Vanishing dispersion limit for Schrödinger-improved Boussinesq system in two space dimensions

Abstract

We study the vanishing dispersion limit of strong solutions to the Cauchy problem for the Schrödinger-improved Boussinesq system in a two dimensional domain. We show an explicit representation of limiting profile in terms of the initial data. Moreover, the first approximation is also represented as a pair of solutions of a linear system with coefficients and forcing term given by the limiting profile.

Keywords

Schrödinger-improved Boussinesq system vanishing dispersion limit

1. Introduction

In this paper, we study the vanishing dispersion limit of strong solutions to the Cauchy problem for the Schrödinger-improved Boussinesq system $\begin{matrix} {(S-iB)}_{ε} \{\begin{matrix} i \partial_{t} u_{ε} + ε Δ u_{ε} = v_{ε} u_{ε}, \\ \partial_{t}^{2} v_{ε} - Δ v_{ε} - Δ \partial_{t}^{2} v_{ε} = Δ | u_{ε} |^{2} \end{matrix} \end{matrix}$ with initial data $\begin{matrix} (u_{ε} (0), v_{ε} (0), \partial_{t} v_{ε} (0)) = (φ, ψ_{0}, ψ_{1}) \end{matrix}$ given at $t = 0$ , where $u_{ε} : R \times Ω \to C$ , $v_{ε} : R \times Ω \to R$ , $ε \in (0, 1]$ , and $Ω \subset R^{2}$ is a domain with smooth boundary. The Laplacian Δ in Ω is understood to be the self-adjoint realization in the Hilbert space $L^{2} (Ω)$ with domain $D (Δ) = (H^{2} \cap H_{0}^{1}) (Ω)$ , where $H^{2} (Ω)$ is the Sobolev space of the second order and $H_{0}^{1} (Ω)$ is the Sobolev space of the first order with Dirichlet zero condition. We already know that for any $(φ, ψ_{0}, ψ_{1}) \in D (Δ) \oplus H_{0}^{1} (Ω) \oplus (H_{0}^{1} \cap {(- Δ)}^{1 / 2} L^{2}) (Ω)$ , ${(S-iB)}_{1}$ has a unique global strong solution $(u_{1}, v_{1}, \partial_{t} v_{1})$ with $\begin{array}{l} u_{1} \in (L_{loc}^{\infty} \cap C_{w}) (R; D (Δ)) \cap C (R; H_{0}^{1} (Ω)) \cap C^{1} (R; H^{- 1} (Ω)), \\ \partial_{t} u_{1} \in (L_{loc}^{\infty} \cap C_{w}) (R; L^{2} (Ω)), \\ v_{1} \in C^{2} (R; H_{0}^{1} (Ω)), \\ {(- Δ)}^{- 1 / 2} \partial_{t} v_{1} \in C (R; L^{2} (Ω)), \end{array}$ and $(u_{1} (0), v_{1} (0), \partial_{t} v_{1} (0)) = (φ, ψ_{0}, ψ_{1})$ , where $H^{- 1} (Ω) = {(H_{0}^{1} (Ω))}^{*}$ (see [20]). The proof of the result with minor modification shows the existence and uniqueness of global solutions $(u_{ε}, v_{ε}, \partial_{t} v_{ε})$ to the Cauchy problem for ${(S-iB)}_{ε}$ in the same class as above for any $ε \in (0, 1)$ .

The problem we consider in this paper is a detailed description of the behavior of $(u_{ε}, v_{ε}, \partial_{t} v_{ε})$ in $L^{2} \oplus L^{2} \oplus L^{2}$ as $ε ↓ 0$ . The problem was first studied in [11] for the whole Eucledian case. It is shown in [11] that for any $(φ, ψ_{0}, ψ_{1}) \in (H^{2} \oplus H^{2} \oplus H^{2}) (R^{2})$ there exists $T_{0} > 0$ such that $\begin{matrix} (1.1) & sup_{t \in [0, T_{0}]} ({‖ u_{ε} (t) - u (t) ‖}_{H^{2}} + {‖ v_{ε} (t) - v (t) ‖}_{H^{2}}) \to 0 \end{matrix}$ as $ε ↓ 0$ , where $u \in C^{1} (R; H^{2} (R^{2}))$ and $v \in C^{2} (R; H^{2} (R^{2}))$ satisfy $\begin{matrix} {(S-iB)}_{0} \{\begin{matrix} i \partial_{t} u = v u, \\ \partial_{t}^{2} v - Δ v - Δ \partial_{t}^{2} v = Δ | u |^{2} \end{matrix} \end{matrix}$ with the same initial condition $(u (0), v (0), \partial_{t} v (0)) = (φ, ψ_{0}, ψ_{1})$ .

The purpose of this paper is to obtain the first approximation $(u^{#}, v^{#})$ of order ε hidden in the convergence (1.1) and present explicit representation of $(u, v)$ and $(u^{#}, v^{#})$ in terms of the initial data $(φ, ψ_{0}, ψ_{1})$ .

We prove:

Theorem 1.
Let $(φ, ψ_{0}, ψ_{1}) \in D (Δ) \oplus D (Δ) \oplus (D (Δ) \cap {(- Δ)}^{1 / 2} L^{2} (Ω))$ . Then there exists $T_{0} \in (0, 1]$ and a constant C depending $‖ φ ‖_{H^{2}}$ , $‖ ψ_{0} ‖_{H^{2}}$ , $‖ ψ_{1} ‖_{H^{2}}$ , and $T_{0}$ such that $\begin{array}{l} (1.2) & sup_{ε \in (0, 1]} sup_{t \in [0, T_{0}]} ({‖ u_{ε} (t) ‖}_{H^{2}} + {‖ v_{ε} (t) ‖}_{H^{2}}) ⩽ C, \\ (1.3) & lim_{ε ↓ 0} sup_{t \in [0, T_{0}]} ({‖ u_{ε} (t) - u (t) ‖}_{H^{2}} + {‖ v_{ε} (t) - v (t) ‖}_{H^{2}}) = 0, \end{array}$ where $(u, v)$ is the unique solution to ${(S - iB)}_{0}$ with $\begin{array}{l} (1.4) & u \in C^{1} (R; D (Δ)), \\ (1.5) & v \in C^{2} (R; D (Δ)), \end{array}$ and $(u (0), v (0), \partial_{t} v (0)) = (φ, ψ_{0}, ψ_{1})$ . Moreover, $(u, v)$ is given explicitly by $\begin{array}{l} (1.6) & u (t) = φ exp (- i \int_{0}^{t} v (t^{'}) d t^{'}), \\ (1.7) & v (t) = (cos t ω) ψ_{0} + (ω^{- 1} sin t ω) ψ_{1} + (cos t ω - I) | φ |^{2}, \end{array}$ where $ω = {(- Δ)}^{1 / 2} {(1 - Δ)}^{- 1 / 2}$ and I is the identity operator.
Remark 1.

The pair $(u, v)$ is regarded as the limiting profile of $(u_{ε}, v_{ε})$ as $ε ↓ 0$ .

The limiting profile $(u, v)$ is represented explicitly in terms of the initial data $(φ, ψ_{0}, ψ_{1})$ as in (1.6) and (1.7).

The operator ω is a bounded self-adjoint operator in $L^{2} (Ω)$ with operator norm bounded by 1.

Theorem 2.
Let $(φ, ψ_{0}, ψ_{1})$ , $T_{0}$ , and $(u, v)$ be as in Theorem 1 . Then:
The convergence in $L^{\infty} (0, T_{0}; L^{2} (Ω))$ $\begin{matrix} (1.8) & sup_{t \in [0, T_{0}]} ({‖ u_{ε} (t) - u (t) - ε u^{#} (t) ‖}_{2} + {‖ v_{ε} (t) - v (t) - ε v^{#} (t) ‖}_{2}) = o (ε) \end{matrix}$ holds, where $(u^{#}, v^{#})$ is a unique solution to $\begin{matrix} {(S - iB)}^{#} \{\begin{matrix} i \partial_{t} u^{#} = v u^{#} + u v^{#} - Δ u, \\ \partial_{t}^{2} v^{#} - Δ v^{#} - Δ \partial_{t}^{2} v^{#} = 2 Δ Re (\overline{u} u^{#}) \end{matrix} \end{matrix}$ with $(u^{#} (0), v^{#} (0), \partial_{t} v^{#} (0)) = (0, 0, 0)$ and $\begin{array}{l} (1.9) & u^{#} \in C^{1} (R; L^{2} (Ω)), \\ (1.10) & v^{#} \in C^{2} (R; L^{2} (Ω)) . \end{array}$

$(u^{#}, v^{#})$ is given by a pair of unique global solutions to the system of linear integral equations $\begin{array}{l} (1.11) & u^{#} (t) = i t Δ φ + \int_{0}^{t} (t - t^{'}) Δ (v u) (t^{'}) d t^{'} - i \int_{0}^{t} (v u^{#} + u v^{#}) (t^{'}) d t^{'}, \\ (1.12) & v^{#} (t) = - 2 \int_{0}^{t} (ω sin (t - t^{'}) ω) Re (\overline{u} u^{#}) (t^{'}) d t^{'} . \end{array}$

Remark 2.
The pair $(u^{#}, v^{#})$ is regarded as the first approximation hidden in (1.3). ${(S-iB)}^{#}$ is (inhomogeneous) linear system of equations of $(u^{#}, v^{#})$ with variable coefficients given by $(u, v)$ and forcing term by u, both of which are written explicitly by $(φ, ψ_{0}, ψ_{1})$ as in Theorem 1.
Remark 3.
We add a remark on motivation of the problem. From mathematical point of view, the problem of vanishing limit of the coefficient of partial differential operators of the highest order is a natural and standard problem in the analysis of nonlinear partial differential equations. In this paper, we study the problem for the first equation in ${(S-iB)}_{ε}$ . The corresponding problem for the second equation has been studied in [20] to compare the Schrödinger-improved Boussinesq system with Zakharov system. From physical point of view, the operator $ε Δ$ appears naturally in the expansion $\begin{matrix} {(m^{2} c^{4} - c^{2} ℏ^{2} Δ)}^{1 / 2} \sim m c^{2} - \frac{ℏ^{2}}{2 m} Δ \end{matrix}$ as $ℏ ↓ 0$ or $m ↑ + \infty$ after the constant term $m c^{2}$ is gauged away via modulation factor $e^{\pm i m c^{2} t}$ . In this setting, the problem in this paper is regarded as a problem of semi-classical limit $ℏ ↓ 0$ or heavy mass limit $m ↑ + \infty$ of the corresponding relativistic equations. We refer the reader to [6,7,10,26] for closely related papers.

We prove Theorems 1 and 2 in Sections 2 and 3, respectively. In the proof of Theorem 2, the uniform $H^{2} \oplus H^{2}$ -bound (1.2) with respect to $ε \in (0, 1]$ plays a key role. The standard and modified energy estimates in the framework of $D (Δ) \oplus H_{0}^{1} (Ω) \oplus (H_{0}^{1} \cap {(- Δ)}^{1 / 2} L^{2}) (Ω)$ as in [1,8,11–13,19,20,23,24] are insufficient for that purpose since they only ensure an $H^{2}$ -control of u proportional to $ε^{- 1 / 2}$ , which diverges as $ε ↓ 0$ . Even in the case $Ω = R^{2}$ where the Strichartz estimate for the free Schrödinger propagator $U (t) = exp (i t Δ)$ is available, the corresponding propagator for ${(S-iB)}_{ε}$ is $U_{ε} (t) = U (ε t)$ with Strichartz norm $L^{4} (R; L^{4} (R^{2}))$ proportional to $ε^{- 1 / 4}$ , which diverges as $ε ↓ 0$ . That is why we require the additional assumption $(ψ_{0}, ψ_{1}) \in D (Δ) \oplus D (Δ)$ for the $H^{2} \oplus H^{2}$ -bound with respect to $ε \in (0, 1]$ .

We refer the reader to [2,9,11,19–21,29–32] for related results on ${(S-iB)}_{1}$ and to [3–5,10,11,14–18,22,25,27,28] for related asymptotic analysis of wavefunctions with respect to parameters appearing as coefficients in nonlinear evolution equations.
2. Proof of Theorem 1

Solutions $(u_{ε}, v_{ε})$ of ${(S-iB)}_{ε}$ satisfy the integral equations $\begin{array}{l} (2.1) & u_{ε} (t) = U (ε t) φ - i \int_{0}^{t} U (ε (t - t^{'})) (v_{ε} u_{ε}) (t^{'}) d t^{'}, \\ (2.2) & v_{ε} (t) = (cos t ω) ψ_{0} + (ω^{- 1} sin t ω) ψ_{1} - \int_{0}^{t} (ω sin (t - t^{'}) ω) | u_{ε} |^{2} (t^{'}) d t^{'}, \end{array}$ where $U (ε t) = exp (i ε t Δ)$ with $ω = {(- Δ)}^{1 / 2} {(1 - Δ)}^{- 1 / 2}$ .

We estimate $u_{ε} (t)$ in $H^{2}$ as $\begin{array}{l} {‖ u_{ε} (t) ‖}_{H^{2}} & ⩽ ‖ φ ‖_{H^{2}} + \int_{0}^{t} {‖ (v_{ε} u_{ε}) (t^{'}) ‖}_{H^{2}} d t^{'} \\ (2.3) & ⩽ ‖ φ ‖_{H^{2}} + C_{0} \int_{0}^{t} {‖ v_{ε} (t^{'}) ‖}_{H^{2}} {‖ u_{ε} (t^{'}) ‖}_{H^{2}} d t^{'} \end{array}$ for any $t ⩾ 0$ , where we have used the unitarity of $U (ε t)$ in $H^{2} (Ω)$ and algebraic structure of $H^{2} (Ω)$ with multiplication bound denoted by $C_{0} ⩾ 1$ . Similarly, we estimate $v_{ε} (t)$ in $H^{2}$ as $\begin{matrix} (2.4) & ‖ v_{ε} ‖_{H^{2}} ⩽ ‖ ψ_{0} ‖_{H^{2}} + t ‖ ψ_{1} ‖_{H^{2}} + C_{0} \int_{0}^{t} {‖ u_{ε} (t^{'}) ‖}_{H^{2}}^{2} d t^{'}, \end{matrix}$ for any $t ⩾ 1$ , where we have used the boundedness of $cos t ω$ and $ω^{- 1} sin t ω$ in $H^{2} (Ω)$ with bounds 1 and t, respectively. We introduce $\begin{array}{l} m_{ε} (t) = {‖ u_{ε} (t) ‖}_{H^{2}} + {‖ v_{ε} (t) ‖}_{H^{2}}, \\ M_{ε} (t) = \int_{0}^{t} m_{ε} {(t^{'})}^{2} d t^{'}, \\ C_{1} = max (‖ φ ‖_{H^{2}} + ‖ ψ_{0} ‖_{H^{2}} + ‖ ψ_{1} ‖_{H^{2}}, C_{0}) . \end{array}$ By (2.3) and (2.4), we have $\begin{matrix} m_{ε} (t) ⩽ C_{1} + C_{1} M_{ε} (t) \end{matrix}$ for any $t \in [0, 1]$ . This gives $\begin{matrix} M_{ε}^{'} (t) = m_{ε} {(t)}^{2} ⩽ C_{1}^{2} {(M_{ε} (t) + 1)}^{2} \end{matrix}$ and therefore $\begin{matrix} (2.5) & {(\frac{1}{M_{ε} + 1})}^{'} (t) = - \frac{M_{ε}^{'} (t)}{{(M_{ε} (t) + 1)}^{2}} ⩾ - C_{1}^{2} . \end{matrix}$ Let $T_{0} > 0$ satisfy $C_{1}^{2} T_{0} < 1$ . Integrating both sides of (2.5) on the time interval $[0, t] \subset [0, T_{0}]$ , we obtain $\begin{matrix} M_{ε} (t) + 1 ⩽ \frac{1}{1 - C_{1}^{2} t} \end{matrix}$ for any $t \in [0, T_{0}]$ . This implies the uniform $H^{2} \oplus H^{2}$ -bound $\begin{matrix} (2.6) & sup_{t \in [0, T_{0}]} ({‖ u_{ε} (t) ‖}_{H^{2}} + {‖ v_{ε} (t) ‖}_{H^{2}}) ⩽ \frac{C_{1}}{1 - C_{1}^{2} T_{0}}, \end{matrix}$ as required.

Next we derive the solution formula (1.6) and (1.7) from ${(S-iB)}_{0}$ . From the first equation of ${(S-iB)}_{0}$ , we have $\begin{matrix} \partial_{t} (u (t) exp (- i \int_{0}^{t} v (t^{'}) d t^{'})) = 0 . \end{matrix}$ Integrating this in t yields (1.6). Since v is real-valued, this gives $| u (t) |^{2} = | φ |^{2}$ , so that the second equation of ${(S-iB)}_{0}$ is rewritten as $\begin{matrix} \partial_{t}^{2} v + ω^{2} v = - ω^{2} | φ |^{2}, \end{matrix}$ which is solved exactly by $\begin{array}{l} v (t) & = (cos t ω) ψ_{0} + (ω^{- 1} sin t ω) ψ_{1} - \int_{0}^{t} (ω sin (t - t^{'}) ω) | φ |^{2} d t^{'} \\ = (cos t ω) ψ_{0} + (ω^{- 1} sin t ω) ψ_{1} + (cos t ω - I) | φ |^{2}, \end{array}$ which is exactly (1.7).

The regularity classes of u and v, namely (1.4) and (1.5), are verified by a direct calculation on (1.6) and (1.7) and strong continuity $cos t ω$ and $ω^{- 1} sin t ω$ in $H^{2} (Ω)$ .

A solution $(u, v)$ of ${(S-iB)}_{0}$ satisfies the integral equations $\begin{array}{l} (2.7) & u (t) = φ - i \int_{0}^{t} (v u) (t^{'}) d t^{'}, \\ (2.8) & v (t) = (cos t ω) ψ_{0} + (ω^{- 1} sin t ω) ψ_{1} - \int_{0}^{t} (ω sin (t - t^{'}) ω) | u |^{2} (t^{'}) d t^{'} \end{array}$ for any $t ⩾ 0$ . Subtracting both sides of (2.1) from those of (2.7), we obtain $\begin{array}{l} u_{ε} (t) - u (t) & = (U (ε t) - I) φ - i \int_{0}^{t} U (ε (t - t^{'})) ((v_{ε} - v) u_{ε} + v (u_{ε} - u)) (t^{'}) d t^{'} \\ (2.9) & + i \int_{0}^{t} (U (ε (t - t^{'})) - I) (v u) (t^{'}) d t^{'} . \end{array}$ Similarly, it follows from (2.2) and (2.3) that $\begin{matrix} (2.10) & v_{ε} (t) - v (t) = - \int_{0}^{t} (ω sin (t - t^{'}) ω) Re (\overline{u_{ε}} + \overline{u}) (u_{ε} - u) (t^{'}) d t^{'} . \end{matrix}$ Estimating (2.9) and (2.10) in the same way as in (2.3) and (2.4) and using (1.4), (1.5), and (2.6), we have $\begin{array}{l} {‖ u_{ε} (t) - u (t) ‖}_{H^{2}} + {‖ v_{ε} (t) - v (t) ‖}_{H^{2}} \\ ⩽ {‖ (U (ε t) - I) φ ‖}_{H^{2}} + \int_{0}^{t} {‖ (U (ε (t - t^{'})) - I) (v u) (t^{'}) ‖}_{H^{2}} d t^{'} \\ (2.11) & + C \int_{0}^{t} ({‖ u_{ε} (t^{'}) - u (t^{'}) ‖}_{H^{2}} + {‖ v_{ε} (t^{'}) - v (t^{'}) ‖}_{H^{2}}) d t^{'} \end{array}$ for any $t \in [0, T_{0}]$ . By the Gronwall argument on (2.11), we obtain $\begin{array}{l} sup_{t \in [0, T_{0}]} ({‖ u_{ε} (t) - u (t) ‖}_{H^{2}} + {‖ v_{ε} (t) - v (t) ‖}_{H^{2}}) \\ ⩽ sup_{t \in [0, T_{0}]} ({‖ (U (ε t) - I) φ ‖}_{H^{2}} \\ (2.12) & + C \int_{0}^{t} e^{C (t - t^{'})} ({‖ (U (ε t^{'}) - I) φ ‖}_{H^{2}} + {‖ (U (ε (t - t^{'})) - I) (v u) (t^{'}) ‖}_{H^{2}}) d t^{'}) \end{array}$ with C independent of $ε \in (0, 1]$ and $t \in [0, T_{0}]$ . The RHS of (2.12) tends to zero as $ε ↓ 0$ since $φ \in H^{2} (Ω)$ and $v u \in C ([0, T_{0}]; H^{2} (Ω))$ . This completes the proof of Theorem 1.

3. Proof of Theorem 2

We first derive (1.11) and (1.12) from ${(S-iB)}^{#}$ . Let $(u^{#}, v^{#})$ satisfy ${(S-iB)}^{#}$ . Then the first equation of ${(S-iB)}^{#}$ with $u^{#} (0) = 0$ is integrated in t to give $\begin{array}{l} u^{#} (t) & = - i \int_{0}^{t} (v u^{#} + u v^{#} - Δ u) (t^{'}) d t^{'} \\ = - i \int_{0}^{t} (v u^{#} + u v^{#}) (t^{'}) d t^{'} + i \int_{0}^{t} Δ (u (0) + \int_{0}^{t^{'}} \partial_{t} u (t^{″}) d t^{″}) d t^{'} \\ = - i \int_{0}^{t} (v u^{#} + u v^{#}) (t^{'}) d t^{'} + i \int_{0}^{t} (Δ φ - i \int_{0}^{t^{'}} Δ (v u) (t^{″}) d t^{″}) d t^{'} \\ = - i \int_{0}^{t} (v u^{#} + u v^{#}) (t^{'}) d t^{'} + i t Δ φ + \int_{0}^{t} (t - t^{'}) Δ (v u) (t^{'}) d t^{'}, \end{array}$ which is precisely (1.11). The second equation of ${(S-iB)}^{#}$ is rewritten as $\begin{matrix} \partial_{t}^{2} v^{#} + ω^{2} v^{#} = - 2 ω^{2} (Re (\overline{u} u^{#})), \end{matrix}$ which is integrated explicitly with $(v^{#} (0), \partial_{t} v^{#} (0)) = (0, 0)$ as $\begin{matrix} v^{#} (t) = - 2 \int_{0}^{t} (ω sin (t - t^{'}) ω) Re (\overline{u} u^{#}) (t^{'}) d t^{'}, \end{matrix}$ which is precisely (1.12). This proves Part (2).

Given $(u, v)$ , the system of integral equations (1.11) and (1.12) has a unique pair of solutions $(u^{#}, v^{#})$ with $u^{#}, v^{#} \in C (R; L^{2} (Ω))$ since the system is (inhomogeneous) linear with coefficients $u, v \in C (R; H^{2} (Ω))$ and forcing term belonging to $C (R; L^{2} (Ω))$ . Moreover, a direct calculation on (1.11) and (1.12) in $C (R; H^{- 2} (Ω))$ shows that $(u^{#}, v^{#})$ represented as (1.11) and (1.12) solves ${(S-iB)}^{#}$ in $H^{- 2} (Ω)$ and satisfies (1.9) and (1.10).

Given $(u, v)$ and $(u_{ε}, v_{ε})$ , we consider the following system of integral equations of $(u_{ε}^{#}, v_{ε}^{#})$ : $\begin{array}{l} (3.1) & u_{ε}^{#} (t) = i t Δ φ - i \int_{0}^{t} U (ε (t - t^{'})) (v u_{ε}^{#} + u_{ε} v_{ε}^{#} - i (t - t^{'}) Δ (v u)) (t^{'}) d t^{'}, \\ (3.2) & v_{ε}^{#} (t) = - \int_{0}^{t} (ω sin (t - t^{'}) ω) Re ((\overline{u_{ε}} + \overline{u}) u_{ε}^{#}) (t^{'}) d t^{'} . \end{array}$ The system (3.1)–(3.2) has a unique pair of solutions $(u_{ε}^{#}, v_{ε}^{#})$ with $u_{ε}^{#}, v_{ε}^{#} \in C (R; L^{2} (Ω))$ since the system is linear with coefficients in $C (R; H^{2} (Ω))$ and forcing term in $C (R; L^{2} (Ω))$ . Let $(u_{ε}^{#}, v_{ε}^{#})$ be a unique global solution to (3.1)–(3.2). We write $u_{ε} - u - ε u_{ε}^{#}$ and $v_{ε} - v - ε v_{ε}^{#}$ as $\begin{array}{l} u_{ε} (t) - u (t) - ε u_{ε}^{#} (t) \\ = (U (ε t) - I - i ε t Δ) φ \\ - \int_{0}^{t} U (ε (t - t^{'})) ((v_{ε} - v - ε v_{ε}^{#}) u_{ε} + v (u_{ε} - u - ε u_{ε}^{#})) (t^{'}) d t^{'} \\ (3.3) & + i \int_{0}^{t} (U (ε (t - t^{'})) - I - i ε (t - t^{'}) Δ) (v u) (t^{'}) d t^{'}, \\ v_{ε} (t) - v (t) - ε v_{ε}^{#} (t) \\ (3.4) & = - \int_{0}^{t} (ω sin (t - t^{'}) ω) Re ((\overline{u_{ε}} + \overline{u}) (u_{ε} - u - ε u_{ε}^{#})) (t^{'}) d t^{'} . \end{array}$ We estimate (3.3) and (3.4) in $L^{2}$ as $\begin{array}{l} {‖ u_{ε} (t) - u (t) - ε u_{ε}^{#} (t) ‖}_{2} \\ ⩽ {‖ (U (ε t) - I - i ε t Δ) φ ‖}_{2} \\ + C \int_{0}^{t} ({‖ v_{ε} (t^{'}) - v (t^{'}) - ε v_{ε}^{#} (t^{'}) ‖}_{2} + {‖ u_{ε} (t^{'}) - u (t^{'}) - ε u_{ε}^{#} (t^{'}) ‖}_{2}) d t^{'} \\ (3.5) & + \int_{0}^{t} {‖ (U (ε (t - t^{'})) - I - i ε (t - t^{'}) Δ) (v u) (t^{'}) ‖}_{2} d t^{'}, \\ (3.6) & {‖ v_{ε} (t) - v (t) - ε v_{ε}^{#} (t) ‖}_{2} ⩽ C \int_{0}^{t} {‖ u_{ε} (t^{'}) - u (t^{'}) - ε u_{ε}^{#} (t^{'}) ‖}_{2} d t^{'} \end{array}$ for any $t \in [0, T_{0}]$ , where $T_{0}$ is as in Theorem 1 and we have used the unitarity of $U (ε (t - t^{'}))$ in $L^{2} (Ω)$ , boundedness of $ω sin (t - t^{'}) ω$ in $L^{2} (Ω)$ with bound 1, Sobolev embedding $H^{2} (Ω) ↪ L^{\infty} (Ω)$ , and (1.2), (1.4), and (1.5).

By the Gronwall argument on (3.5) and (3.6), we obtain $\begin{array}{l} sup_{t \in [0, T_{0}]} ({‖ u_{ε} (t) - u (t) - ε u_{ε}^{#} (t) ‖}_{2} + {‖ v_{ε} (t) - v (t) - ε v_{ε}^{#} (t) ‖}_{2}) \\ ⩽ sup_{t \in [0, T_{0}]} ({‖ (U (ε t) - I - i ε t Δ) φ ‖}_{2} \\ + C \int_{0}^{t} e^{C (t - t^{'})} ({‖ (U (ε t) - I - i ε t Δ) φ ‖}_{2} \\ (3.7) & + {‖ (U (ε (t - t^{'})) - I - i ε (t - t^{'}) Δ) (v u) (t^{'}) ‖}_{2}) d t^{'}) \end{array}$ with constant C independent of $ε \in (0, 1]$ and $t \in [0, T_{0}]$ . By (3.7), we obtain $\begin{matrix} (3.8) & sup_{t \in [0, T_{0}]} ({‖ ε^{- 1} (u_{ε} (t) - u (t)) - u_{ε}^{#} (t) ‖}_{2} + {‖ ε^{- 1} (v_{ε} (t) - v (t)) - v_{ε}^{#} (t) ‖}_{2}) \to 0 \end{matrix}$ as $ε ↓ 0$ , since $φ \in D (Δ)$ and $v u \in C (R; D (Δ))$ .

We now turn to the proof of uniform $L^{2}$ -boundedness of $(u_{ε}^{#}, v_{ε}^{#})$ . We estimate (3.1) and (3.2) as $\begin{array}{l} (3.9) & \begin{matrix} {‖ u_{ε}^{#} (t) ‖}_{2} & ⩽ t ‖ Δ φ ‖_{2} + \int_{0}^{t} ({‖ v (t^{'}) ‖}_{\infty} {‖ u_{ε}^{#} (t^{'}) ‖}_{2} + {‖ u_{ε} (t^{'}) ‖}_{\infty} {‖ v_{ε}^{#} (t^{'}) ‖}_{2} \\ + (t - t^{'}) {‖ Δ (v u) (t^{'}) ‖}_{2}) d t^{'} \\ ⩽ C + C \int_{0}^{t} ({‖ u_{ε}^{#} (t^{'}) ‖}_{2} + {‖ v_{ε}^{#} (t^{'}) ‖}_{2}) d t^{'}, \end{matrix} \\ (3.10) & \begin{matrix} {‖ v_{ε}^{#} (t) ‖}_{2} & ⩽ \int_{0}^{t} ({‖ u_{ε} (t^{'}) ‖}_{\infty} + {‖ u (t^{'}) ‖}_{\infty}) {‖ u_{ε}^{#} (t^{'}) ‖}_{2} d t^{'} \\ ⩽ C \int_{0}^{t} {‖ u_{ε}^{#} (t^{'}) ‖}_{2} d t^{'} \end{matrix} \end{array}$ for any $t \in [0, T_{0}]$ , where the constant C is independent of $ε \in (0, 1]$ and $t \in [0, T_{0}]$ and we have used the Sobolev embedding $H^{2} (Ω) ↪ L^{\infty} (Ω)$ , the algebraic structure of $H^{2} (Ω)$ , and (1.2), (1.4), and (1.5). Adding both sides of (3.9) and (3.10) and applying the Gronwall argument on the resulting integral inequality, we obtain $\begin{matrix} (3.11) & sup_{ε \in (0, 1]} sup_{t \in [0, T_{0}]} ({‖ u_{ε}^{#} (t) ‖}_{2} + {‖ v_{ε}^{#} (t) ‖}_{2}) ⩽ C, \end{matrix}$ as was to be shown.

Finally, we write the differences $u_{ε}^{#} - u^{#}$ and $v_{ε}^{#} - v^{#}$ as $\begin{array}{l} (3.12) & \begin{matrix} u_{ε}^{#} (t) - u^{#} (t) & = - i \int_{0}^{t} U (ε (t - t^{'})) (v (u_{ε}^{#} - u^{#}) + u_{ε} (v_{ε}^{#} - v^{#}) \\ + (u_{ε} - u) v^{#}) (t^{'}) d t^{'}, \end{matrix} \\ (3.13) & v_{ε}^{#} (t) - v^{#} (t) = - \int_{0}^{t} (ω sin (t - t^{'}) ω) Re ((\overline{u_{ε}} - \overline{u}) u_{ε}^{#} + 2 \overline{u} (u_{ε}^{#} - u^{#})) (t^{'}) d t^{'} . \end{array}$ By the unitarity of $U (ε (t - t^{'}))$ in $L^{2} (Ω)$ , the boundedness of $ω sin (t - t^{'}) ω$ in $L^{2} (Ω)$ with bound 1, the Sobolev embedding $H^{2} (Ω) ↪ L^{\infty} (Ω)$ , and (1.2), (1.4), (1.5), and (3.11), we estimate (3.12) and (3.13) as $\begin{array}{l} (3.14) & \begin{matrix} {‖ u_{ε}^{#} (t) - u^{#} (t) ‖}_{2} & ⩽ C \int_{0}^{t} ({‖ u_{ε}^{#} (t^{'}) - u^{#} (t^{'}) ‖}_{2} + {‖ v_{ε}^{#} (t^{'}) - v^{#} (t^{'}) ‖}_{2} \\ + {‖ u_{ε} (t^{'}) - u (t^{'}) ‖}_{H^{2}}) d t^{'}, \end{matrix} \\ (3.15) & {‖ v_{ε}^{#} (t) - v^{#} (t) ‖}_{2} = C \int_{0}^{t} ({‖ u_{ε} (t^{'}) - u (t^{'}) ‖}_{H^{2}} + {‖ u_{ε}^{#} (t^{'}) - u^{#} (t^{'}) ‖}_{2}) d t^{'} \end{array}$ for any $t \in [0, T_{0}]$ . Adding both sides of (3.14) and (3.15) and applying the resulting integral inequality, we have $\begin{array}{l} sup_{ε \in (0, 1]} sup_{t \in [0, T_{0}]} ({‖ u_{ε}^{#} (t) - u^{#} (t) ‖}_{2} + {‖ v_{ε}^{#} (t) - v^{#} (t) ‖}_{2}) \\ (3.16) & ⩽ C \int_{0}^{T_{0}} {‖ u_{ε} (t) - u (t) ‖}_{H^{2}} d t . \end{array}$ The RHS of (3.16) tends to zero as $ε ↓ 0$ , due to (1.3). Therefore, (1.8) follows from (3.8) and (3.16). This completes the proof of Theorem 2.

Footnotes

Acknowledgements

The authors thank the referee for important comments. This work is partially supported by Grant-in Aid for Scientific Research (A) 19H00644, JSPS.

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