Growth of Sobolev norms for the Maxwell–Dirac and Dirac–Klein

Abstract

We obtain the growth of Sobolev norms of the solution to the Maxwell–Dirac equations in $R^{1 + 1}$ by applying elementary techniques. In particular, we estimate $L^{\infty}$ bound of the solution by making use of a local energy conservation. The similar idea can be applied to the Dirac–Klein–Gordon equations.

Keywords

Maxwell–Dirac Dirac–Klein–Gordon growth of Sobolev norms

1. Introduction

We are interested in the growth of Sobolev norms of the solution to the Maxwell–Dirac equations in $R^{1 + 1}$ . $\begin{array}{l} (1.1) & \begin{matrix} \partial_{t} U_{1} + \partial_{x} U_{1} + i M U_{2} = i A_{0} U_{1} + i A_{1} U_{1}, \\ \partial_{t} U_{2} - \partial_{x} U_{2} + i M U_{1} = i A_{0} U_{2} - i A_{1} U_{2}, \\ \partial_{t t} A_{0} - \partial_{x x} A_{0} = - (| U_{1} |^{2} + | U_{2} |^{2}), \\ \partial_{t t} A_{1} - \partial_{x x} A_{1} = | U_{1} |^{2} - | U_{2} |^{2} . \end{matrix} \end{array}$ The unknowns are the Dirac spinor field $U_{j} : R^{1 + 1} \to C$ and electromagnetic potential components $A_{μ} : R^{1 + 1} \to R$ . $M > 0$ is a mass constant. The system (1.1) is complemented with initial data $\begin{array}{l} U_{j} (x, 0) = u_{j} (x) \in H^{s} (R), A_{μ} (x, 0) = a_{μ} (x) \in H^{r} (R), \\ \partial_{t} A_{μ} (x, 0) = b_{μ} (x) \in H^{r - 1} (R), \end{array}$ where $H^{s} = {(1 - \partial_{x}^{2})}^{- s / 2} L^{2} (R)$ is the standard Sobolev space of order s. We also use the homogeneous Sobolev space $\begin{matrix} {‖ | \nabla |^{s} f ‖}_{L^{2} (R)} = {(\int_{R} | ξ |^{2 s} {| \hat{f} (ξ) |}^{2} d ξ)}^{1 / 2}, \end{matrix}$ where $\hat{f}$ is Fourier transform of f.

The initial value problem of the system (1.1) has been studied in [2,4,6,8,12]. The global well-posedness of the system (1.1) with $s = r = 1$ was proved in [2] by using the conservation of $L^{2}$ norm. An explicit representation of solutions was found in [4] for the massless case $M = 0$ . It was proved in [6] that the system (1.1) is locally well-posed for $s > 0$ , $r > \frac{1}{2}$ , $s ⩽ r ⩽ min (s + 1, 2 s + \frac{1}{2})$ . The global well-posedness of the coupled Maxwell–Dirac-Thirring-Gross–Neveu equations was proved with data for the Dirac spinor in the critical space $L^{2} (R)$ .

If we apply the usual energy estimate, the $L^{2}$ conservation and Gronwall’s inequality to the solution of (1.1), then the following exponential bound can be obtained. $\begin{matrix} {‖ \partial_{x} U_{j} (\cdot, t) ‖}_{L^{2} (R)} ≲ exp (t^{2}) . \end{matrix}$ We are interested in obtaining a polynomial bound which is far less than exponential one. Making use of the local charge conservation, we estimate $L^{\infty}$ norm of $U_{j}$ . Then we obtain growth of $H^{s}$ norm of the solution to (1.1).

Theorem 1.1.
For the initial data $u_{j} \in H^{1}$ and $a_{μ} \in H^{2}$ , $b_{μ} \in H^{1}$ , the global solution of ( 1.1 ) satisfies the bound $\begin{array}{l} {‖ | \nabla |^{α} U_{j} (\cdot, t) ‖}_{L^{2} (R)} ≲_{D} {(1 + t)}^{3 α} for 0 ⩽ α ⩽ 1, \\ {‖ | \nabla |^{β} A_{μ} (\cdot, t) ‖}_{L^{2} (R)} ≲_{D} {(1 + t)}^{3 β - \frac{3}{2}} for 1 ⩽ β ⩽ 2, \end{array}$ where $A ≲_{D} B$ is used for denoting initial data dependence.

We also consider the following Dirac–Klein–Gordon equations in $R^{1 + 1}$ . $\begin{array}{l} (1.2) & \begin{matrix} \partial_{t} ψ_{1} + \partial_{x} ψ_{1} + i M ψ_{2} = i ϕ ψ_{2}, \\ \partial_{t} ψ_{2} - \partial_{x} ψ_{2} + i M ψ_{1} = i ϕ ψ_{1}, \\ \partial_{t t} ϕ - \partial_{x x} ϕ + m^{2} ϕ = 2 Re (ψ_{1} {\bar{ψ}}_{2}), \end{matrix} \end{array}$ where $ψ_{j} : R^{1 + 1} \to C$ , $ϕ : R^{1 + 1} \to R$ and $M, m > 0$ are constants. The above system is complemented with initial data $\begin{array}{l} (1.3) & ψ_{j} |_{t = 0} = ψ_{j 0} \in H^{s}, (ϕ, \partial_{t} ϕ) |_{t = 0} = (ϕ_{0}, ϕ_{1}) \in H^{r} \times H^{r - 1} . \end{array}$

The local and global well-posedness of the system (1.2) has been studied in [1,7,10]. In particular, it was proved in [10] that for initial data (1.3) with Sobolev exponent $(s, r) \in R$ , where $\begin{matrix} R = {(s, r) \in R \times R : s ⩾ 0, r > 0, s ⩽ r ⩽ 1 + s}, \end{matrix}$ the system (1.2) admits a global solution $\begin{matrix} (ψ_{1}, ψ_{2}, ϕ, \partial_{t} ϕ) \in C ([0, \infty); H^{s} \times H^{s} \times H^{r} \times H^{r - 1}) . \end{matrix}$ Growth-in-time of higher Sobolev norms of solutions to the system (1.2) was studied in [11]. Let us use the notation $ψ = {(ψ_{1}, ψ_{2})}^{T}$ . Several bounds for $‖ ψ ‖_{H^{s}}$ and $‖ ϕ ‖_{H^{r}}$ were obtained according to $(s, r) \in R$ . They used the so called upside-down I-method combined with an energy estimate in $X^{s, b}$ spaces, null form estimates and the product law of Sobolev spaces. Let us recall some related results in [11].

$(1)$ For the initial data $ψ_{j 0} \in H^{1}$ and $ϕ \in H^{r}$ ( $1 < r ⩽ 2$ ), the global solution of (1.2) satisfies the bound $\begin{array}{l} {‖ ψ (t) ‖}_{H^{1}} ≲_{D} {(1 + t)}^{5 +} and {‖ ϕ (t) ‖}_{H^{r}} ≲_{D} {(1 + t)}^{5 r - \frac{3}{2} +}, \end{array}$ where the notation $a + : = a + ε$ is used for a sufficiently small $ε > 0$ .

$(2)$ Assume that $(ψ_{10}, ψ_{20}, ϕ_{0}, ϕ_{1})$ belong to the Schwarz class. Then the global solution of (1.2) satisfies the bound $\begin{array}{l} {‖ ψ (t) ‖}_{H^{1}} ≲_{D} {(1 + t)}^{5 +} and {‖ ϕ (t) ‖}_{H^{1}} ≲_{D} {(1 + t)}^{5 - \frac{3}{2} +} . \end{array}$ Making use of the local charge conservation, we estimate $L^{\infty}$ norm for $ψ_{j}$ and ϕ. Then we consider the growth of $H^{s}$ norm of the solution to (1.2).
Theorem 1.2.
For the initial data $ψ_{j 0} \in H^{1}$ and $ϕ_{0} \in H^{2}$ , $ϕ_{1} \in H^{1}$ , the global solution of ( 1.2 ) satisfies the bound $\begin{array}{l} {‖ | \nabla |^{α} ψ (t) ‖}_{L^{2}} ≲_{D} {(1 + t)}^{5 α} for 0 ⩽ α ⩽ 1, \\ {‖ | \nabla |^{β} ϕ (t) ‖}_{L^{2}} ≲_{D} {(1 + t)}^{5 β - \frac{5}{2}} for 1 ⩽ β ⩽ 2 . \end{array}$
Remark 1.3.
1. We can obtain, as in [8], growth of $‖ | \nabla |^{β} ϕ (t) ‖_{L^{2}}$ for $0 ⩽ β < 1$ by applying the energy estimate and the product estimate $\begin{matrix} ‖ f g ‖_{H^{- c}} ⩽ c ‖ f ‖_{H^{a}} ‖ g ‖_{H^{b}} . \end{matrix}$ 2. As far as we know, there is no known result on the optimality of α, β in Theorem 1.1 and 1.2. We remark that there may be room for improvement.

We prove Theorem 1.1 and Theorem 1.2 in Section 2 and 3 respectively. We conclude this section by giving a few notations. We use c to denote various constants and $A ≲ B$ to denote an estimate of the form $A ⩽ c B$ . Especially, $A ≲_{D} B$ is used for denoting initial data dependence.
2. Proof of Theorem 1.1

The Maxwell–Dirac system (1.1) can be rewritten as $\begin{array}{l} (2.1) & D_{0} U_{1} + D_{1} U_{1} + i M U_{2} = 0, \\ (2.2) & D_{0} U_{2} - D_{1} U_{2} + i M U_{1} = 0, \\ (2.3) & \partial_{t t} A_{0} - \partial_{x x} A_{0} = - (| U_{1} |^{2} + | U_{2} |^{2}), \\ (2.4) & \partial_{t t} A_{1} - \partial_{x x} A_{1} = | U_{1} |^{2} - | U_{2} |^{2} 2, \end{array}$ where the covariant derivatives are defined by $\begin{matrix} D_{μ} U_{j} = \partial_{μ} U_{j} - i A_{μ} U_{j} . \end{matrix}$ The spacetime derivatives are denoted by $\partial_{0} = \partial_{t}$ , $\partial_{1} = \partial_{x}$ . The Maxwell–Dirac system enjoys gauge invariance and the system (2.1)–(2.4) appears when the Lorenz gauge condition $\partial_{t} A_{0} - \partial_{x} A_{1} = 0$ is imposed. We refer to [4,9] for the formulation (2.1)–(2.4).

It is well known that (2.1)–(2.4) obey the conservation of $L^{2}$ norm: $\begin{array}{l} (2.5) & \int_{R} {| U_{1} (x, t) |}^{2} + {| U_{2} (x, t) |}^{2} d x = \int_{R} {| u_{1} (x) |}^{2} + {| u_{2} (x) |}^{2} d x : = L . \end{array}$ Using the idea in [5], we derive local version of conservation of $L^{2}$ norm. Multiplying (2.1), (2.2) by ${\bar{U}}_{1}$ , ${\bar{U}}_{2}$ respectively and taking the real part, we have $\begin{array}{l} (2.6) & \begin{matrix} (\partial_{t} + \partial_{x}) | U_{1} |^{2} & = 2 M Im ({\bar{U}}_{1} U_{2}), \\ (\partial_{t} - \partial_{x}) | U_{2} |^{2} & = - 2 M Im ({\bar{U}}_{1} U_{2}), \end{matrix} \end{array}$ where we used the basic algebra associated with covariant derivative. $\begin{array}{l} \partial_{α} (U \bar{V}) = D_{α} U \bar{V} + U \overline{D_{α} V}, \end{array}$ where U and V are complex-valued functions. Then we have $\begin{array}{l} (2.7) & \partial_{t} (| U_{1} |^{2} + | U_{2} |^{2}) + \partial_{x} (| U_{1} |^{2} - | U_{2} |^{2}) = 0 . \end{array}$ Integrating (2.7) on the domain $\begin{array}{l} Ω (x_{0}, t_{0}) = {(x, t) | 0 < t < t_{0}, x_{0} - t_{0} + t < x < x_{0} + t_{0} - t}, \end{array}$ we have by applying Green’s Theorem $\begin{array}{l} (2.8) & \begin{matrix} 2 \int_{0}^{t_{0}} | U_{1} |^{2} (x_{0} + t_{0} - s, s) d s + 2 \int_{0}^{t_{0}} | U_{2} |^{2} (x_{0} - t_{0} + s, s) d s \\ = \int_{x_{0} - t_{0}}^{x_{0} + t_{0}} {| u_{1} (s) |}^{2} + {| u_{2} (s) |}^{2} d s ⩽ L . \end{matrix} \end{array}$ Taking into account $\begin{matrix} \frac{d}{d t} | U_{1} |^{2} (x + t, t) = \partial_{t} | U_{1} |^{2} + \partial_{x} | U_{1} |^{2}, \frac{d}{d t} | U_{2} |^{2} (x - t, t) = \partial_{t} | U_{2} |^{2} - \partial_{x} | U_{2} |^{2}, \end{matrix}$ we integrate (2.6) along the outgoing and ingoing characteristics to obtain $L^{\infty}$ bounds of $U_{j}$ as follows. $\begin{array}{l} (2.9) & \begin{matrix} | U_{1} (x, t) | ⩽ | u_{1} (x - t) | + M t^{\frac{1}{2}} {(\int_{0}^{t} {| U_{2} (x - t + s, s) |}^{2} d s)}^{\frac{1}{2}}, \\ ≲_{D} {(1 + t)}^{\frac{1}{2}}, \\ | U_{2} (x, t) | ≲_{D} {(1 + t)}^{\frac{1}{2}}, \end{matrix} \end{array}$ where (2.8) is used.

Now we are ready to prove Theorem 1.1. Combined with (2.5) and (2.9), the energy estimates for (2.3), (2.4) lead us to $\begin{array}{l} ‖ \partial_{x} A ‖_{L^{2}} & ≲_{D} 1 + \int_{0}^{t} ‖ U ‖_{L^{\infty}} ‖ U ‖_{L^{2}} d s \\ ≲_{D} {(1 + t)}^{\frac{3}{2}}, \end{array}$ where we denote $‖ \partial_{x} U ‖_{L^{2}} = ‖ \partial_{x} U_{1} ‖_{L^{2}} + ‖ \partial_{x} U_{2} ‖_{L^{2}}$ and $‖ \partial_{x} A ‖_{L^{2}} = ‖ \partial_{x} A_{0} ‖_{L^{2}} + ‖ \partial_{x} A_{1} ‖_{L^{2}}$ . Taking a spatial derivative $\partial_{x}$ , multiplying $\partial_{x} {\overline{U}}_{1}$ , $\partial_{x} {\overline{U}}_{2}$ on (2.1), (2.2) respectively and taking real parts, we have $\begin{array}{l} (2.10) & \begin{matrix} \partial_{t} | \partial_{x} U_{1} |^{2} + \partial_{x} | \partial_{x} U_{1} |^{2} \\ = 2 M Im (\partial_{x} {\overline{U}}_{1} \partial_{x} U_{2}) + 2 Im (\partial_{x} (A_{0} {\overline{U}}_{1}) \partial_{x} U_{1}) + 2 Im (\partial_{x} (A_{1} {\overline{U}}_{1}) \partial_{x} U_{1}), \\ \partial_{t} | \partial_{x} U_{2} |^{2} - \partial_{x} | \partial_{x} U_{2} |^{2} \\ = 2 M Im (\partial_{x} {\overline{U}}_{2} \partial_{x} U_{1}) + 2 Im (\partial_{x} (A_{0} {\overline{U}}_{2}) \partial_{x} U_{2}) + 2 Im (\partial_{x} (A_{1} {\overline{U}}_{2}) \partial_{x} U_{2}) . \end{matrix} \end{array}$ Note that $\begin{array}{l} Im (\partial_{x} {\overline{U}}_{1} \partial_{x} U_{2}) + Im (\partial_{x} {\overline{U}}_{2} \partial_{x} U_{1}) = 0, \\ Im (\partial_{x} (A_{μ} {\overline{U}}_{k}) \partial_{x} U_{k}) = Im (\partial_{x} A_{μ} {\overline{U}}_{k} \partial_{x} U_{k}), \end{array}$ where we use $Im (A_{μ} | \partial_{x} U_{k} |^{2}) = 0$ for $μ = 0, 1$ and $k = 1, 2$ . Then adding two equations in (2.10) and integrating on $R$ , we have $\begin{array}{l} \frac{d}{d t} \int_{R} | \partial_{x} U_{1} |^{2} + | \partial_{x} U_{2} |^{2} d x \\ ≲ ‖ \partial_{x} U_{1} ‖_{L^{2}} ‖ \partial_{x} A ‖_{L^{2}} ‖ U_{1} ‖_{L^{\infty}} + ‖ \partial_{x} U_{2} ‖_{L^{2}} ‖ \partial_{x} A ‖_{L^{2}} ‖ U_{2} ‖_{L^{\infty}}, \end{array}$ which leads us to $\begin{array}{l} (2.11) & ‖ \partial_{x} U_{1} ‖_{L^{2}} + ‖ \partial_{x} U_{2} ‖_{L^{2}} ≲_{D} {(1 + t)}^{3}, \end{array}$ where (2.5) and (2.9) are used. Considering (2.9) and (2.11), we have $\begin{array}{l} {‖ \partial_{x}^{2} A ‖}_{L^{2}} & ≲_{D} 1 + \int_{0}^{t} ‖ U ‖_{\infty} ‖ \partial_{x} U ‖_{L^{2}} \\ ≲_{D} {(1 + t)}^{\frac{9}{2}} . \end{array}$ Making use of interpolation inequality, we have $\begin{array}{l} {‖ | \nabla |^{α} U ‖}_{L^{2}} ≲ ‖ U ‖_{L^{2}}^{1 - α} ‖ \partial_{x} U ‖_{L^{2}}^{α} ≲_{D} {(1 + t)}^{3 α} for 0 ⩽ α ⩽ 1, \\ {‖ | \nabla |^{β} A ‖}_{L^{2}} ≲ ‖ \partial_{x} A ‖_{L^{2}}^{2 - β} {‖ \partial_{x}^{2} A ‖}_{L^{2}}^{β - 1} ≲_{D} {(1 + t)}^{3 β - \frac{3}{2}} for 1 ⩽ β ⩽ 2 . \end{array}$

3. Proof of Theorem 1.2

The Dirac–Klein–Gordon equations in $R^{1 + 1}$ can be written as $\begin{array}{l} (3.1) & \partial_{t} ψ_{1} + \partial_{x} ψ_{1} + i M ψ_{2} = i ϕ ψ_{2}, \\ (3.2) & \partial_{t} ψ_{2} - \partial_{x} ψ_{2} + i M ψ_{1} = i ϕ ψ_{1}, \\ (3.3) & \partial_{t t} ϕ - \partial_{x x} ϕ + m^{2} ϕ = 2 Re (ψ_{1} {\bar{ψ}}_{2}), \end{array}$ where $ψ_{j} : R^{1 + 1} \to C$ , $ϕ : R^{1 + 1} \to R$ and $M, m > 0$ are constants. It is well known that (3.1)–(3.3) obey the conservation of $L^{2}$ norm. $\begin{array}{l} (3.4) & \int_{R} {| ψ_{1} (x, t) |}^{2} + {| ψ_{2} (x, t) |}^{2} d x = \int_{R} {| ψ_{10} (x) |}^{2} + {| ψ_{20} (x) |}^{2} d x : = L . \end{array}$

Considering the solution formula of Klein–Gordon equation (see [3, Section 11.2]), the solution ϕ in (3.3) can be given by $\begin{array}{l} ϕ (x, t) = ϕ_{lin} + \int_{0}^{t} \int_{x - t + τ}^{x + t - τ} J_{0} (m \sqrt{{(t - τ)}^{2} - {(x - ξ)}^{2}}) Re (ψ_{1} {\bar{ψ}}_{2}) d ξ d τ, \end{array}$ where $J_{0}$ is Bessel function of oder 0 and $ϕ_{lin}$ is the solution to the linear equation $\begin{array}{l} \partial_{t t} ϕ - \partial_{x x} ϕ + m^{2} ϕ = 0, (ϕ, \partial_{t} ϕ) |_{t = 0} = (ϕ_{0}, ϕ_{1}) . \end{array}$ Then we have, combined with (3.4), $\begin{array}{l} (3.5) & \begin{matrix} | ϕ (x, t) | & ≲ | ϕ_{lin} | + \int_{0}^{t} \int_{x - t + τ}^{x + t - τ} | ψ_{1} |^{2} + | ψ_{2} |^{2} d ξ d τ \\ ≲_{D} 1 + t . \end{matrix} \end{array}$

We derive local version of conservation of $L^{2}$ norm. Multiplying (3.1), (3.2) by ${\bar{ψ}}_{1}$ , ${\bar{ψ}}_{2}$ respectively and taking the real part, we have $\begin{array}{l} (\partial_{t} + \partial_{x}) | ψ_{1} |^{2} = 2 M Im ({\bar{ψ}}_{1} ψ_{2}) - 2 ϕ Im ({\bar{ψ}}_{1} ψ_{2}), \\ (\partial_{t} - \partial_{x}) | ψ_{2} |^{2} = 2 ϕ Im ({\bar{ψ}}_{1} ψ_{2}) - 2 M Im ({\bar{ψ}}_{1} ψ_{2}), \end{array}$ from which we have $\begin{array}{l} \partial_{t} (| ψ_{1} |^{2} + | ψ_{2} |^{2}) + \partial_{x} (| ψ_{1} |^{2} - | ψ_{2} |^{2}) = 0 . \end{array}$ Applying the same argument as in Section 2, we have $\begin{array}{l} (3.6) & \begin{matrix} 2 \int_{0}^{t_{0}} | ψ_{1} |^{2} (x_{0} + t_{0} - s, s) d s + 2 \int_{0}^{t_{0}} | ψ_{2} |^{2} (x_{0} - t_{0} + s, s) d s \\ = \int_{x_{0} - t_{0}}^{x_{0} + t_{0}} ({| ψ_{10} (s) |}^{2} + {| ψ_{20} (s) |}^{2}) d s ⩽ L . \end{matrix} \end{array}$ Integrating (3.1), (3.2) along the characteristic lines, we have $\begin{array}{l} (3.7) & \begin{matrix} | ψ_{1} (x, t) | ⩽ | ψ_{10} (x - t) | + \int_{0}^{t} (M | ψ_{2} | + | ϕ ψ_{2} |) (x - t + s, s) d s \\ ≲ | ψ_{10} (x - t) | + M L^{\frac{1}{2}} t^{\frac{1}{2}} + L^{\frac{1}{2}} {(\int_{0}^{t} {| ϕ (x - t + s, s) |}^{2} d s)}^{\frac{1}{2}} \\ ≲_{D} {(1 + t)}^{\frac{3}{2}}, \\ | ψ_{2} (x, t) | ≲_{D} {(1 + t)}^{\frac{3}{2}}, \end{matrix} \end{array}$ where we used (3.5) and (3.6).

Now we are ready to prove Theorem 1.2. We make use of energy estimate of (3.3) to obtain $\begin{array}{l} (3.8) & \begin{matrix} ‖ \partial_{x} ϕ ‖_{L^{2}} & ≲ ‖ \partial_{x} ϕ_{0} ‖_{L^{2}} + ‖ ϕ_{1} ‖_{L^{2}} + \int_{0}^{t} {‖ Re (ψ_{1} {\bar{ψ}}_{2}) ‖}_{L^{2}} d s \\ ≲ ‖ \partial_{x} ϕ_{0} ‖_{L^{2}} + ‖ ϕ_{1} ‖_{L^{2}} + \int_{0}^{t} ‖ ψ ‖_{L^{\infty}} ‖ ψ ‖_{L^{2}} d s \\ ≲_{D} {(1 + t)}^{\frac{5}{2}}, \end{matrix} \end{array}$ where (3.7) is used. Taking $\partial_{x}$ and multiplying $\partial_{x} {\bar{ψ}}_{1}$ , $\partial_{x} {\bar{ψ}}_{2}$ on (3.1), (3.2) respectively, we have $\begin{array}{l} (3.9) & \begin{matrix} \partial_{t} | \partial_{x} ψ_{1} |^{2} + \partial_{x} | \partial_{x} ψ_{1} |^{2} = 2 M Im (\partial_{x} {\overline{ψ}}_{1} \partial_{x} ψ_{2}) + 2 Im (ϕ \partial_{x} {\overline{ψ}}_{2} \partial_{x} ψ_{1}) + 2 Im (\partial_{x} ϕ {\overline{ψ}}_{2} \partial_{x} ψ_{1}), \\ \partial_{t} | \partial_{x} ψ_{2} |^{2} - \partial_{x} | \partial_{x} ψ_{2} |^{2} = 2 M Im (\partial_{x} {\overline{ψ}}_{2} \partial_{x} ψ_{1}) + 2 Im (ϕ \partial_{x} {\overline{ψ}}_{1} \partial_{x} ψ_{2}) + 2 Im (\partial_{x} ϕ {\overline{ψ}}_{1} \partial_{x} ψ_{2}) . \end{matrix} \end{array}$ Noting that $\begin{array}{l} Im (\partial_{x} {\overline{ψ}}_{1} \partial_{x} ψ_{2}) + Im (\partial_{x} {\overline{ψ}}_{2} \partial_{x} ψ_{1}) = 0, \\ Im (ϕ \partial_{x} {\overline{ψ}}_{2} \partial_{x} ψ_{1}) + Im (ϕ \partial_{x} {\overline{ψ}}_{1} \partial_{x} ψ_{2}) = 0, \end{array}$ we have from (3.9) $\begin{array}{l} \partial_{t} (| \partial_{x} ψ_{1} |^{2} + | \partial_{x} ψ_{2} |^{2}) + \partial_{x} (| \partial_{x} ψ_{1} |^{2} - | \partial_{x} ψ_{2} |^{2}) \\ = 2 \partial_{x} ϕ Im ({\bar{ψ}}_{1} \partial_{x} ψ_{2}) + 2 \partial_{x} ϕ Im ({\bar{ψ}}_{2} \partial_{x} ψ_{1}) . \end{array}$ Integrating on $R$ , we arrive at $\begin{array}{l} \frac{d}{d t} \int | \partial_{x} ψ_{1} |^{2} + | \partial_{x} ψ_{2} |^{2} d x ⩽ 2 ‖ \partial_{x} ϕ ‖_{L^{2}} (‖ ψ_{1} ‖_{L^{\infty}} ‖ \partial_{x} ψ_{2} ‖_{L^{2}} + ‖ ψ_{2} ‖_{L^{\infty}} ‖ \partial_{x} ψ_{1} ‖_{L^{2}}), \end{array}$ which leads to $\begin{array}{l} (3.10) & \begin{matrix} ‖ \partial_{x} ψ_{1} ‖_{L^{2}} + ‖ \partial_{x} ψ_{2} ‖_{L^{2}} \\ ≲ ‖ \partial_{x} ψ_{10} ‖_{L^{2}} + ‖ \partial_{x} ψ_{20} ‖_{L^{2}} + \int_{0}^{t} ‖ \partial_{x} ϕ ‖_{L^{2}} (‖ ψ_{1} ‖_{L^{\infty}} + ‖ ψ_{2} ‖_{L^{\infty}}) d s \\ ≲_{D} {(1 + t)}^{5}, \end{matrix} \end{array}$ where (3.7) and (3.8) are used. Make use of (3.7) and (3.10), we obtain $\begin{array}{l} {‖ \partial_{x}^{2} ϕ ‖}_{L^{2}} & ≲ {‖ \partial_{x}^{2} ϕ_{0} ‖}_{L^{2}} + ‖ \partial_{x} ϕ_{1} ‖_{L^{2}} + \int_{0}^{t} {‖ \partial_{x} Re (ψ_{1} {\bar{ψ}}_{2}) ‖}_{L^{2}} d s \\ ≲ {‖ \partial_{x}^{2} ϕ_{0} ‖}_{L^{2}} + ‖ \partial_{x} ϕ_{1} ‖_{L^{2}} + \int_{0}^{t} ‖ ψ ‖_{L^{\infty}} ‖ \partial_{x} ψ ‖_{L^{2}} d s \\ ≲_{D} {(1 + t)}^{\frac{15}{2}} . \end{array}$ Considering interpolation inequality, we have $\begin{array}{l} {‖ | \nabla |^{α} ψ (t) ‖}_{L^{2}} ≲_{D} {(1 + t)}^{5 α} for 0 ⩽ α ⩽ 1, \\ {‖ | \nabla |^{β} ϕ (t) ‖}_{L^{2}} ≲_{D} {(1 + t)}^{5 β - \frac{5}{2}} for 1 ⩽ β ⩽ 2 . \end{array}$

Footnotes

Acknowledgement

This research was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government(MSIT) (2020R1F1A1A01072197).

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