Sharp global well-posedness of the BBM equation in $L^p$ type Sobolev spaces

The global well-posedness of the BBM equation is established in $H^{s,p}(\textbf{R})$ with $s\geq \max\{0,\frac{1}{p}-\frac{1}{2}\}$ and $1\leq p<\infty$. Moreover, the well-posedness results are shown to be sharp in the sense that the solution map is no longer $C^2$ from $H^{s,p}(\textbf{R})$ to $C([0,T];H^{s,p}(\textbf{R}))$ for smaller $s$ or $p$. Finally, some growth bounds of global solutions in terms of time $T$ are proved.

1. Introduction. In this paper we are concerned with the Cauchy problem for the Benjamin-Bona-Mahony (BBM) equation where u : R × R → R is a real-valued function, u 0 is a given initial data. The BBM equation was studied in [4] for modeling the propagation of unidirectional, one-dimensional, small-amplitude long waves in nonlinear dispersive media. The well-posedness and ill-posedness of Eq. (1) are studied extensively in references [5,7,19,21,27]. In [5], Bona and Tzvetkov showed that the BBM equation is globally well-posed in H s (R) with s ≥ 0. The same result holds in periodic case, see [21,27]. The well-posedness is sharp in the sense that Eq. (1) is ill-posed in H s (R) if s < 0. In fact, Bona and Tzvetkov [5] also illustrated that the solution map u 0 → u(t) of Eq.(1) is not C 2 from H s (R) to C([0, T ]; H s (R)) for s < 0. Later, Panthee [19] improved this to that the solution map is not continuous at the origin from H s (R) to even D (R) at any fixed t > 0 small enough. Very recently, Carvajal and Panthee [7] considered the following generalized BBM equation where L is defined by Fourier transform Lu(ξ) = |ξ| α+1 u(ξ), α > 0. They proved that Eq. (2) is locally well-posed in H s (R) for some s depending on α and k. Moreover, Eq. (2) is ill-posed in H s (R) if s < max{0, 1 2 − α k } in the sense that the solution map is not C k+1 −differentiable at the origin from H s (R) to C([0, T ]; H s (R)) for any T > 0. In the case α = 1, some global well-posedness results of (2) with periodic boundary condition were obtained in [27].
(1) is conservative, namely As a consequence, one concludes that Eq. (1) is globally well-posed in H 1 (R). However, the conservation law is no longer valid if the solution is not regular enough, say the solution only belongs to H s,p (R) with s < 1 and p < 2. Here and below, the inhomogeneous Bessel potential spaces H s,p (R n ), 1 ≤ p < ∞, s ∈ R, are defined as the completion of Schwartz class with respect to the norm where Λ s is the fractional differential operator defined by the symbol (1 + |ξ| 2 ) s/2 . Thus, a natural question arises, is Eq. (1) globally well-posed in H s,p (R) for such s and p? We answer the question in the following theorem. The most interesting fact given by Theorem 1.1 is that Eq. (1) is globally wellposed in low regularity spaces, in particular it allows s < 1 and p < 2. A nice way to prove global well-posedness in low regularity spaces is the I−method, which is introduced in [9] and widely used to establish global well-posedness in rough spaces for many dispersive equations, see e.g. [16,25,28]. The I−method is very efficient when the phase space is L 2 −based Sobolev spaces H s . But we are working in L p type Sobolev spaces, it's not obvious how to use the I−method directly. To overcome this difficulty, we split and w solves Here χ k (x) is a smooth function such that χ = 1 if |x| ≤ k and χ = 0 if |x| ≥ 2k. To illustrate our ideas, we consider the simple case u 0 ∈ L p (R), p ∈ (2, ∞). On one hand, since v(0, ·) L p goes to zero as k goes to infinity, one can show that, for any T > 0, Eq. (4) has a unique solution v ∈ C([0, T ]; L p (R)) provided k is large enough (see Lemma 3.3). On the other hand, for any k > 0 fixed, w(0, x) belongs to L p (R) L 2 (R), then one can adapt the I− method to show that the solution w of Eq.(5) does not blow up in L p (R) norm at any T > 0. Combining these together gives the global well-posedness of Eq.(1) in L p (R)(p > 2). The more general case follows similarly, see Section 3 for details.
In section 4, we shall show that theorem 1.1 is sharp in the sense that Eq. (1) is ill-posed in H s,p for smaller s. Precisely, we have The definition of ill-posedness is given in section 4. The ill-posedness extends the corresponding results in [5,7,19] to an L p setting. The proof of Theorem 1.2 is essentially the same to disproving with the same s and p, where ϕ(D) is the Fourier multiplier with symbol ξ/(1 + ξ 2 ). The idea is to find a sequence {f n } of smooth functions such that f n H s,p goes to zero as n goes to infinity, and at the same time f 2 n → δ in distribution sense, where δ is the Dirac function. Then the right hand side of (6) goes to zero but the left hand side will never be trivial as n goes to infinity. In Section 4, we make the formal argument rigorous and adapt the approach to obtain the ill-posedness.
In Section 5, we are devoted to the growth of norm of solutions to Eq.(1). Precisely, we are interested in the estimates of u(T ) H s,p in terms of T as T grows. It turns out that the picture of our results in the case p ≤ 2 and p > 2 are different. We shall show a polynomial growth bound of u(T ) H s,p for all s ≥ 1 2]. Note that the range of s coincides to that of well-posedness, thus the study in the case p ≤ 2 is more or less complete. However, if u 0 ∈ L p (R) with p > 2, no growth bounds of u(T ) L p are available so far due to technical reasons in this paper. Nevertheless, an exponential growth bound of u(T ) L p can be proved if we assume further some decay of u 0 at infinity, say u 0 belongs to the weighted Lebesgue space L p ( x α dx), α > 0. Finally, we investigate the persistence of the BBM equation in weighted spaces L p ( x α dx).
Notations. We denote by s + (−) that a constant equals s plus (minus) a small enough positive number, A B means A ≤ CB for some absolute constant C, A ∼ B means A B and B A, and A B means A/B is very big, say A/B ≥ 1000.

Preliminaries.
2.1. Some facts from harmonic analysis. We start with the classical theory of Fourier multiplier. Let m be a measurable function on R n and define the operator where S (R n ) denotes the Schwartz space, f (ξ) = F f, and F −1 denote the Fourier transform and the inverse transform, respectively. We shall call m(ξ) the symbol of the operator m(D). Let 1 ≤ p ≤ ∞, we say m is an Fourier multiplier on L p (R n ) if m(D) can be extended to a bounded operator on L p (R n ), namely The set of all L p (R n ) multipliers is denoted by M p (R n ). We list some basic facts on M p (R n ) as follows: The following theorem is a powerful tool to determine if a bounded function is an L p (R n ) multiplier. From now on, we use the notation x = (1 + |x| 2 ) 1/2 . Theorem 2.1. Let k ∈ N, k > n 2 + 1, m ∈ C k (R n \{0}), and there exists ≥ 0 such that for any |α| ≤ k, Proof. The case = 0 is Mihlin multiplier theorem [24], while the case > 0 is a consequence of Bernstein theorem, see e.g. [10].
It follows from the Theorem 2.1 that ξ − , > 0, belongs to M 1 (R n ). It turns out that M 1 (R n ) 1<p<∞ M p (R n ). A typical example is the Riesz potential operator R j with symbol ξ j /|ξ|. In fact, R j is bounded on L p (R n ) for all 1 < p < ∞ but not on L 1 (R n ). It should be noted that R j is bounded from L 1 (R n ) to L 1,∞ (R n ), where the weak−L p space (0 < p < ∞) is defined as a set of all measurable functions f such that We now recall the Sobolev embedding theorem. Then holds for q ∈ [p, np n−sp ] if 1 < p < ∞, q ∈ [p, np n−sp ) if p = 1. (b) Let sp = n, 1 ≤ p < ∞. Then (7) holds for all q ∈ [p, ∞).
Proof. If sp < n, it follows from Corollary 6.1.6 and Theorem 6.2.4 of [12] that the operator is bounded. Then by Marcinkiewicz interpolation theorem n−sp ] is bounded. This proves (a). Since (b) is a consequence of (a), the proof is complete.
The following fractional Leibniz rule will be used in this paper, see [15] for 1 < p < ∞ and [3,13] for p = 1, respectively. Proposition 1. If s ≥ 0, 1 ≤ p < ∞, then for all f, g ∈ S (R n ) with p 1 , p 2 , q 1 , q 2 ∈ (1, ∞) satisfying 1 SHARP GLOBAL WELL-POSEDNESS OF THE BBM 5767 2.2. Bilinear estimates. Now we prove some a priori estimates that will be used in this paper.
provided that the right hand side is finite.
Proof. We first consider the case p ≥ 2. Write Since the symbol of ∂ x Λ 1 p −2 is iξ ξ 1 p −2 , by Theorem 2.1 it is an L 1 multiplier, and of course an L p multiplier. Moreover, by Sobolev embedding, Λ − 1 where we used Proposition 1 in the last step.
Now we turn to the case 1 ≤ p < 2. Write By Theorem 2.1 and 2.2 again, we find ∂ x Λ −1− is bounded on L p and Λ By Proposition 1 again, the above quantity is bounded by We have used the fact s ≥ 1 p − 1 2 in the first case of the last step. This completes the proof.
From the proof of Proposition 2, we have the following variants of bilinear estimates. These inequalities will be used to obtain the global well-posedness of (1) and growth of norms of solutions.
Now we introduce the I−operator used in this paper. Let 0 ≤ s < 1, N 1. Define the I−operator where Iφ(ξ) = m(ξ) φ(ξ), m(ξ) is a positive smooth even function satisfying It's easy to show that the following assertions hold: provided that the right hand side is finite, the implicit constant is independent of N .
Proof. By a limit process, it suffices to prove the proposition for f, g ∈ S (R). We first observe that the inequality is equivalent to By Parseval identity and duality, it suffices to show . In order to estimate A, we write where χ is a smooth even function such that

by properties of Fourier transform and convolution
where * denotes the convolution operator, (·, ·) the inner product of L 2 , ϕ(x) = ϕ(−x). By Hölder inequality, we have with 1 p + 1 2 + 1 q = 1. Since q > 2, it follows from Hausdorff-Young's inequality that holds for 1 Thanks to (11)-(13), we find that the contribution of A 1 is bounded by the right hand side of (9). For A 2 , similar to A 1 , we have with the same q as in (11). Since the support Moreover, it follows from Hausdorff-Young's inequality again that (16) Note that N > 1, it follows from (14)-(16) that For A 3 , using the formula similar to (10), by Hölder inequality where p 1 , p 2 are given by , and by Theorem 2.1 we know ( ξ |ξ| ) 1−s (1 − χ(ξ)) ∈ M p1 , then using the embedding H 1−s → L p1 we obtain Combining this and Since the proof of the case 1 2 < s < 1 is similar, we give only a sketch of it. In fact, and H s → L p2 , then we obtain (18).
Finally, the desired conclusion follows from the estimates of A 1 , A 2 , A 3 .

Corollary 1. It holds that
provided that the right hand side is finite, the implicit constant is independent of N .
Proof. It follows from Proposition 3 in the case p = 2 that Moreover, in light of (8), we find Thus the conclusion follows. 3. Well-posedness for BBM.
It's convenient to rewrite (19) in an integral form as Now we can state the main result in this subsection.
Proof. Let Γ be the map defined by (25), namely The strategy is to prove Γ has a fixed point in the ball where C is the implicit constant of Lemma 3.1, and X T is the space of bounded functions on [0, T ] with values in H s,p equipped with norm Let u, v ∈ B, it follows from Lemma 3.1 and Proposition 2 that Thus, Γ is a contraction on B if This proves the existence of solution in B. The solution, thanks to (19), satisfies further that u t X T < ∞. Thus the solution is continuous from [0, T ] to H s,p (R). Finally, the analytic property of the solution map Ξ(t) follows from a standard argument, see e.g. [5].

3.2.
Global well-posedness. Let u be the solution of (1), we split while the reminder w solves Here χ k (x) = χ( x k ), k 1, χ is a smooth function such that χ = 1 if |x| ≤ 1 and provided that k is large enough.
Proof. The lemma follows from Theorem 3.2 if This is possible for k large enough since the support of (1 − χ k )u 0 is contained in {x ∈ R : |x| ≥ k}.
Remark 2. For any ε > 0, we may obtain a solution v satisfying sup 0≤t≤T v(t) H s,p ≤ ε/ T 2 provided that k is large enough depending on T and ε.
Now we deal with the w part of the solution. Applying I−operator on both sides of (27) yields that The equivalent integral equation form of (28) is (29) where S(t) is the free evolution as before. Note that the support of χ k u 0 is contained in {x ∈ R : |x| ≤ 2k}, by Hölder inequality and (8), we have with life span τ ∼ ( Iw 0 H 1 + 1) −1 .
In order to extend the local solution to a global one, we need to prove a bound of sup 0≤t≤T Iw(t) H 1 for any T > 0. To this end, we decompose estimate Iw(t) H 1 on each interval and then add those estimates together. Multiplying (28) with Iw and integrating we find for t ∈ (0, τ ) To proceed, we need to deal with the terms on right hand side of (32).

Thus, it suffices to bound
Since ξ i m(ξ i ) N , the quantity can be controlled by as desired.
To obtain a bound of sup 0≤t≤T w(t) H s,p , we rewrite (27) as where S(t) is the free group. Thanks to Lemma 3.1, we deduce from (40) that Moreover, by Lemma 2 and Remark 2, we have provided that k is large enough. Now inserting (42) and (43) into (41) yields that Absorbing the last term of (44) in the left hand side, and using (39), we obtain Thus, u = v + w does not blow up on the interval [0, T ] in H s,p (R). Since T can be arbitrary large, we obtain the following theorem.
Based on this result, we are ready to prove Theorem 1.1.
Proof of Theorem 1.1. If p ≥ 2, then by Theorem 3.5, it suffices to consider the subcase s ≥ 1. Since u 0 ∈ H s,p (R) → L p (R), by Theorem 3.5 again, for any T > 0, there exists a unique solution u of (1) such that sup 0≤t≤T u(t) L p < ∞.   Following [5,7,19], the ill-posedness may reduce to disproving a bilinear estimate. Let ε > 0, consider the Cauchy problem Rewriting (45) into an integral equation form, we obtain where S(t) denotes the free evolution exp{−itϕ(D)} as before. Then a formal calculation yields that ∂u ∂ε (0, t, x) = S(t)u 0 := U (t, x) and Thus, we obtain the following sufficient condition for ill-posedness of Cauchy problem (1) in H s,p (R).
Proposition 4. For any T > 0, there exists a u 0 ∈ H s,p (R) such that fails, and consequently the Cauchy problem (1) is ill-posed in H s,p (R).
Remark 3. It's easy to see that if {f j (t, x)} is a t−uniform approximate identity then for any τ > 0 The following property of t−uniform approximate identities is important in the proof of our ill-posedness results.
Proposition 5. Let 1 ≤ p < ∞, g ∈ L p (R), and {f j (t, x)} ∈ L ∞ (0, T ; L 1 (R)) be a t−uniform approximate identity. Then we have uniformly for t ∈ [0, T ], where * denotes the convolution operator. In other words, Proof. Since translations are continuous on L p (R), for any ε > 0, there exists τ > 0 such that for |y| ≤ τ g(· − y) − g(·) L p ≤ ε. Now by Minkowski inequality we find  The idea of the proof is to construct a sequence {u 0j } such that the right hand side of (47) goes to zero but the left hand side has a positive lower bound. Let u 0j be smooth functions defined by Fourier transform as

By Definition 4.2 and Remark 3 we obtain
where j 1, γ = ln j and 1 A denotes the characteristic function of A. It follows from Plancherel theorem that u 0j L 2 = 1. Since u 0j is even, u 0j is real. Moreover, using Fourier inversion, we arrive at Then it is easy to see that for 1 < p < ∞ Let χ : R → R be a smooth function such that χ(x) = 0 if |x| ≤ 1 2 or |x| ≥ 2 and Thus, for any s ∈ R and p ∈ [2, ∞) Here we have used the fact that j −s ξ s χ j (ξ) is an L p multiplier, namely with an upper bound independent of j, which can be verified by Theorem 2.1. It follows from Lemma 3.1 that for 1 < p < ∞ In particular, we find S(t)u 0j ∈ L ∞ (0, T ; L 2 ). Moreover, we have 5778 MING WANG Lemma 4.4. For any T > 0, (S(t)u 0j ) 2 ∈ L ∞ (0, T ; L 1 (R)) is a t−uniform approximate identity.
Proof. Since S(t) is a unitary group on L 2 (R), we find for all t ∈ [0, T ] So it suffices to show that (S(t)u 0j ) 2 satisfies (3) of Definition 4.2. Note that S(t)u 0j is the solution of the linear equation Since −iϕ(D) is bounded on L p (R), using (50) we obtain for 1 < p < ∞ Let ψ : R → R be a smooth function such that ψ(x) = 1 for |x| ≥ 1 and ψ(x) = 0 for |x| ≤ 1 2 . Set ψ τ = ψ(·/τ ) and U τ = ψ τ U . Then U τ satisfies the following equation By Duhamel principle we find Thanks to Lemma 3.1, (51) and the fact that ϕ(D) is bounded from L 4 3 to L 2 , we have as j → ∞. Similarly, we obtain as j → ∞. This completes the proof.
Proof of Theorem 4.3. Since the Green function of (1 − ∂ 2 x ) −1 is e −|x| , one can show that (see [14]) where sgn(x) is the standard sign function. Then it follows from Proposition 5 and Lemma 4.4 that For s < 0, we have which gives, in light of (56), that Since the Fourier transform of K(x) is ϕ(ξ), it is easy to check that Thus, for any T > 0, there exists a constant c T > 0 depending only on T such that Collecting (57) and (58) gives On the other hand, since s < 0, by (49) we find u 0j H s,p → 0 as j goes to infinity. Hence  In particular, let s = 0 in Theorem 4.5 we obtain the following In order to prove Theorem 4.5, we need the following interpolation inequalities between Bessel potential spaces H s,p (R). Lemma 4.6. Let s < s < s ∈ R. If 1 < p < ∞, then We only prove the case p = 1, the case 1 < p < ∞ is similar. Let ψ : R → R be a smooth function such that ψ(x) = 1 for |x| ≥ 1 and ψ(x) = 0 for |x| ≤ 1 2 . Set ψ N = ψ(·/N ). Then using Theorem 2.1 one obtains that Then Let φ ≥ 0 be a smooth function with compact support such that φ L 2 = 1. For j 1, we set Then u 0j L 2 = 1 and we have the following Proof. It's easy to see that for α = 0, 1, 2, An application of Lemma 4.6 implies that as j goes to infinity for 1 < p < 2. If p = 1, then the restriction on s becomes s < 1 2 . By Lemma 4.6 again we have for any ε > 0 In particular, for any t > 0 the norms u(t) H s,p are finite if u(t) is the solution of (1) associated with initial data u 0 ∈ H s,p (R). However, the growth of u(t) H s,p in terms of time t is not known. This is an interesting topic since the growth of norms quantify the transfer of energy from low to high frequencies, see Bourgain [6] and Sohinger [23]. Due to some technical reasons, we divide our exploration in two cases.
The growth of norm in this case is a byproduct of proving global well-posedness of (1) by I−method.
Theorem 5.1. Let 0 ≤ s ≤ 1, u 0 ∈ H s (R), then the solution u(t) of (1) satisfies that for any T > 0 where the implicit constant depends only on s and u 0 H s .
Proof. The case s = 1 follows from the conservation law (E). So suppose 0 ≤ s < 1.
One can proceed with the same argument in Section 3 dealing with the w−part, and find that (39) becomes This implies the desired conclusion.
Now we turn to the growth of norms in H s (R), s > 1. The strategy is different from that in Theorem 5.1. We follow the idea of Bourgain [6] and Sohinger [23] to deduce an iteration bound as which holds for all times t and some constants r ∈ (0, 1]. Here τ, C depend only on s and the initial data. Then (59) implies that

MING WANG
Theorem 5.2. Let s > 1, u 0 ∈ H s (R), then the solution u(t) of (1) satisfies that for any T > 0 where the implicit constant depends only on s and u 0 H s .
Proof. Let D be the operator defined by the symbol |ξ|. Multiplying (1) with D 2s−2 u and integrating yield that Let s − 1 = m + α where m ≥ 0 is an integer and 0 ≤ α < 1. Then One can rewrite I 1 as . On one hand, applying integration by parts, Hölder inequality and Theorem 2.2 we find Here and below Q(·) is an increasing function which may be changed in different places. On the other hand, thanks to a commutator estimate (see Appendix, Theorem A.12 [15]) Observe that both 2m+α−1 s−1 and 2m+α−1 s−1 are less than 2(1 − 3 4(s−1) ). Thus, conditions on u 0 at infinity. Let α > 0, denote by L p ( x α dx) the space of functions satisfying that Theorem 5.4. Let 2 < p < ∞, α > 0, u 0 ∈ L p ( x α dx), then the solution of (1) satisfying the bound for all T > 0, and the implicit constant depends only on u 0 L p ( x α dx) .
Proof. The theorem can be proved in the same way as Theorem 3.5. Split u = v+w, where v, w solves (26) and (27), respectively. It follows from (40) and Lemma 3.1 that Absorbing the first term by the second term on RHS of (73), applying Lemma 2.3 yield that Thanks to Remark 2, one can choose k ∼ T This implies the desired conclusion. In order to obtain a growth bound of solutions in L p (R), the assumption u 0 ∈ L p ( x α dx) is imposed in Theorem 5.4. An interesting problem is that whether the BBM equation persists in weighted spaces L p ( x α dx). In other words, does the solution u(t) of (1) belongs to L p ( x α dx) for any t > 0? We give an affirmative answer to this question in the following theorem.
As a direct consequence of Theorem 5.5, we have the following results.
Remark 6. Nahas proved in [17] that if u(t, x) is the solution of u t + u xxx + (u 3 ) x = 0, u(0, x) = u 0 (x) with u 0 ∈ H s L 2 ( x α dx), where s ≥ max{ 1 4 , α}, then u(t) ∈ H s L 2 ( x α dx) for all t > 0 in the lifespan of u. Comparing Corollary 3 with the persistence of KdV equation in weighted Sobolev spaces, the smoothing assumptions on solutions are removed. This reflects the regularized property of the BBM equation to some extent.