WELL-POSEDNESS AND ASYMPTOTIC BEHAVIOR OF THE DISSIPATIVE OSTROVSKY EQUATION

. In this paper we study the global well-posedness and the large-time behavior of solutions to the initial-value problem for the dissipative Ostrovsky equation. We show that the associated solutions decay faster than the solutions of the dissipative KdV equation.


Introduction. The Ostrovsky equation
(u t + u xxx + uu x ) x + γu = 0, γ = ±1, (1.1) was originally derived first by Ostrovsky [25] (see also [8,9]) to model internal waves in the ocean or surface waves in a shallow channel with an uneven bottom and also capillary waves on the surface of the liquid or for oblique magneto-acoustic waves in plasma. The sign of γ is related to the type of dispersion. See also [20] for the first rigorous derivation of the Ostrovsky equation. A model of the propagation of long internal waves in a deep rotating fluid can be found in [6]. The structure of (1.1) is very similar to that of the KdV equation but unlike the KdV equation, the Ostrovsky equation is evidently nonintegrable by the method of inverse scattering transform [9]. When the effects of friction is considered in modeling (1.1), it turns into (see [11,Chapter 1] and [21]) (u t + u xxx + βD α u + uu x ) x + γu = 0, (x, t) ∈ M = R × (0, +∞), u(x, 0) = u 0 (x), (1.3) The space X s will be defined below. Isaza and Mejía proved in [14], by using Bourgain spaces and the technique of elementary calculus inequalities, introduced by Kenig, Ponce, and Vega in [16], that if s > −3/4, the Ostrovsky equation (1.1) is locally well-posed in H s (R), and it is not quantitatively well-posed (see [2,Definition 1]) in H s (R), if s < −3/4. The index s c = −3/4 is critical in studying the wellposedness of the KdV equation. One can see [18] to study a well-posedness result of (1.1) at the critical regularity. By working on suitable Bourgain-type spaces, equation (1.4) was proved in [12,29] to be globally well-posed in H s (R), s > s α,c , where s α,c = − 3 4 , 0 < α ≤ 1, − 3 5−α , 1 ≤ α ≤ 2. We note that the index s α,c shows that the dissipation somehow is weaker than dispersion when α ≤ 1, while in the case α ≥ 1 the well-posedness index s α,c is improved, compared with the KdV case. This will be expected for equation (1.3).
Here we borrow the ideas of [23] and introduce a Bourgain-type space. This space is in fact the intersection of the space introduced in [4] and of a Sobolev space. The advantage of this space is that it contains both the dissipative and dispersive parts of the linear symbol of (1.3). By using Strichartz's type estimates injected into the framework of Bourgain spaces and some techniques introduced in [16,27], we derive a bilinear estimate in these spaces which yields the local well-posedness of (1.3) in H s (R) for s > s α , where The main difficulty of proving the associated bilinear estimate arises from the rotation term which contains somehow a singularity.
Concerning the asymptotic behavior of the solution of (1.4) with α = 2, Amick, Bona and Schonbek used the Hopf-Cole transformation and proved in [1] that the solution of (1. if u 0 ∈ L 1 (R) ∩ H 2 (R). Later, Karch in [15] improved this result by applying the standard scaling argument to show that for u 0 ∈ L 1 (R) ∩ L 2 (R) The constant K is uniquely determined as a function of M using the condition R U M (x, 1)dx = M . This means that the dispersion is negligible compared to dissipation and nonlinear effects. Vento extended the results of [1] to (1.4) and showed that u(t) L 2 (R) (1 + t) − 1 2α . We will study the effects of the rotation in the asymptotic behavior of the solutions of (1.3). We prove that the solutions of (1.3) satisfies Estimate (1.5) shows that the solutions of (1.3) decay faster than the solutions of (1.4). To do that, by using the properties of the generalized heat-type kernel, we give some asymptotic estimates of the free solution. Then we derive decay rates estimate of the solution of the nonlinear problem. This paper is organized as follows: After stating our main results, we prove the bilinear estimate in Section 2. Section 3 is devoted to obtain some decay estimates of the linear equation of (1.3). Finally the decay estimates of the nonlinear problem is established in Section 4.
1.1. Notations. We introduce some notations which are standard. For s ≥ 0 and 1 ≤ p ≤ ∞, the Sobolev spacesẆ s,p =Ẇ s,p (R) and W s,p = W s,p (R) are respectively endowed with the norms f Ẇ s,p = D s f L p (R) and f W s,p = f L p (R) + D s f L p (R) . When p = 2, we simplify by the notationsḢ s =Ḣ s (R) and H s = H s (R). We also denote the Fourier transform of f byf or F(f ). Define More precisely, (1.6) is obtained by combining the classical inequality and the elementary inequality . See also [3]. Remark 2. The dissipative part in (1.3), when becoming small enough, has no effect on the low regularity of the equation. In the sequel, we shall prove wellposedness for (1.3) in the case of α ∈ [1,2]. To completeness, in the appendix, we use the ideas of [22] and sketch the proof of the global well-posedness of (1.3) when α ∈ [0, 1].
Related to the symbol of the linear equation, we define the Bourgain-type space where U is the unitary group extracted from (1.1) and H s,b with s, b ∈ R is the space-time version of H s (R) defined by the norm By Duhamel's principle, the solution of (1.3) can be locally written in the integral form as where ψ ∈ C ∞ 0 (R) is a time cut-off function satisfying supp(ψ) ⊂ [−2, 2] and ψ ≡ 1 on [−1, 1], ψ T (t) = ψ(t/T ) and the S α is the semigroup associated with the free evolution of (1.3), which can be extended to a linear operator on R by setting The following local existence is a consequence of Theorem 1.2 together with linear estimates of Section 2 (see the proof of Theorem 1.3 in [29]). Remark 3. It is worth noting that it was proved in [30] that the pure dissipative equation and that the solution map fails to be smooth when s < s d,α . Hence, Theorem 1.3 can be reduced to s α < s < s d,α . We also should note by the triangle inequality that s can be small enough and we can obtain the same result for large s. See [23] for the details.
The following conserved quantities of (1.3) will be very important in proving the global well-posedness of (1.3) and some estimates of the solution. The proof is similar to [23]. where The global well-posedness will follow from Theorems 1.3 and 1.4 (see the proofs of Theorem 1.1 in [23] and Theorem 1.3 in [29]). Theorem 1.5. Let u 0 ∈ H s (R) with s > s α . The existence time of the solution u of (1.7) can be extended to infinity and u ∈ C((0, +∞); H ∞ (R)).
Our main result on the decay estimates of the solutions of (1.3) reads as follows. Although the global well-posedness is guaranteed by Theorem 1.5, the results of Theorems 1.1 and 1.4 are sufficient for our purposes. Theorem 1.6. Let α ≥ 1, ≥ 0, 2 ≤ p ≤ ∞ and u 0 ∈ X s with s > 3/2 such that ∂ −1 x u 0 ∈ L 1 (R). Then the associated solution u of (1.3) satisfies 2. Linear and bilinear estimate. Before proving Theorem 1.2, we state several linear estimates (for the free and forcing terms) and some well-known Strichartztype estimates. The linear estimates are principally contained in [5,23]. Because these proofs are dependent with dissipative term |ξ| α , and independent with the dispersive term. So the factor 1/ξ have no effect in linear estimates. But in the nonlinear estimates, the factor 1/ξ is important. See also [24], where a factor similar to 1/ξ appears in the associated semigroup. (a) For all u ∈ S (R 2 ), [23]. The result (c) will imply a gain of regularity for the nonhomogeneous part of (1.7), and can be proved by an argument similar to the proof of Proposition 2.4 in [23] or Proposition 2.6 in [24].
Using the Cauchy-Schwarz inequality and Lemma 2.3, we easily obtain In Ω 1 , one has |ξ| 2 and thus if K 1 denotes the term between brackets in (2.2), then without any restriction on s. Estimate in Ω 2 .

2.2.
Bilinear estimate for the case γ = −1. We divide the integral domain into four subregions as the case γ = 1. The estimate of Ω 1 is the same as the case γ = 1. The sets A j andΩ j are defined the same as the case γ = 1. Let u = ξ1 ξ , and denote When |ξ| 100, then |ξ 2 | |ξ| + |ξ 1 | 101 and so arguing as in Ω 1 we obtain the required estimate.

−2s
C. Also, we have |u| = ξ1 ξ < 1 50 , and therefore Cξ 2 . On the other hand, the minimum of |λ| is attained when A = 1, that is, the minimum is of the order of ξ. Therefore, there is C > 0 such that |λ | C|ξ|. Making the change of variable λ = φ(ξ 1 ), we conclude that If |σ| |σ 1 |, then If |σ 1 | |σ|, then Estimate in Ω 3 . In this region, by symmetry, the required estimate is obtain in the same way as Ω 2 .
Remark 4. The quantitatively well-posedness was defined in [2]: for u 0 ∈ H s , and T ∈ (0, 1], we say that u ∈ X s,b is a solution of (1.3) in [0, T ] with initial datum u 0 , if there is an extension v ∈ X s,b of u such that where L(u 0 ) := ψ(· t )S α (· t )u 0 and B(u, v) = t 0 S α (t−τ )∂ x (uv) (t )dt . Let (s, · S ) be a Banach space of space-time functions such that (i) S ∩ X is dense in S, where X = ∩ s∈R {u ∈ S (R 2 ); ξ sû (λ) L 2 ξ L 1 τ < +∞}; (ii) S → C bd (R + t ; H s ); (iii) for u 0 ∈ H s , L(u 0 ) ∈ S and L(u 0 S ≤ C u 0 H s ; (iv) for u, v ∈ S ∩ X, B(u, v) ∈ S and B(u, v) S ≤ C u S v S , and thus B has a unique continuous extension to S × S.
If such S exists, we say that the problem u = L(u 0 ) + B(u, u), for given u 0 ∈ H s is quantitatively well posed in H s . Using the ideas of [2], it was proved in [14] that the Ostrovsky equation (1.1) is not quantitatively well-posed in H s if s < −3/4. But (1.3) contains the dissipation term which does not allow us to use the ideas of [2,14]. On the other hand, it was proved in [29] that the associated bilinear estimate for the dissipative KdV equation is optimal in H s for s > −3/4. Unfortunately, because of the singular term 1/ξ in the symbol of (1.3), we are not able to show that our well-posedness result is sharp. However, based on the proof of Theorem 1.2, our conjecture is that the critical index is α−4 2(3−α) .

Decay estimates of linear equation.
In this section we will obtain some decay estimates of the solution of the linear equation Then the solution u of (3.1) satisfies
Proof. We consider the case = 0. The case > 0 is similar.
First we note that the map ξ → ∂ −1 x u 0 (ξ) is continuous over R, so for any > 0 there is δ ∈ (0, 1) such that Then, we have from the assumptions of lemma, the properties of S α and the Plancherel theorem that Hence, Under the assumption of Lemma 3.1, the solution u of (3.1) satisfies We have from the Gagliardo-Nirenberg inequality and Lemma 3.1 that

4.
Decay of nonlinear problem. In this section, attention is turned to decay estimates of the nonlinear problem. Proof. This is a direct consequence of the conservation law (1.11). Proof. First we note from the Cauchy-Schwarz inequality, inequality (1.6) and Lemma 4.1 that . Therefore we have from (1.12) that Hence we deduce from (1.6) that This proves (i). To prove (ii), we multiply (1.3) by ∂ 3 x u and integrate over R we obtain after some integration by parts that Now by using the Gagliardo-Nirenberg inequality we get after some integration by parts that (4.5) Therefore we obtain from (4.4) and (4.5) that (4.6) And this proves the case (ii).
Proof. Some estimates, similar to ones in Lemma 4.2, hold for the case γ = −1, however the proof is slightly different because the coefficient of the third term of the left hand side of (1.12) is not positive. (i) If u 0 ∈ X 1 then the corresponding solution u of (1.3) satisfies We multiply (1.3) by 2u x and integrate over R and then over the time interval [0, t] to obtain We derive by subtracting (4.9) from (1.12) that (4.10) We have from the Plancherel theorem, the fractional Leibniz rule [7] and Lemma 4.1 that (4.11) Moreover we have from Lemma 4.1 that To estimate the third term of the right hand side of (4.10), we consider two cases. If α = 2, then (1.11) implies that Hence we deduce by inserting (4.11)-(4.13) into (4.10) that Now we consider the case α < 2. First we observe that sup t u(t) L ∞ (R) < C. Indeed we have from the integral form (1.7), the properties of S α and Lemma 4.1 that (4.14) Next we get by the Cauchy-Schwarz inequality, the fractional Leibniz rule, the Sobolev interpolation, (4.14) and Lemma 4.1 that By combining Equations (4.11), (4.12), (4.15) with (4.10), we complete the proof of (i) in this case. The proof of (ii) is similar to the case (ii) of Lemma 4.2.
The higher order estimates can be established by an argument similar to Lemma 4.2 and Lemma 4.3. Proof. Applying the operator Λ s = (I − ∂ 2 x ) s/2 to equation (1.3) and multiplying by Λ s u and integrating over R, we obtain that By Corollary 2, the integrand in the right hand side of (4.17) is in L 1 (R + ), so that lim t→+∞ u(t) L 2 (R) = 0.
Proof of Theorem 1.6. We first treat the case p = 2. Multiplying equation (1.3) by D 2 −3 u and then integrating over R, one obtains In the case ≥ 1, similar to (4.19) with j = − 1, we can show from the Plancherel theorem, the fractional Leibniz rule and an interpolation that that . Hence, it follows from the Sobolev embedding and Lemma 4.4 that d dt u(t) 2Ḣ −1 + β u 2Ḣ The case = 0 is a little bit more delicate. Indeed, we have from the following Gagliardo-Nirenberg inequality , and a Sobolev interpolation that Again Lemma 4.4 implies that Appendix. Here, we sketch the proof the global well-posedness of (1.3) in the case α ∈ [0, 1]. The proofs proceed along the same lines of the argument provided in [22,23]. Indeed, the idea is to work in the Bourgain space associated with the Ostrovsky equation, i.e. related only to the dispersive part of the symbol of the dissipative Ostrovsky equation. Let U (t) be the unitary group which defined the free evolution of the Ostrovsky equation (1.1), i.e., U (t)f = R e ixξ e itm(ξ)f (ξ)dξ, t ∈ R, f ∈ S (R). (4.22) We denote by Y s,b the Bourgain-type space associated with H s,b for (1.1) by the norm For T ≥ 0, the localized version of Y s,b is defined by the norm u Y s,b T = inf g Y s,b , g ∈ Y s,b and g(t) = u(t) on [0, T ] .
To prove the local well-posedness result, a fixed point argument shall be applied to the integral form (1.7). The following linear estimates for the free and forcing terms are derived by mimicking the argument provided in [22], and we omit the details.