Schrödinger-improved Boussinesq system in two space dimensions

We study the Cauchy problem for the Schrödinger-improved Boussinesq system in a two-dimensional domain. Under natural assumptions on the data without smallness, we prove the existence and uniqueness of global strong solutions. Moreover, we consider the vanishing “improvement” limit of global solutions as the coefficient of the linear term of the highest order in the equation of ion sound waves tends to zero. Under the same smallness assumption on the data as in the Zakharov case, solutions in the vanishing “improvement” limit are shown to satisfy the Zakharov system.


Introduction
In this paper, we study the Cauchy problem for the Schrödinger-improved Boussinesq system (S-iB) (S-iB) i∂ t u + u = vu, with initial data (u(0), v(0), ∂ t v(0)) = (ϕ, ψ 0 , ψ 1 ) given at t = 0, where u : R × → C, v : R × → R, and ⊂ R 2 is a domain with smooth boundary ∂ . The Laplacian is understood to be the self-adjoint realization in the Hilbert space L 2 ( ) with domain D( ) = (H 2 ∩ H 1 0 )( ), where H 2 ( ) is the Sobolev space of the second order and H 1 0 ( ) is the Sobolev space of the first order with vanishing condition on ∂ . The system (S-iB) is regarded as a substitute for the Zakharov system (Z) and the "improvement" has been made on the second equation of (S-iB) by modifying the dispersion relation of ion-sound waves of Boussinesq type. See [27] for details.
is the natural energy space for (S-iB) in the sense that E(t) makes sense as a real number by (1.3).
(2) No smallness condition is required on ϕ 2 . This is a striking difference in view of the corresponding cases of the Zakharov system and cubic nonlinear Schrödinger equation [2,8,17,33,34,47].
is an improvement in an estimate in [34], where the RHS on (1.1) is replaced by C (1 + t 2 ) exp(2C 2 0 ϕ 2 2 t 2 ). The H 1 -bound of exponential type (1.1) is a natural consequence of a simple Gronwall type argument, which is much simpler than that of [34].
is a natural consequence of a simple Gronwall type argument on a modified energy for H 2 ⊕ H 1 ⊕ H 1 [7,15,23,37,38] with coefficient of exponential contribution by (1.1). The H 2 ⊕ H 1 ⊕ H 1 -bound of double exponential type (1.2) arises also in the Brezis-Gallouet argument [2,6,37], whereas our proof is independent of the Brezis-Gallouet inequality. See the proof of Theorem 1. We prove Theorem 1 in Sect. 3. The idea of the proof is similar to that of [14,21,22,33] in the sense that we prove that solutions of some regularized system by the Yosida approximation form a bounded sequence in H 2 ⊕ H 1 ⊕ H 1 and a Cauchy sequence in H 1 ⊕ L 2 ⊕ L 2 . For the H 1 ⊕ L 2 ⊕ L 2 -control (1.1) by the energy without smallness of ϕ 2 , we employ the Gronwall argument on v to avoid a direct use of the sharp Gagliardo-Nirenberg inequality (1.3). For the H 2 ⊕ H 1 ⊕ H 1 -control (1.2), we introduce a modified energy for H 2 ⊕ H 1 ⊕ H 1 which closes another Gronwall argument on (∂ t u, ∂ t v). Those two strategies require a specific technique, which is totally different from that of [33].
The second purpose of this paper is to study the vanishing "improvement" limit problem for (S-iB) as the coefficient of the linear term of the highest order tends to zero. For that purpose, we introduce (S-iB) ε with ε ∈ (0, 1) as follows: with the same initial data. The proof of Theorem 1 with minor modifications implies the global existence of solutions to the Cauchy problem for (S-iB) ε in the same class as in Theorem 1. We denote by (u ε , v ε , ∂ t v ε ) the solutions to (S-iB) ε . We prove that, under the same smallness condition as in (Z), solutions (u ε , v ε , ∂ t v ε ) of (S-iB) ε tend to the solution (u, v, ∂ t v) of (Z) as ε ↓ 0. Specifically, we prove: where C depends on ϕ H 1 , ψ 0 2 , ψ 1 2 , (− ) −1/2 ψ 1 2 and C depends on as ε ↓ 0.

Remark 2.
(1) The smallness assumption ϕ 2 < √ 2/C 0 is necessary to ensure the global existence of solutions to (Z) as well as the uniform estimate (1.5) with respect to ε ∈ (0, 1) and t ∈ R.

Preliminaries
In this section, we collect basic estimates to be used in the proof of the main theorems below. We use the following Gagliardo-Nirenberg inequalities the standard elliptic estimate and the elementary equalities By (2.1), we have with the same constant C 0 . We also use the following lemma.
We summarize basic properties of the Yosida approximation of the identity.
(3) The following estimates hold for any n ∈ Z >0 .

Proof of Theorem 1
In this section, we prove Theorem 1. We introduce the following regularized system for (S-iB): Then, (S-iB) n are converted into the integral equations By the standard fixed point argument [9], (S-iB) n have unique global solutions We first show the boundedness of the sequence ((u n , v n , ∂ t v n ); n ∈ Z >0 ) of solutions to (S-iB) n with values in H 2 ⊕ H 1 ⊕ H 1 . From now on, we restrict our attention to the case t > 0 for definiteness. Omitting explicit dependence of the time variable, we have ≤ C 0 J n v n 2 J n u n 2 ∇ J n u n 2 Similarly, By (3.4),(3.5),(3.6), and (3.7), we obtain The Gronwall argument on (3.8) with (3.9) implies The estimate ( and calculate its time derivative as = 2Re((∂ t J n v n )J n u n |J n (− u n + J n (J n v n · J n u n ))) + (J n ∂ t v n | |J n u n | 2 ) = (J n ∂ t v n | |J n u n | 2 − 2Re(J n u n J n u n )) + 2Re((J n ∂ t v n )J n u n |J 2 n (J n v n · J n u n )) = 2(J n ∂ t v n ||∇ J n u n | 2 ) + 2Re((J n ∂ t v n )J n u n |J 2 n (J n v n · J n u n )).
from which we have Similarly, By (3.10),(3.11), and (3.14), we have for any ε > 0 Integrating both sides of (3.15) and letting ε ↓ 0, we obtain where and we have added the last term for the Gronwall argument below. By (3.10),(3.12), and (3.16), for any ϕ = 0 we obtain where we have used the following inequality and F n (0) is bounded uniformly in n as By (3.16) and (3.18), we obtain the following uniform bound for The proof of the convergence of (u n , v n , ∂ t v n ) on compact intervals with values in is almost similar to that of [33] and omitted. Moreover, the existence and uniqueness of global solutions (u, v, ∂ t v) to (S-iB) in the class and the convergence of (u n , v n , ∂ t v n ) to (u, v, ∂ t v) on compact time-intervals with values in H 1 ⊕ L 2 ⊕ (L 2 ∩ (− ) 1/2 L 2 ) with weak convergence in H 2 ⊕ H 1 ⊕ H 1 follow in the same way as in the proof of Theorem 1 of [33]. Therefore, Part (2) of Theorem 1 follows from (3.3) and (3.4) and Part (3) follows from (3.3),(3.10), (3.18), and (3.19). It remains to prove that ∂ t v ∈ C 1 (R; H 1 0 ( )). Let t 0 ∈ R and let I = [a, b] an interval with t 0 ∈ I . By (S-iB), we write and estimate both sides in H 1 as which tends to zero as t → t 0 . This completes the proof of Theorem 1.