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The initial-boundary value problem for the Korteweg–de Vries equation. (English) Zbl 1111.35062

The author studies the following initial-boundary value problem for the Korteweg-de Vries equation on the right (left) half-plane \(\mathbb{R}^+\) \((\mathbb{R}^-)\):
\[ \begin{aligned} \partial_tu+\partial_t^3u+u\partial_xu&=0,\, (x,t)\in (0,+\infty)\times (0,T)\, ((x,t)\in (-\infty,0)\times (0,T)); \\ u(0,t)&=f(t),\, t\in (0,T)\, (g_1(t),\; t\in (0,T)) \\ u(x,0)&=\varphi(x),\,x\in (0,+\infty)\, (\varphi(x),\,x\in (-\infty,0)) \\ (\partial_x u(0,t)&=g_2(t),\,t\in (0,T)) \end{aligned} \tag{\(1^+(1^-)\)} \]
and on the segment \(0<x<L\):
\[ \begin{aligned} \partial_tu+\partial_x^3u+u\partial_xu&=0,\, (x,t)\in (0,L)\times (0,T); \\ u(0,T)&=f(t),\, u(L,t)=g_1(t),\, \partial_xu(L,t)=g_2(t),\, t\in (0,T); \\ u(x,0)&=\varphi(x),\,x\in (0,L). \end{aligned} \tag{2} \]
On \(\mathbb{R}\) the \(L^2\)-based inhomogeneous Sobolev spaces \(H^s=H^s(\mathbb{R})\) are defined by the norm \(\| \varphi\| _{H^s}=\| \langle \xi \rangle^s \hat{\varphi}(\xi) \| _{L^2_{\xi}}\), where \(\langle\xi\rangle=(1+| \xi| ^2)^{1/2}\) and \(e^{-t\partial_x^3}\) denotes the linear homogeneous solution group on \(\mathbb{R}\) defined by \(e^{-t\partial_x^3}\varphi(x)=\frac{1}{2\pi}\int_{\xi}e^{it\xi^3}\hat{\varphi}(\xi)d\xi\), so that \((\partial_t+\partial_x^3)e^{-t\partial_x^3}\varphi(x)=0\) and \(e^{-t\partial_x^3}\varphi(x)|_{t=0}=\varphi(x)\). Then \((1^+)\) is considered for \(-\frac{3}{4}<s<\frac32\), \(s\neq\frac12\) in the setting
\[ \varphi\in H^s(\mathbb{R}^+),\, f\in H^{\frac{s+1}{3}}(\mathbb{R}^+)\quad\text{and for}\quad \frac{1}{2}<s<\frac{3}{2},\, \varphi(0)=f(0) \tag{3} \] \((1^-)\) – in the setting \[ \varphi\in H^s(\mathbb{R}^-),\, g_1\in H^{\frac{s+1}{3}}(\mathbb{R}^+),\, g_2\in H^{\frac{s}{3}}(\mathbb{R}^+)\quad\text{and for}\quad \frac12<s<\frac32,\,\varphi(0)=g_1(0) \tag{4} \] and (2) – in the setting
\[ \begin{aligned} \varphi\in H^s(0,L),&\quad f\in H^{\frac{s+1}{3}}(\mathbb{R}^+),\, \quad g_1\in H^{\frac{s+1}{3}}(\mathbb{R}^+),\quad g_2\in H^{\frac{s}{3}}(\mathbb{R}^+),\\ \text{and for }&\frac{1}{2}<s<\frac{3}{2},\, \varphi(0)=f(0),\, \varphi(L)=g_1(0) \end{aligned}\tag{5} \] The function \(u(x,t)\) is a distributional solution of \((1^+)\), (3) (\((1^-)\), (4)) on \([0,T]\) if:
(a)
well-defined nonlinearity: \(u\) belongs to some space \(X\) with the property \(u\in X\Rightarrow\partial_xu^2\) is a well-defined distribution;
(b)
\(u(x,t)\) satisfies the equation \((1^+)\) (\((1^-)\)) in the sense of distributions;
(c)
space traces: \(u\in C([0,T]; H_x^s)\) and in this sense \(u(\cdot,0)=\varphi\) in \(H^s(\mathbb{R}^+)\) (in \(H^s(\mathbb{R}^-)\));
(d)
time traces: \(u\in C(\mathbb{R}_x;H^{\frac{s+1}{3}}(0,T))\) and in this sense \(u(0,\cdot)=f\) in \(H^{\frac{s+1}{3}}(0,T)\) (\(u(0,\cdot)=g_1\) in \(H^{\frac{s+1}{3}}(0,T)\));
(e)
derivative time traces: \(\partial_xu\in C(\mathbb{R}_x; H^{\frac{s}{3}}(0,T))\) and only for \((1^-)\)(4) in the sense \(u(0,\cdot)=g_2\) in \(H^{\frac{s}{3}}(0,T)\).
The function \(u(x,t)\) is a mild solution of \((1^+)\) (\((1^-)\)) on \((0,T)\), if \(\exists\) a sequence \[ \{u_n\}\in C([0,T]; H^3(\mathbb{R}^+_x))\cap C^1([0,T]; L^2(\mathbb{R}^+_x)) \] such that:
(a)
\(u_n(x,t)\) solves \((1^+)\) in \(L^2(\mathbb{R}^+_x)\) (\((1)^-\) in \(L^2(\mathbb{R}^-_x)\)) for \(0<t<T\);
(b)
\(\lim_{n\to+\infty}\| u_n-u \| _{C([0,T]; H^s(\mathbb{R}^+_x))}=0\) (\(\lim_{n\to+\infty}\| u_n-u \| _{C([0,T]; H^s(\mathbb{R}^-_x))}=0\));
(c)
\(\lim_{n\to+\infty}\| u_n(0,\cdot)-f \| _{H^{\frac{s+1}{3}}(0,T)}=0 (\lim_{n\to+\infty}\| u_n(0,\cdot)-g_1 \| _{H^{\frac{s+1}{3}}(0,T)}=0, \newline \lim_{n\to\infty}\| \partial_xu_n(0,\cdot)-g_2 \| _{H^{\frac{s}{3}}(0,T)}=0\)).
The main result is contained in the following theorem: Let \(-\frac{3}{4}<s<\frac{3}{2}\); (a) Given \((\varphi,f)\) satisfying (3). \(\exists\) \(T>0\) depending only on the norms of \(\varphi\), \(f\) in (3) and \(\exists\) \(u(x,t)\) that is both a mild and distributional solution to (\(1^+\)) (3) on \([0,T]\); (b) Given \((\varphi,g_1,g_2)\) satisfying (4). \(\exists\) \(T>0\) depending only on the norms of \(\varphi\), \(g_1\), \(g_2\) in (4) and \(\exists\) \(u(x,t)\) that is both a mild and distributional solution to \((1^-)\)(4) on \([0,T]\); (c) Given \((\varphi,f,g_1,g_2)\) satisfying (5). \(\exists\) \(T>0\) depending only on the norms of \(\varphi\), \(f\), \(g_1\), \(g_2\) in (5) and \(\exists\) \(u(x,t)\) that is both a mild and distributional solutions to (2), (5) on \((0,T)\).
The main feature of this work is the low regularity requirements for \(\varphi\) and \(f\).

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35A07 Local existence and uniqueness theorems (PDE) (MSC2000)
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References:

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