The Navier–Stokes equations on the plane with time-dependent external forces

We are concerned with the non-stationary Navier–Stokes equations on the whole plane with external forces which are non-decaying in time, and give a sufficient condition on the external forces for the existence of a solution which exists for whole time. Typical examples are time-periodic solutions and solutions almost periodic in time. The stability under perturbation is also verified.


Introduction
We are concerned with the non-stationary Navier-Stokes equations on the whole plane ∂u ∂t (x, t) − Δ x u(x, t) + u(x, t) · ∇ u(x, t) + ∇π(x, t) = f (x, t), (0.1) ∇ · u(x, t) = 0 ( 0 . 2 ) for x ∈ R 2 and t ∈ R, where u(x, t) = u 1 (x, t), u 2 (x, t) is a vector-valued unknown function standing for the velocity, π(x, t) is a scalar-valued function standing for the pressure, and f (x, t) = f 1 (x, t), f 2 (x, t) is a given function standing for the external force.
More precisely, we consider two cases. One is to consider the system (0.1)-(0.2) for (x, t) ∈ R 2 × R; namely, we consider the external force is defined for all t ∈ R. Typical examples are time-periodic functions or almost periodic functions in time. The other one is the Cauchy problem; namely, we consider (0.1)-(0.2) for (x, t) ∈ R 2 × (0, ∞), together with the initial condition with a function ∇ · u 0 (x) = 0. In particular, we are concerned with the existence and uniqueness in the small, together with the stability under small initial perturbations, for small external force (and initial datum) in appropriate classes. Throughout this paper we assume that f (x, t) is given by the potential functions F j,k (x, t) with the formula Maremonti [18,19] first considered problems of this type on unbounded domains. Then Salvi [25], Kozono and Nakao [14], Maremonti and Padula [20], the author [27], Galdi and Sohr [10], Kubo [15], Galdi and Silvestre [8,9], Crispo and Maremonti [3], Farwig and Okabe [5], Silvestre [26], Iwabuchi and Takada [11], Nguyen [23], Kozono et al. [13], Kyed [16], Lemarié-Rieusset [17], Farwig et al. [4] considered similar problems, mostly for time-periodic solutions. However, their results are limited to the case where the space dimension is larger than or equal to three. On the other hand, Kobayashi [12] considered the two-dimensional case, but his solutions correspond to weak solutions and the uniqueness is not known. As far as the author knows, only Galdi [6,7] obtained the unique existence for two-dimensional case, but the method relies on the periodicity. The purpose of this paper is to solve the problems above on the whole plane R 2 in the class of critically decreasing solutions or supercritically decreasing functions introduced by the author in [28,29]. Then we obtain the unique existence of small solutions, together with the continuous dependence on the datum, in these classes under appropriate assumptions on F(x, t). Our approach does not rely on the periodicity, and hence we can consider almost periodic solutions in time, and more general solutions. We also obtain the stability of this solution by showing the global solvability and show the asymptotic behavior of the Cauchy problems with small initial disturbance.
Since our problems is a generalization of the stationary problems with solutions in the classes above with time-independent external forces, we need to impose some symmetry condition, such as where x ⊥ = (−x 2 , x 1 ). This condition, introduced in the author [29] for the unique existence of small stationary solution with stationary external force, implies that the solution is equivariant under the action of the cyclic group C 4 = {id, σ, σ 2 , σ 3 } of order 4, where σ (x) = x ⊥ . The viewpoint above is first introduced by Brandolese [2], which is concerned with sharp decay rate of the solution of the Cauchy problems.
In the critical case u(x) decays like 1/|x|. Hence, in order to treat the nonlinear terms u 1 (x)u 2 (x) and u 1 (x) 2 − u 2 (x) 2 , it seems natural to introduce Hardy space or some modification, as in Miyakawa [21,22] and Okabe and Tsutsui [24]. However, in order to apply the fixed point argument, we need to keep symmetry, and hence we cannot rely merely on the theory of Hardy spaces. Hence we employ weighted L p -space and show the weighted estimates of the Stokes semigroup instead, by regarding the Gauss kernel as a singular integral and applying its boundedness on weighted spaces. Moreover, particularly in the treatment of critically decreasing functions, we need to consider improper integrals which is not Bochner integrable. We get around this difficulty by showing the convergence in the weak- * topology, which need to consider the dual problem and our solutions are obtained as very weak solutions, as is done in [27]. However, it is not clear whether the solutions thus obtained are strong solutions or not. Hence we start with the study of strong solutions of linear problems. Moreover, to employ this argument we need to consider the Lorentz spaces in place of the standard L p -spaces.
In order to state the assumption on F(x, t), we introduce the functions and impose the conditions t) for k = 1, 2 and every x ∈ R 2 , t ∈ R (0.5) and G 3 [F](x ⊥ , t) = G 3 [F](x, t) for every x ∈ R 2 , t ∈ R. (0.6) The outline of this paper is as follows. In Sect. 1 we rewrite the system, fix notations and state the results, Next we reduce the problems into integral equations in Sect. 2. Then we introduce linear estimate in Sect. 3, Finally in Sect. 4 we prove the theorems in Sect. 2.

Results
We first rewrite the system in order to state the assumptions. Since ∇ · u(x, t) = 0, we can write Moreover, we have the equalities Substituting these equalities into (1.1) we have . Thus our problems are to find the solution (1.2)- In order to removeπ(x, t), we introduce the Helmholtz projection P defined by the matrix (δ j,k + R j R k ), where R j denotes the Riesz operator. Then we can rewrite (1. In order to treat the critical case, we introduce the Lorentz space. Suppose that K is a Lebesgue measurable subset of R n and f (x) is a Lebesgue measurable function on K . Then we put For detailed properties of the Lorentz spaces and real interpolation, see Bergh and Löfström [1] for example. In the following we list the properties needed in this paper. For p ∈ [1, ∞), the space L p, p (K ) coincides with the standard Lebesgue space L p (K ) equipped with the norm f p . In general, the function f p,ρ does not satisfy the triangle inequality. However, well-defined for every t > 0, and the functional satisfies the triangle inequality and the estimate Hence L p,ρ (K ) becomes a Banach space with norm f * p,ρ . Further, let L p,∞− (K ) denote the closure of L p (K ) in L p,∞ (K ). Then, for p ∈ (1, ∞) and 1 ≤ ρ < ∞ or ρ = ∞, the space (1.5) Another important property is the real interpolation: Suppose that p 0 , p 1 ∈ (0, ∞), Moreover, we introduce the class of solenoidal vector fields. Let This space is well-defined independent of the choice of p 0 and p 1 up to equivalent norms.
We next introduce the class of potential functions and solutions. Suppose that 2 < p < 4 and that 1 − 2/ p ≤ a < 1/2. We consider external forces F(x, t) satisfying · 2a F(·, t) ∈ BC I , L p/2,ρ (R 2 ) 4 , and consider the solution u(x, t) satisfying Since · −a ∈ L 2/a,∞ ∩ L ∞ , it follows that the function u(x, t) in the class above satisfies BC I , We thus see that u(·, t) ∈ BC I , L 2,1 (R 2 ) Then our main theorem on the whole time is the following.
Then there exist a positive constant ε 0 and a strictly monotoneincreasing function ω(s) on [0, ε 0 ] such that ω(0) = 0 and that the following assertion holds.
Then there exists a solution u(·, t) of the equation (1.4) in R in the sense of distributions and the symmetry condition (C4E) for every t ∈ R such that This solution is the unique solution of (1.4) in R satisfying the symmetry above and the condition (1.7) with the estimate sup t∈R · a u(·, t) p,ρ < ω(ε 0 ). (1.9) Furthermore, for every ε ∈ (0, ε 0 ), the mapping · 2a G[F](·, t) → · a u(·, t) is uniformly continuous from the closed ball in BC R, L p/2,ρ (R 2 ) 3 with center 0 and radius ε to In order to state an immediate consequence of Theorem 1.1, we recall the definition of almost periodic function. A mapping f from R to a metric space X is called almost periodic if, for every ε > 0, there exists a positive number L such that, for every t 0 there exists a number is also periodic with respect to t in L p,ρ (R 2 ) 2 .
We also have the result on the Cauchy problem in the same function space, which implies the stability of the solutions constructed in Theorem 2.3 under small perturbation. In order to state the theorem, let X denote the direct product L p,ρ ( Then our result is the following. Then there exists a solution u(·, t) of (1.4) such that the symmetry condition (C4E) holds for every t > 0 and that with the estimate sup t>0 · a u(·, t) p,ρ ≤ ω(ε). (1.13) Moreover, if · a u 0 (·) ∈ L p,ρ (R 2 ) 2 as well, wherẽ (1.14) This solution is the unique solution of (1.4) in (0, ∞) satisfying (1.11), the symmetry above and the condition (1.12) with the estimate Furthermore, for every ε ∈ (0, ε 0 ), the mapping · a u 0 (·), · 2a G[F](·, t) → · a u(·, t) is uniformly continuous from the closed ball in X with center 0 and radius ε to Observe that, if the condition · a u 0 (·) ∈ L p,∞ (R 2 ) 2 \ L p,∞− (R 2 ) 2 holds, then the convergence · a u(·, t) → · a u 0 (·) in L p,∞ (R 2 ) 2 as t → +0 does not hold in general, since the space of test functions is not dense in the space u(x) x a u(x) ∈ L p,∞ (R 2 ) 2 .

Reduction to the integral equation
In order to prove the main results, we rewrite (1.4) into an integral equation. Put Then we can formally rewrite (1.4) and (1.4) respectively. We first introduce a proposition concerning the estimate of heat kernel on weighted L pspaces.
Proposition 2.1 Suppose that 1 < p < ∞ and that either 1 ≤ q < ∞, q = ∞− or q = ∞. If a satisfies −n/ p < a < n(1 − 1/ p), then we have the following assertions: (i) There exists a positive constant C a, p such that the estimate · a exp(tΔ)u p,q ≤ C a, p · a u p,q holds for every t ≥ 0 and every u(x) such that · a a ∈ L p,q (R 2 ).
Proof We first show Assertion (i) for q = p. In this case −n < ap < n( p − 1), and hence · ap is an A p weight. Then weighted boundedness of the Fourier multiplier implies the estimate from which the conclusion for q = p follows at once. We next consider the general case of Assertion (i). Choose p 0 , p 1 such that p 0 < p < p 1 and that −n/ p j < a < n(1 − 1/ p j ) for j = 0, 1. Then Assertion (i) for q j = p j yields the estimate · a exp(tΔ)u p j ≤ C a, p j · a u p j for j = 0, 1. This fact implies that the mapping v → · a exp(tΔ) · −a v is bounded from L p j (R 2 ) into itself with norm independent of t ≥ 0. Hence, by real interpolation, we see that the mapping above is bounded from L p,q (R 2 ) into itself with norm independent of t ≥ 0. This fact implies · a exp(tΔ)u p,q ≤ C a, p · a u p,q with a positive constant C a, p independent of q. This yields the Assertion (i) for general case. We turn to the Assertion (ii). From the assumption we see that, for every fixed where Then Assertion (i) implies I 1 ≤ C a, p · a u − ϕ p,q . It follows that On the other hand, we have We Since ε > 0 is arbitrary, this completes the proof of Assertion (ii).
Then the following proposition, which will be proved at the end of this section, justifies the transformation above rigorously. Although some of the assumptions are unnecessary in the proof, the statement is designed so that it is directly applicable in later sections Then we have the following assertions: Then the function u(x, t) defined by the formula in the sense of distributions. Moreover, if u 0 ∈ L r ,ρ (R 2 ) 2 as well, whereρ is given by (1.14), then we have is a solution of (2.6)-(2.7), then u(x, t) is given by (2.5).
2 . and that, for every t ∈ R, the limit exists in the weak- * topology of L r ,ρ (R 2 ) 2 . Then the limit u(·, t) belongs to

and is a solution of the problem
is a solution of (2.10), then (2.9) exists in the weak- * topology for every t ∈ R, and the limit coincides with u(·, t).
In order to solve (2.1) and (2.2), consider the mappings and respectively, and assume the conditions (0.5) and (0.6), and u(x, t) satisfies (C4E). Then we have for I = R, together with the existence of the limit with respect to the weak- * topology of L p/2,ρ (R 2 ) 2 and the continuous dependence of with the estimate (1.6). We moreover assume that G 1 [F](·, t), G 2 [F](·, t) satisfy (0.5) and that G 3 [F](·, t) satisfies (0.6) for every t ∈ R.
Then there exists a solution u(·, t) of (2.1) such that the symmetry condition (C4E) holds for every t ∈ R and that (1.7) with the estimate (1.8). This solution is the unique solution of (2.1) satisfying the symmetry above and the condition (1.7) with the estimate (1.9).
Furthermore, for every ε ∈ (0, ε 0 ), the mapping · 2a G[F](·, t) → · a u(·, t) is uniformly continuous from the closed ball in BC R, L p/2,ρ (R 2 ) 3 with center 0 and radius Theorem 2.4 Let p, a and ρ be the same as in Theorem 2.3. Suppose that u 0 (x) satisfies (C4E) and ∇ · u 0 (x) = 0, and that Then there exists a solution u(·, t) of (2.2) such that the symmetry condition (C4E) holds for every t > 0 and that (1.12) with the estimate (1.13) holds. This solution is the unique solution of (2.2) with satisfying the symmetry above and the condition (1.12) with the estimate (1.15). Furthermore, for every ε ∈ (0, ε 0 ), the mapping is uniformly continuous from the closed ball in X with center 0 and radius ε to In the same way as in (2.2), we can formally rewrite (1.17) into Moreover, if we put r = p and q = 2 p/(2 + p), we see that implies the assumptions of Proposition 2.2, (i). Hence Theorem 1.4 reduces to the following theorem.
This solution is the unique solution of (2.12) satisfying (1.19).
Furthermore, for every δ ∈ (0, δ 0 ), the mapping w 0 (x) → u(·, t), t 1/2−1/ p u(·, t) is uniformly continuous from the closed ball in L 2,∞ σ (R 2 ) with center 0 and radius δ to Proof of Proposition 2. 2 We first prove Assertion (i). Suppose that u(x, t) is a solution of (2.5) in (s, ∞). We first prove the continuity. Fix t 1 > s arbitrarily. Then we have It follows that exp (t 1 − s)Δ u 0 belongs to the closure of D(Δ) as a closable operator in (2.14) Next we prove the continuity of the term Then we have where Then we have in the same way. Hence these two terms can be taken arbitrarily small if we take δ > 0 sufficiently small. Next we have Once t 0 and δ are fixed, the term I 1 r ,ρ can be taken arbitrarily small if This completes the proof of the required continuity. We next show that u(x, t) satisfies (2.6)-(2.7). Fix t 0 , t 1 such that s < t 1 < t 0 , and assume that where In the sequel the limits are taken either t 2 = t 0 , t 3 → t 0 + 0 or t 2 → t 0 − 0 and t 3 = t 0 . We first consider J 1 . We can write (2.17) On the other hand, we have We next consider J 2 . We can write We now split J 2 = J 2,1 + J 2,2 , where and Then we have Substituting this equality into (2.20) and (2.21) we obtain and Substituting this into (2.22) we conclude that Next, (2.23) implies On the other hand, we have We finally rewrite J 3 = J 3,1 + J 3,2 , where Then we can estimate Since t 1 < t 0 can be taken arbitrarily close to t 0 , we see that in the sense of distributions. Since t 0 > s is arbitrary, we conclude that u(x, t) is a solution of (2.6) in the sense of distributions in (s, ∞). The condition (2.7) follows from (2.14) and the fact 2 as well, the fact (2.8) follows from (2.13) and (2.14).
Next, suppose that u(x, t) is the solution of (2.6)-(2.7). Put Then v(x, t) is also the solution of (2.6)-(2.7). Putting w( = 0 for all t > s and w(·, t) → 0 as t → s + 0. It follows that w(x, t) ≡ 0, and hence u(x, t) = v(x, t) is the solution of (2.5). We turn to the proof of Assertion (ii). Suppose that u(t, x) is defined as the limit (2.9). Fix s ∈ R. Then, for t > s, we have (2.28) This implies that u(x, t) is the solution of (2.5) with u 0 (x) = u(x, s) ∈ L r ,ρ (R 2 ) 2 . Hence Assertion (i) implies that u(x, t) is a solution of (2.6). Since s ∈ R is arbitrary, u(x, t) is a solution of (2.10).
On the other hand, suppose that u(x, t) is a solution of (2.10). Then Assertion (i) implies that the equality (2.28) holds for every s, t ∈ R such that s < t. Fix t ∈ R. Then we have This implies that exp (t − s)Δ ϕ r /(r −1),ρ → 0 as s → −∞. On the other hand, the assumption implies that u(·, s) r ,ρ is bounded. Substituting these facts into (2.29) we see that exp (t − s)Δ u(·, s), ϕ → 0 as s → −∞. This implies the required weak- * convergence. From this fact and (2.28) we see that converges * -weakly in L r ,ρ (R 2 ) 2 to u(·, t) as s → −∞. This implies that u(x, t) is the limit (2.9) in the sense of weak- * topology.

Linear estimates
We first have the following proposition on the weighted Lorentz boundedness.
Next, the integral I [G](x, s, t) satisfies the following propositions.
We next show the result on symmetry. Then we have the following proposition.

Proposition 3.4 In addition to the assumptions of Proposition 3.2, we assume that the functions G k (x, t) satisfy the equalities (0.5) and (0.6). Then we have
for every x ∈ R 2 , t ∈ R and s > 0.
Proof We have the equality This completes the proof.
Corollary 3.7 Suppose that 0 < a < 1/2 and that 1 < q < 4. Suppose also that either 1 ≤ ρ < ∞ or ρ = ∞−. Then there exists a positive constant C such that the following assertions hold.
(i) We have the estimate (ii) Put s = 2q/(2 − q). Then we can choose C so that we have the estimate Proof We first prove Assertion (i). Lemma 3.6 for k = 0, 1 yields the estimate for q ∈ (1.2). Assertion (i) follows from this estimate and real interpolation.
We turn to the proof of Assertion (ii). Lemma 3.6 for k = 1, 2 implies (1, 2). Assertion (ii) follows from this estimate and real interpolation.

Integrating (3.20) on [T , S], we conclude (3.19).
We next give a proposition to treat the critical case.
We now prove the main results in this section.
Theorem 3.10 Suppose that 0 < a < 1/2, 2 < p < 2/(1 − a) and 1 ≤ ρ ≤ ∞. Then there exists a positive constant C such that the following assumption holds. In addition to the assumption of Proposition 3.2, the functions G 1 (x, t), G 2 (x, t) satisfy (0.5) and the function G 3 (x, t) satisfies (0.6). Then the limit I 1 [G](x, t) exists in the sense that, for every t ∈ R, the function · a I 0 [G](·, t, T ) converges to · a I 1 [G](·, t) in the weak- * topology of L p,ρ (R 2 ) 2 . The limit I 1 [G](x, t) satisfies the symmetry property (C4E) for every t ∈ R, and x a I 1 [G](·, t) belongs to BC R, L p,ρ (R 2 ) 2 with the estimate Proof If we prove the convergence, the property (C4E) for I 0 [G](x, t, T ) yields the property (C4E) for I 1 [G](x, t). We next put We put D ± = {x ∈ R 2 | ±x 1 ≤ 0} and w(x, t) = x a I 0 [G](x, t, T ). Then, for every ψ(x) as above, we have Since w(x) satisfies (C4E), it follows that w(x) satisfies (C2E). This implies In the same way we have Then Proposition 3.2 with T = 1 implies that It follows that In order to estimate K 2 (t, T ), we put ϕ(x) = P x a ψ − (x). Then ϕ(x) satisfies (C2E). Furthermore, for every α ∈ (1, 2/a), the weight x −aα is an A α weight. It follows that Hence we have for α ∈ (1, 2/a) and every ρ by real interpolation.
On the other hand, we have the equality where P j,k (∇) is a homogeneous differential operator of order 1. Then, for every t ∈ R, Proposition 3.8 implies that for every S, T such that 1 ≤ T ≤ S < ∞. This estimate, together with (3.25), implies Since the the right-hand side tends to 0 as T → ∞, the limitK 2 (t) = lim T →∞ K 2 (t, T ) exists and satisfies the estimate Summing up this estimate and (3.24) we see that the limitK (t) = lim T →∞ K (t, T ) exists and satisfies the estimates with a positive constant C. It follows that Since t ∈ R is arbitrary, we see that This implies that · a I 0 [G](·, t, T ) converges in L p,ρ (R 2 ) 2 as T → ∞, and the conver- For the critical case p = 2/(1 − a) we have the following theorem. Proof We proceed in the same way as in the proof of Theorem 3.10. We can prove (3.24) in this case with q = 2/(1 + a), ρ = 1, p = 2/(1 − a) and ρ = ∞. On the other hand, we have for every S, T such that 1 ≤ T ≤ S < ∞. It follows that Since the right-hand side tends to 0 as T → ∞, the limitK 2 (t) = lim T →∞ K 2 (t, T ) exists and satisfies the estimate Applying Proposition 3.9 together with (3.25), we obtain Summing up this estimate and (3.24) we see that the limitK (t) = lim T →∞ K (t, T ) exists and satisfies the estimate Since ψ ∈ L 2/(1+a),1 (R 2 ) 2 is arbitrary, we conclude that · a I 0 [G](·, t, T ) converges to We finally show the continuity. Fix t 0 ∈ R, and let t 1 and t 2 be real numbers satisfying t 0 − 1/2 < t 1 ≤ t 0 ≤ t 2 < t 0 + 1/2 and t 1 < t 2 . Then we can write Then we have It follows that · a J 2 (·) p,ρ → 0 as t 1 → t 0 − 0 and t 2 → t 0 + 0.
Next, suppose that t ≥ 1. Then we have and

Proof of results
We first prove Theorem 2.4. Define the function space X by   ( j) ](·, t) ∈ X , and for u ( j) (x, t) ∈ X for j = 1, 2, we have the estimate with a positive constant C by virtue of Proposition 3.1 and Theorem 3.12.
Since H j [0, u](x, t) is a linear combinations of products of the components of u(x, t), we have Substituting this estimate into (4.1) we obtain the conclusion.
Suppose that · a u 0 (·) p,ρ + sup t∈R · 2a G[F](·, t) p/2,ρ ≤ ε < ε 0 and that u ∈ X satisfies u X ≤ ω(ε). Then Proposition 4.1 with u (2) ] = 0 and u (2) Hence the mapping T maps B into itself, where B denotes the closed ball in X with center 0 and radius ω(ε). Next, for u (1) , u (2) ∈ B, Proposition 4.1 with u is a contraction mapping from B into itself. Since X is complete, there uniquely exists a fixed point of T in B. Furthermore, for every R ∈ 0, ω(ε 0 ) , we can choose ε ∈ (0, ε 0 ) such that ω(ε) = R. This implies the uniqueness of the fixed point u such that u X < ω(ε 0 ). This is the required solution.
We can prove Theorem 2.3 in the same way as Theorem 2.4 with u 0 (x) = 0, by setting X = u(·, t) u(·, t) satisfies (C4E), · a u(·, t) ∈ BC R, L p,ρ (R 2 ) 2 equipped with the norm u X = sup t∈R · a u(·, t) p,ρ and making use of Theorem 3.10 or 3.11 in place of Theorem 3.12 and Proposition 3.1. We turn to the proof of Corollary 1.2. Suppose that F(x, t) is periodic with respect to t with period T . Then we have Then u(x, t + T ) is a solution corresponding to G[F](x, t + T ), and it satisfies the estimate sup t∈R · a u(·, t + T ) p,ρ = ω(ε ) < ω(ε 0 ).
Hence the estimate corresponding to (4.3) implies That is, for every ε > 0, there exists a positive number L such that, for every a ∈ R, there exists T ∈ [a, a + L] satisfying u(·, t + T ) − u(·, t) X < ε. Hence · a u(·, t) is almost periodic in L p,ρ (R 2 ) 2 . This completes the proof of Assertion (ii).
Proof of Proposition 4.2 By the standard L p -L q estimate and real interpolation we have exp(tΔ)w for r ∈ [2, p] with a constant C depending only on p ∈ (2, ∞). Next, put w(x, t) = w 1 (x, t) − w 2 (x, t). Then we have in the same way as (4.2). It follows that We thus conclude In order to estimate S 0 [w](t) p,∞ , we employ the duality argument. For ϕ(x) ∈ C ∞ 0,σ (R 2 ), we have the duality  The standard L p -L q estimate and real interpolation imply that Form (4.7), (4.8) and this estimate we see that From this estimate and (4.4), (4.6) we conclude that S[w 0 , w](·, t) 2,∞ + t 1/2−1/ p S[w 0 , w](·, t) p,∞ This completes the proof.

Conflict of interest
The author declares that he has no conflict of interest.
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