=L^{r'}_{\sigma }\) and \(A_r^*\)(the adjoint operator of \(A_r\))\(=A_{r'}, \) where \(1/r+1/r'=1.\) It is shown by Giga [23], Borchers–Sohr [5], Giga–Sohr [25], Borchers–Miyakawa [2] and Farwig–Sohr [13, 15] that \(-A_r\) generates a holomorphic semigroup \(\{e^{-tA_r} ; t\ge 0\}\) of class \(C_0\) in \(L^r_{\sigma }.\) Since \(P_r u=P_q u\) for all \(u\in L^r \cap L^q\) \((1<r,q<\infty )\) and since \(A_r u=A_q u\) for all \(u\in D(A_r) \cap D(A_q ),\) for simplicity, we shall abbreviate \(P_r u,P_q u\) as Pu for \(u\in L^r \cap L^q \) and \(A_r u,A_q u\) as Au for \(u\in D(A_r) \cap D(A_q ),\) respectively. By real interpolation, we define \(L^{p,q}_{\sigma }(\Omega )=L^{p,q}_{\sigma }\) by
where \(1<p_0<p<p_1<\infty ,\) \( \theta \in (0,1),\) \(q\in [1,\infty ]\) satisfy \(1/p=(1-\theta )/p_0+\theta /p_1.\) In the case where \(\Omega \) is an aperture domain, since \(L^{p}_{\sigma }=\overline{C^{\infty }_{0,\sigma }(\Omega )}^{\Vert \cdot \Vert _p}\) is characterized as
see [15], and since \(L^{p,q}_{\sigma }(\Omega )\subset L^{p_1}_{\sigma }+L^{p_2}_{\sigma },\) all u belonging to \(L^{p,q}_{\sigma }\) satisfy the vanishing flux condition \(\phi (u)=0.\) Here \(\phi (u)=\int _{M}N\cdot u\ dS\) and N is the unit normal vector on M directed to \(\Omega _-.\)
Recall that, for \(1<p<\infty \) and for a measurable set D, the weak-\(L^p(D)\) norm is defined by
which is equivalent to \(\Vert f\Vert _{L^{p,\infty }(D)},\) as mentioned before. It is known that, for \(1<p<\infty ,\)
where the supremum is taken over all measurable subsets E of the domain D with \(0<\mu (E)<\infty ,\) see e.g. [4, 24].
Now, we define mild \(L^{3,\infty }\)-solutions to (N-S) according to [55]. A similar definition was introduced in [31] for mild \(L^3\)-solutions.
Definition 1
Let \(T\le \infty \) and \(f\in L^{1}_{loc}(-\infty ,T; D(A_{p})^*+D(A_{q})^*)\) for some \(1<p,q<\infty .\)
A function \(v\in C((-\infty ,T);L^{3,\infty }_{\sigma })\) is called a mild \(L^{3,\infty }\)-solution to (N-S) on \((-\infty ,T)\) if v satisfies
for all \( \phi \in C^{\infty }_{0,\sigma }\) and all \(-\infty<s<t<T.\)
Mild \(L^{3,\infty }\)-solutions to the initial-boundary value problem for (N-S) on the interval [0, T) are defined similarly, so we do not write its definition here. For a moment let us consider the case where \(\int _s^t(f(\tau ),e^{-(t-\tau )A}\phi )d\tau \) converges as \(s\rightarrow -\infty \) for all \(\phi \in C^{\infty }_{0,\sigma }.\) E.g., this holds true by (2.8) below when \(f=\nabla \cdot F\) with \(F=(F_{ij})_{i,j=1,2,3}\in L^{\infty }(-\infty ,T;L^{3/2,\infty }).\) Since moreover \(\lim _{s\rightarrow -\infty }e^{-(t-s)A}\phi =0\) in \(L^{3/2,1}_{\sigma },\) we conclude from Lemma 7 below that in this case (2.2) for \(v\in L^{\infty }(-\infty ,T;L^{3,\infty }_{\sigma })\) is equivalent to
for all \(\phi \in C^{\infty }_{0,\sigma }\) and all \(t<T.\) Note that this holds for all \(\phi \in L^{3/2,1}_{\sigma }.\) Furthermore, we see that (2.3) yields (2.2), if \(v\in L^{\infty }(-\infty ,T; L^{3,\infty }_{\sigma }),\) \(f=\nabla \cdot F\) and if \(F\in L^{\infty }(-\infty ,T;L^{3/2,\infty }).\) Hence, Definition 1 is equivalent to the definition given in [55, Definition 1], if we assume \(v\in BUC(\mathbb R; L^{3,\infty }_{\sigma })\) and \(F\in BUC(\mathbb R; L^{3/2,\infty }).\) We also note that (2.3) is a weak form of (1.1).
In order to prove our main result, we recall properties of the Lorentz spaces, estimates of the Stokes semigroup and several uniqueness theorems for mild solutions.
Lemma 5
(Kozono–Yamazaki [33]) Let \(p_1,p_2\in (1,\infty )\) with \(1/r:=1/p_1+1/p_2<1\) and let \(q\in [1,\infty ].\) Then, for all \(f\in L^{p_1,\infty }(\Omega )\) and \(g\in L^{p_2,q}(\Omega ),\) it holds that
where \(C=C(p_1,p_2,q).\)
For \(u\in \dot{W}^{1,2}_{0}(\Omega ) = \overline{C_0^{\infty }(\Omega )}^{\Vert \nabla \cdot \Vert _2}\) it holds with an absolute constant \(C>0\) that
Lemma 6
(Shibata [48, 49]) For all \(t>0\) and \(\phi \in L^{q,s}_{\sigma },\) the following inequalities are satisfied :
In the case where \(\Omega \) is an exterior domain, Shibata [48, 49] proved (2.6) and (2.7) for all \(r=s.\) If \(q<p,\) his estimates (2.6)–(2.7) with \(r=s\) and real interpolation yield (2.6)–(2.7) even for \(r=1, s=\infty .\) In the restricted case \(r=1,\) Yamazaki [55] obtained (2.7) also by a method different from [48, 49]. In the case where \(\Omega \) is \(\mathbb R^3,\) \(\mathbb R^3_+,\) a perturbed halfspace or an aperture domain, the usual \(L^q\)-\(L^p\) estimates for the Stokes semigroup and real interpolation directly yield (2.6)–(2.7), since in this case the \(L^q\)–\(L^p\) estimates hold for all \(1<q\le p<\infty .\) For details of \(L^q\)–\(L^p\) estimates for the Stokes semigroup, see also [2, 3, 25,26,27, 29, 34, 36, 48, 54].
Lemma 7
(Yamazaki [55]) The following estimates
hold for all \(F\in L^{\infty }(s,t;L^{3/2,\infty }),\) \(u,w\in L^{\infty }(s,t;L^{3,\infty }),\) \(\phi \in L^{3/2,1}_{\sigma }(\Omega )\) and all \(-\infty \le s<t,\) where the constant C depends only on \(\Omega .\)
In the case where \(\Omega \) is an exterior domain, the whole space or halfspace, Yamazaki [55] proved Lemma 7 by real interpolation. His proof is also valid in the case where \(\Omega \) is a perturbed halfspace or an aperture domain. In the case where \(\Omega =\mathbb R^3,\) Meyer [41] obtained estimates similar to Lemma 7 by a method different from [55].
The following lemma is direct consequence of Lemma 7 using the duality \(L^{3,\infty }_{\sigma }=(L^{3/2,1}_{\sigma })^*.\)
Lemma 8
(Yamazaki [55]) There exists a constant \(\epsilon _0=\epsilon _0(\Omega )\) with the following property : Let \(T\le \infty ,\) \(u,v,w\in BC((-\infty ,T);L^{3,\infty }_{\sigma })\) and let w satisfy
for all \(\phi \in L^{3/2,1}_{\sigma }\) and all \(-\infty<t<T.\) Assume that
Then, \(w(t)=0\) for all \(t\in (-\infty ,T).\)
Lemma 9
[10, Lemma 2.6] There exists a constant \(\epsilon _1(\Omega )>0\) such that if \(T\le \infty ,\) u, v are mild \(L^{3,\infty }\)-solutions to (N-S) on \((-\infty ,T)\) for the same force f,
then
Lemma 9 can be proven by Lemma 8, the uniqueness of mild solutions in \(C([0,T); \tilde{L}^{3,\infty })\) to IBVP, see [10, Lemma 2.5], and the continuity of \(w(t):=u(t)-v(t)\) with respect to the time-variable t in \(\tilde{L}^{3,\infty }.\) For the detail, see [10, Lemma 2.6].
Finally, we come to the key lemma of the proof of Theorem 1. If u and v are solutions to the Navier–Stokes equations, then \(w:=u-v\) satisfies
Hence, if \(\Omega \) is a bounded domain and if u, v belong to the Leray–Hopf class, under the hypotheses of Theorem 1, the usual energy method and the Poincaré inequality yield \(\Vert w(t)\Vert _2^2\le e^{-c(t-s)}\Vert w(s)\Vert _2^2\) for \(t>s.\) Letting \(s\rightarrow -\infty ,\) we get \(w(t)=0\) for all t. Consequently, in the case of bounded domains, Theorem 1 is obvious. In the case where \(\Omega \) is an unbounded domain, u and v do not belong to the energy class in general and the Poincaré inequality does not hold in general. Hence, since we cannot use the energy method, we will use the dual equations of the above system. It is notable that several researchers utilized the dual equation argument to prove several uniqueness theorems and a-priori estimates for solutions to (N-S). See e.g. Foias [18], Maremonti [38], Lions–Masmoudi [37]. Very recently, Crispo–Maremonti [7] used the dual equation argument to prove uniqueness theorems for suitable weak solutions to (N-S) in the sense of Caffarelli–Kohn–Nirenberg.
Here we will use a similar argument as in Lions–Masmoudi [37]. We recall the dual equations of the above system (U), namely,
Lemma 10
[10] There exists an absolute constant \(\delta _0>0\) with the following property: Let \(u,v \in BC((-\infty ,0];\tilde{L}^{3,\infty }_{\sigma }),\) \(h\in L^2_{loc}((-\infty ,0]; L^{6/5}\cap L^2)\) and
Then there exists a unique solution \(\Psi \in L^2_\mathrm{{loc}}((-\infty ,0];D(A_2))\cap W^{1,2}_\mathrm{{loc}}((-\infty ,0];L^2_{\sigma })\) to (D) such that
for all \(t<0.\) Here C is an absolute constant.
Remark 5
In [10, Lemma 2.7], Lemma 10 was proven in the case \(h\in BC((-\infty ,0]; L^{6/5}\cap L^2).\) In the same way as in [10], we easily see that this lemma holds even for the case \(h\in L^2_{loc}((-\infty ,0]; L^{6/5}\cap L^2).\)
In the rest of this section, we prove some properties of weak \(L^r\)-spaces \((L^r_w).\) From now on, for simplicity, we use the following notations:
etc.
Lemma 11
Let \(\Omega \subset \mathbb R^3\) be a measurable set and \(1<r<\infty .\) If \(f\in L^r_w(\Omega )\) and a constant \(A> 0\) satisfy
then there exists a sequence \(\{j_n\}_{n=1}^{\infty }\) of natural numbers such that
where \(L=L(r)=\frac{1}{4}\left( \frac{2^r-1}{2}\right) ^{1/r}.\)
Proof
We use a proof by contradiction. Assume that there exists \(n_0\in {\mathbb {N}}\) such that
Let
Since
we have
Hence, for \(N>n_0,\)
which implies
Letting \(N_0(\ge n_0)\) be sufficiently large so that
by (2.14) we have
This implies
On the other hand, from (2.12) with \(j=N_0\) we obtain
This contradicts (2.16). Therefore, (2.13) cannot be true and we conclude that for each natural number n, there is a natural number \(j_n(\ge n)\) such that
which proves Lemma 11. \(\square \)
Lemma 12
Let \(\Omega (\subset \mathbb R^3)\) be a measurable set and \(U\in L^3_w(\Omega ).\) Assume that there exist a real number \(\omega >0,\) \(p_0\in {\mathbb {N}}\) and a sequence \(\{g_n\}\) such that
Then, there exists \(k_0\in {\mathbb {N}}\) such that
Here \(C_2: =2^{1/3}4/L(3) .\)
Proof
We use a proof by contradiction. Assume that
Then, from Lemma 11 we observe that there exists a sequence \(\{j_n\}\) of natural numbers such that
From (2.21) we obtain
which yields
On the other hand, by using (2.17) with n replaced by \(k+1,\) we have for all \(\tau >0\) and all \(k\ge p_0-1\)
Letting \(\tau =2^{-k-2},\) we obtain
Since \(j_n\ge p_0\) for all sufficiently large n, we have
Then from (2.23) and (2.24) we obtain
Since
by (2.25) we see
This and (2.20) yield
which contradicts (2.18). This proves Lemma 12. \(\square \)
Recall that, for any measurable set D, \(L^{3}_w(D)=L^{3,\infty }(D)\) and
where c and \(c'\) are absolute constants. Then, since \(2^{-n-2}(\mu \{2^{-n-2}<|g_{n+1}-U|<\lambda \})^{1/3} \le \Vert g_{n+1}-U\Vert _{L^3_w(\{|g_{n+1}-U|< \lambda \})}\) for all \(\lambda >2^{-n-2},\) Lemma 12 directly yields the following lemma with \(L^3_w\) replaced by \(L^{3,\infty }\):
Lemma 13
Let \(\Omega (\subset \mathbb R^3)\) be a measurable set and \(U\in L^{3,\infty }(\Omega ).\) Assume that there exist a real number \(\omega >0,\) \(p_0\in {\mathbb {N}}\) and a sequence \(\{g_n\}\) such that
Then, there exists \(k_0\in {\mathbb {N}}\) such that
where \(C_3\) is an absolute constant.
3 Proof of Theorem 1
In this section, we prove Theorem 1 by using a similar argument given in [10] and Lemma 13.
Proof of Theorem 1
Let \(\delta :=\frac{\min \{\epsilon _1,\delta _0\}}{6+3C_3},\) where \(\epsilon _1,\) \(\delta _0\) and \(C_3\) are the constants given in Lemmata 9, 10 and 13. By (1.5) and (1.7), there exists \(s_0\in (-\infty ,T)\) such that
Since \(\lim _{t\rightarrow -\infty }\frac{|t|^{1/2}}{|t-s_0|^{1/2}}=1,\) by (1.6) we have
Without loss of generality, we may assume \(T>0\) and \(s_0=0,\) i.e.,
Let \( \epsilon \in (0,1]\) be an arbitrary fixed number, which will be chosen suitably small later on,
Since \(w=u-v\in BC((-\infty ,0];L^{3,\infty }),\) there exists a sequence \(\{t_k\}\) such that
Indeed, since \(w\in BUC([-n-1,-n];L^{3,\infty }),\) for each \(n=0,1,2,\ldots \) there exist a number \(m=m(n)\) and a finite sequence \(\{t^n_i\}_{i=1}^m\subset [-n-1,-n]\) such that
Then, arranging all members of \(\{t^n_{i}\}_{n,i}\) in order from the largest, we have the sequence \(\{t_k\}\) satisfying (3.4). Then, letting
where \(1_{S}\) denotes the characteristic function of a set S, we have
Let \(\{D_k\}_{k=0}^{\infty }(\subset \Omega )\) be an arbitrary sequence of measurable sets with \(0<\mu (D_k)<\infty \) for \(k=0,1,2,\ldots ,\) which will be suitably defined later on. Using this sequence, we define \(\tilde{D}(t)\subset \Omega \) for each \(t\le 0\) as follows:
(Step 1) We will first show that
for all \(j=1,2,\ldots ,\) where \(C_0=C_0(\Omega )\) is a constant depending only on \(\Omega .\) Note that the functions \(\mu (\tilde{D}(t))\) and \(\Vert w(\tau )\Vert _{L^{3,\infty }(\tilde{D}(\tau ))}^2\) are continuous on \((t_{k+1},t_k]\) for each \(k\in {\mathbb {N}} \cup \{0\}\) and hence these functions are piecewise continuous on \([-j,0],\) so that both sides of (3.8) are well-defined. Also note that, as we will show below, (3.8) holds for an arbitrary choice of measurable sets \(\{D_k\}\) with \(0<\mu (D_k)<\infty \) for \(k=0,1,2\ldots .\)
By (2.1) and the Hölder inequality, it holds that, for all measurable sets \(F\subset \Omega ,\)
where \(C^*\) is an absolute constant. Hence, for each \(k\in {\mathbb {N}}\cup \{0\},\) there exists a measurable set \(E_k\subset \tilde{D}(t_k)(=D_k)\) such that
Using the sequence \(\{E_k\},\) we define a set \(\tilde{E}(t)\subset \Omega \) for each \(t\le 0\) as follows:
Then \(\tilde{E}(t)\subset \tilde{D}(t)\) for each \(t\le 0.\) Let, for \(\tau \le 0,\)
Note that for \(t_{k+1}<t\le t_k,\) \(h(t)=h(t_k)=w(t_k)\mu (E_k)^{-\frac{1}{3}}1_{E_k}.\) Also note that
where we used the interpolation inequality:
for \(1<q<\infty \) and \(1\le r\le \infty .\) Similarly, by Lemma 5 we have
Since \(0<\mu (\tilde{E}(t))=\mu (E_k)<\infty \) for all \(t_{k+1}< t\le t_k\) and all \(k=0,1,\ldots ,\) by (3.13) and (3.14) we see \(h\in L^{\infty }_{loc}((-\infty ,0]; L^2\cap L^{6/5,1}).\)
Furthermore, by the interpolation inequality, (3.13) and (3.14), we have
Since for \(t_{k+1}<t\le t_{k}\) it holds that
from (3.4), (3.10), (3.12), (3.15) and \(L^{3,\infty }=(L^{3/2,1})^*,\) we obtain for \(t_{k+1}<t\le t_k\)
Since the above estimate holds for all \(t\in (t_{k+1}, t_{k}]\) and all \(k=0,1,2,\ldots ,\) we have for all \(j\in {\mathbb {N}}\)
Hence in order to show (3.8), it suffices to show
Let \(j\in {\mathbb {N}}\) be fixed. For \(-3j<t<0,\) let
Then, it holds that
for all \(\phi \in C^{\infty }_{0,\sigma }.\) Since \(C^{\infty }_{0,\sigma }\) is dense in \(L^{3/2,1}_{\sigma },\) from Lemma 7, we see that the above equality holds for all \(\phi \in L^{3/2,1}_{\sigma }.\) By the duality \(L^{3/2,\infty }=(L^{3,1})^*,\) Lemma 5 and Lemma 6, we have for \(\varphi \in L^{3/2,1}\cap L^2\)
which implies \(w_1(t)\in L^2\) and
Furthermore we observe that \(w_1\) satisfies
for all \(\psi \in W^{1,2}(-j,0;L^2_{\sigma })\cap L^2(-j,0; D(A_2)).\) For the detail of the proof of (3.21), see [10, Proof of (3.8)].
In order to show (3.17), since \(w=w_0+w_1,\) we decompose into two terms as follows:
We estimate \(I_0\) and \(I_1\) separately. By \(L^{6,\infty }=(L^{6/5,1})^*,\) Lemma 6 and (3.13), we obtain
Let \(\Psi \) be the solution to (D) with right-hand side \(h(x,\tau )= \tilde{w}(x,\tau ) \mu (\tilde{E}(\tau ))^{-\frac{1}{3}}1_{\tilde{E}(\tau )}(x) \) and initial value \(\Psi (0)=0,\) cf. Lemma 10. Note that \(h\in L^{\infty }_{loc}((-\infty , 0]; L^{6/5}\cap L^{2}).\) Then,
Since \(\Psi (0)=0\) and since \(w_1\in L^2(-j,0;L^2_{\sigma })\) implies that \(\int _{-j}^0(w_1,\nabla \pi )d\tau =0,\) by (3.21) we observe that
By (2.11), (3.13) and (3.20), we have
Furthermore, by Lemmata 5 and 6, (2.11), (3.13) and the duality \(L^{6,2}=(L^{6/5,2})^*,\) we have
Similarly, we observe that
Hence, we obtain \(|I_1|=|J_0+J_1+J_2|\le \frac{CK^3}{j^{3/4}}\Big \{\int _{-j}^0\mu (\tilde{D}(\tau ))^{1/3} \,d\tau \Big \}^{1/2}\) so that by (3.22)
which is the desired estimate (3.17). Thus from (3.16) and (3.17) we get (3.8).
(Step 2) Here we will show \( \liminf _{t\rightarrow -\infty }\Vert w(t)\Vert _{L^{3,\infty }(\Omega )}<\epsilon _1.\) Let us define \(\{D_k\}.\) Let
\(y_0\in \Omega \) be a fixed arbitrary point and
Note that \(\mu (D_k)\ge \mu (D_k^2)>0.\) Then, since \(\mu (\{x\in \Omega \ ; \ |f(x)|>s\})\le Cs^{-3}\Vert f\Vert _{3,\infty }^3,\) we have
Recall \(\tilde{D}(t)=D_k\) for \(t_{k+1}<t\le t_{k}.\) Since \(|t_{k}|\le |t|\) for \(t_{k+1}<t\le t_{k},\) it holds that
for all \(t\le 0.\) Then, (3.8) implies
where \(\tilde{C}_0\) is a constant depending only on \(\Omega .\)
Now we choose \(\epsilon \) such that
Then, by (3.26), for all \(j\in {\mathbb {N}}\) it holds that
Thus, it is straightforward to see that there exists a sequence \(\{s_j\}\) such that
for all \(j\in {\mathbb {N}}.\)
Next, we will estimate \( \Vert w(s_j)\Vert _{L^{3,\infty }(\Omega \backslash \tilde{D}(s_j))} .\) Since \(t_{k}>t_{k+1}\rightarrow -\infty ,\) we can choose a sequence \(\{k(j)\}_{j=1}^{\infty }\subset {\mathbb {N}}\) such that
By (3.7), we see \(\tilde{D}(s_j)=D_{k(j)}.\) Thus, by \(\sup _{s\le 0}\Vert u(s)\Vert _{3,\infty }\) \(\le \) \(\delta ,\)
since (3.4) and (3.30) imply \(\Vert w(s_j)-w(t_{k(j)})\Vert _{L^{3,\infty }(\Omega \backslash D_{k(j)})}\le \epsilon \le \delta .\)
Now let
Note that, since \(K>\delta ,\) we see \((2\tilde{C}_0+1+\delta )K>2\tilde{C}_0K+\delta +\delta ^2\) and hence by (3.27)
Then (3.3),(3.23) and (3.33) imply
Thus, by (3.2) and (3.31) yield
We turn to estimate the last term of (3.34) i.e. \(\limsup _{j\rightarrow \infty }\Vert V\Vert _{L^{3,\infty }(\Omega \backslash D_{k(j)})}\) by using Lemma 13, (3.2) and (3.29). Choose \(m\in {\mathbb {N}}\) such that
Since \(|t_{k(j)}|+1\ge |t_{k(j)}|+|t_{k(j)+1}-t_{k(j)}|\ge |t_{k(j)+1}|\ge |s_j|,\)
Since \(j\le |s_j|\le 2j\) for all \(j\in {\mathbb {N}},\) we have \(2^{i}\le |s_{2^{i}}|\le 2^{i+1}\) for all \(i\in {\mathbb {N}}.\) Let \(\{\tilde{s}_n\}\) be the following subsequence of \(\{s_j\}\):
then
Thus,
Set
Since (3.29) and (3.38) implies
for all \(n\in {\mathbb {N}},\) we see that \(g_n(=v(\tilde{s}_n))\) and \(\omega \) satisfy (2.27).
Since (3.3), (3.33),(3.35) and (3.37) imply
by (3.2) we see that \(g_n(=v(\tilde{s}_n))\) and \(U(=V)\) satisfy (2.28) with \(\omega =3\delta .\)
Thus, since \(a_{k(j)}\searrow 0\) as \(j\rightarrow \infty ,\) from Lemma 13 we observe
Here \(C_3\) is the constant given in Lemma 13.
Therefore, from (3.29), (3.34) and (3.39) we obtain
which implies
Since \(\delta =\frac{\min \{\epsilon _1,\delta _0\}}{6+3C_3},\) we conclude that
which with the help of Lemma 9 yields
This proves Theorem 1. \(\square \)
4 Proof of Theorem 3
Proof
In this section we will prove Theorem 3, by using the same methods as in the proof of Theorem 1. Let \(\delta \) be the same number given in the proof of Theorem 1. Without loss of generality, we may assume \(T>0\) and
Let \(\epsilon \in (0,1]\) be an arbitrary fixed number, K be the same number given in the proof of Theorem 1, i.e. \(K:=\sup _{t\le 0}\Vert v(t)\Vert _{3,\infty }+1+\delta ,\) and \(\{t_k\}\) be a sequence such that
Let \(y_0\in \Omega \) be a fixed arbitrary point and
Then we define a sequence \(\{D_{k}\}\) of measurable subsets of \(\Omega \) as follows.
and also define \(\tilde{D}(t)\) in the same way as in the proof of Theorem 1:
Note that, as we have proven in (Step 1) of the proof of Theorem 1, (3.8) holds. In the same way as in the proofs of (3.25)–(3.26), we see that
where the constants C(N) and \( \tilde{C}_0(N)\) are depending only on N and \(\Omega .\) Now we choose \(\epsilon \) such that
Then, in the same way as in (3.29)–(3.30), there exist sequences \(\{k(j)\}\subset {\mathbb {N}}\) and \(\{s_j\}\) such that
for all \(j\in {\mathbb {N}}.\) Since
we can choose subsequences \(\{s_{j(n)}\}_{n=1}^{\infty }\) and \(\{t_{k(j(n))}\}_{n=1}^{\infty }\) of \(\{s_j\}\) and \(\{t_{k(j)}\}\) respectively and find a suitable \(l_0\in \{1,2,\ldots , N\}\) such that
For simplicity, we denote \(\{k(j(n))\}\) by \(\{k(n)\}.\) Note \(t_{k(n)+1}<s_{j(n)}\le t_{k(n)},\) \(t_{k(n)}\rightarrow -\infty \) and \(b_{k(n)}\rightarrow 0\) as \(n \rightarrow \infty .\) Since
and since
for sufficiently large n, by (4.2) we have
Set
Then, by (4.1), (4.3), (4.5), (4.6) and (4.8) we have
Since \(g_n-U=v(s_{j(n)})-V(s_{j(n)}),\) by (4.2) and (4.12) we see that \(\{ g_{n}\}\) and \(U(=V_{l_0})\) satisfy (2.27) and (2.28). Thus, by Lemma 13 we have
Hence, from (4.6), (4.11) and (4.13) we obtain
Therefore, since \(\delta =\frac{\min \{\epsilon _1,\delta _0\}}{6+4C_3},\) we conclude that
which with the help of Lemma 9 proves Theorem 3. \(\square \)
Data availability
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The author would like to thank the anonymous referees for their valuable comments. This work was supported by JSPS KAKENHI Grant no. 21K03323.
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Appendix
Appendix
Here we will show that (1.12)–(1.14) guarantee \(\frac{1}{\sqrt{-t}}v(\frac{x}{\sqrt{-t}})\in BC((-\infty ,0);\tilde{L}^{3,\infty }_{\sigma }(\mathbb R^3_{+})).\) Let
for \(t<0\) and let
Since \(\Vert S_t v\Vert _{3,\infty }=\Vert v\Vert _{3,\infty }\) and \(S_tv\in \tilde{L}^{3,\infty }(\mathbb R^3_+)\) for \(v\in \tilde{L}^{3,\infty }(\mathbb R^3_+)\) and for \(t<0,\) it suffices to show the following lemma:
Lemma 14
Let functions v, R, V, Q satisfy
Then
Proof
(Case 1) We first consider the case \(v\in L^{3,\infty }(\mathbb R^3_+)\cap L^{\infty }(\mathbb R^3_+).\) Since \(S_tV=V,\) we have
Let \(\epsilon >0\) be an arbitrary fixed number and let \(r<q<3.\) Since \(R\in L^{3,\infty }(\mathbb R^3_+)\cap L^{r,\infty }(\mathbb R^3_+),\) we see that \(R\in L^{q}(\mathbb R^3_+)\) and hence there exists a function \(\psi _{\epsilon }\in C_0^{\infty }(\mathbb R^3_+)\) such that
It is straightforward to see that \(S_t\psi _{\epsilon }\in C((-\infty ,0); L^q(\mathbb R^3_+)).\) Then, since
we have \(\limsup _{t_1\rightarrow t_2}\Vert S_{t_1}R-S_{t_2}R\Vert _{q} \le 2|t_2|^{\frac{1}{2}(\frac{3}{q}-1)}\epsilon \) and, letting \(\epsilon \rightarrow 0,\)
for all \(t_2<0.\) Since \(\Vert S_{t_1}v-S_{t_2}v\Vert _{\infty }\le (\frac{1}{|t_1|^{1/2}}+\frac{1}{|t_2|^{1/2}})\Vert v\Vert _{\infty },\) by the interpolation, (A.4) and (A.5), we have
for all \(t_2<0.\) Therefore, (A.3) holds.
(Case 2) Next we consider the case \(v\in \tilde{L}^{3,\infty }(\mathbb R^3_+).\) In this case, it holds that
Indeed, by (A.1), there exists a sequence \(\{\phi _k\}\) such that
Then
which implies \(\limsup _{n\rightarrow \infty }\Vert v1_{\{|v|>n\}}\Vert _{3,\infty }\le 1/k\) for all \(k\in {\mathbb {N}}\) and hence (A.7). Now let
Clearly,
Since
Then,
Since \(v_n-R_n=v-R=V,\) we have
Hence, as demonstrated in (Case 1), (A.9), (A.10) and (A.11) imply
Thus,
for all \(t_2<0.\) Therefore, letting \(n\rightarrow \infty ,\) from (A.7) we conclude that \(\lim _{t_1\rightarrow t_2}\Vert S_{t_1}v-S_{t_2}v\Vert _{3,\infty }=0\) and (A.3) holds. \(\square \)
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Taniuchi, Y. On uniqueness of mild \(L^{3,\infty }\)-solutions on the whole time axis to the Navier–Stokes equations in unbounded domains. Math. Ann. 389, 2561–2594 (2024). https://doi.org/10.1007/s00208-023-02702-x
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DOI: https://doi.org/10.1007/s00208-023-02702-x