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数学代写|数学物理代写Mathematical Physics代考|PHZ3113C Limitations of Local Smoothing

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数学代写|数学物理代写Mathematical Physics代考|Limitations of Local Smoothing

The theorem of Doi shows if there is no trapping there is a gain of $1 / 2$ derivative, and if there is trapping you must lose something but does not show what you lose. One hyperbolic trapped orbit or a very “thin” trapped set loses a “trivial” $\epsilon$. A stable or elliptic geodesic loses everything (no smoothing effect).

An important question is to ask if there is something in between trivial loss and total loss.

We now introduce a class of asymptotically Euclidean examples with a degenerate hyperbolic orbit. We consider the manifold $X=\mathbb{R}x \times \mathbb{R}\theta / 2 \pi \mathbb{Z}$, equipped with a metric of the form
$$g=d x^2+A^2(x) d \theta^2,$$
where $A \in \mathcal{C}^{\infty}$ is a smooth function, $A \geq \epsilon>0$ for some epsilon. We are primarily interested in the case $A(x)=\left(1+x^{2 m}\right)^{1 / 2 m}, m \in \mathbb{Z}_{+}$, in which case the manifold is asymptotically Euclidean (with two ends). Clairaut’s theorem implies the only periodic geodesic is at $x=0 . A(x)^{-2}$ has a critical point of order $x^{2 m}$ at $x=0$, which is degenerate for $m>1$. The Gaussian curvature is nonpositive, asymptotically 0 as $x \rightarrow \pm \infty$, and vanishes to order $2 m-2$ at $x=0$.
We prove the following theorem.

数学代写|数学物理代写Mathematical Physics代考|Sketch of Proof Ideas

We use a positive commutator idea: let $B=\arctan (x) \partial_x$ and compute
$$[\Delta, B]=2\langle x\rangle^{-2} \partial_x^2+2 A^{\prime} A^{-3} \arctan (x) \partial_\theta^2+\text { l.o.t. }$$
Here “l.o.t.” can be absorbed into $H^{1 / 2}$ energy. Everything looks good except the coefficient of $\partial_\theta^2$ has $A^{\prime} \arctan (x)$, which vanishes to order $2 m$ at $x=0$, so the commutator is not strictly positive! Integrations by parts yields
$$\int_0^T\left(\left|\langle x\rangle^{-1} \partial_x u\right|_{L^2}^2+\left||x|^m\langle x\rangle^{-m-3 / 2} \partial_\theta u\right|_{L^2}^2\right) d t \leq C\left|u_0\right|_{H^{1 / 2}}^2 .$$
In order to estimate near $x=0$, we separate variables:
$$u(t, x, \theta)=\sum_k e^{i k \theta} u_k(t, x),$$
and
$$u_0(x, \theta)=\sum_k e^{i k \theta} u_{0, k}(x)$$
and try to estimate on each mode $u_k$. By orthogonality, it suffices to show
$$\int_0^T\left|\chi(x) k u_k\right|_{L^2(\mathbb{R})}^2 d t \leq C\left(\left|\langle k\rangle^{m /(m+1)} u_{0, k}\right|_{L^2}^2+\left|u_{0, k}\right|_{H^{1 / 2}}^2\right.$$
for some $\chi \in \mathcal{C}_c^{\infty}(\mathbb{R})$ with $\chi(x) \equiv 1$ near $x=0$.
By a duality argument, energy cutoff, and Fourier transform $t \mapsto \tau$, it suffices to show the following (sharp) cutoff resolvent estimate.

数学代写|数学物理代写数学物理代考|局部平滑的局限性

.

Doi的定理表明，如果没有陷阱，就会有$1 / 2$导数的增益，如果有陷阱，你一定会失去一些东西，但没有显示你失去了什么。一个双曲圈闭轨道或一个非常“薄”的圈闭集失去了一个“微不足道的”$\epsilon$。一个稳定的或椭圆测地线失去了一切(没有平滑效果)

$$g=d x^2+A^2(x) d \theta^2,$$

数学代写|数学物理代写数学物理代考|证明思想的草图

$$[\Delta, B]=2\langle x\rangle^{-2} \partial_x^2+2 A^{\prime} A^{-3} \arctan (x) \partial_\theta^2+\text { l.o.t. }$$

$$\int_0^T\left(\left|\langle x\rangle^{-1} \partial_x u\right|{L^2}^2+\left||x|^m\langle x\rangle^{-m-3 / 2} \partial\theta u\right|{L^2}^2\right) d t \leq C\left|u_0\right|{H^{1 / 2}}^2 .$$

$$u(t, x, \theta)=\sum_k e^{i k \theta} u_k(t, x),$$

$$u_0(x, \theta)=\sum_k e^{i k \theta} u_{0, k}(x)$$

$$\int_0^T\left|\chi(x) k u_k\right|{L^2(\mathbb{R})}^2 d t \leq C\left(\left|\langle k\rangle^{m /(m+1)} u{0, k}\right|{L^2}^2+\left|u{0, k}\right|_{H^{1 / 2}}^2\right.$$

Matlab代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。