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| Convergence of Solutions of General Dispersive Equations Along Curve |
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Citation： |
Yong DING,Yaoming NIU.Convergence of Solutions of General Dispersive Equations Along Curve[J].Chinese Annals of Mathematics B,2019,40(3):363~388 |
| Page view： 836
Net amount： 715 |
Authors： |
Yong DING; Yaoming NIU |
Foundation： |
This work was supported by the National Natural Science
Foundation of China (Nos.11571160, 11661061, 11761054) and the
Inner Mongolia University Scientific Research Projects (Nos.
NJZZ16234, NJZY17289). |
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| Abstract： |
In this paper, the authors give the local $L^{2}$ estimate of the
maximal operator $S^\ast_{\phi,\gamma}$ of the operator family
$\{S_{t,\phi,\gamma}\}$ defined initially by
$$
S_{t,\phi,\gamma}f(x):=\rme^{\rmi t\phi(\sqrt{-\Delta})}
f(\gamma(x,t))=(2\pi)^{-1} \int_{\mathbb{R}} \rme^{\rmi
\gamma(x,t)\cdot\xi+\rmi t\phi(|\xi|)}\wh{f}(\xi)\rmd\xi,\quad
f\in\mathcal{S}(\mathbb{R}),
$$
which is the solution (when $n=1$) of the following dispersive
equations ($\ast$) along a curve $\gamma$:
$$
\bigg\{\!\!\!\begin{array}{ll}
\rmi\partial_{t}u+\phi(\sqrt{-\Delta})u=0, & (x,t)\in \mathbb{R}^n\times \mathbb{R},\u(x,0)=f(x), & f\in \mathcal{S}(\mathbb{R}^{n}),
\end{array}\eqno(\ast)
$$
where $\phi: \mathbb{R}^{+}\rightarrow\mathbb{R}$ satisfies some
suitable conditions and $\phi(\sqrt{-\Delta})$ is a
pseudo-dif\/ferential operator with symbol $\phi(|\xi|)$. As a
consequence of the above result, the authors give the pointwise
convergence of the solution (when $n=1$) of the equation ($\ast$)
along curve $\gamma$.
Moreover, a global $L^{2}$ estimate of the maximal operator
$S^\ast_{\phi,\gamma}$ is also given in this paper. |
Keywords： |
$L^{2}$ estimate, Global maximal operator, Dispersive equation, Curve |
Classification： |
42B20, 42B25, 35S10 |
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