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# 数学代写|偏微分方程代考Partial Differential Equations代写|MATH402

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## 数学代写|偏微分方程代考Partial Differential Equations代写|The Cauchy-Kovalevskaya Theorem

The theorem of Cauchy and Kovalevskaya quite generally asserts the local existence of solutions to a system of partial differential equations with initial conditions on a noncharacteristic surface. The coefficients in the equations, the initial data and the surface on which they are prescribed are required to be analytic. This is a severe restriction which, in general, cannot be removed. Moreover, we shall see that the theorem does not distinguish between well-posed and ill-posed problems; it covers situations where a small change in the data leads to a large change in the solution. For these reasons, the theorem has little practical importance. Historically, however, it is the first existence theorem for a general class of PDEs and it is one of very few such theorems which can be proved without the tools of functional analysis.

We shall state and prove the theorem for quasilinear first-order systems of the form
$$\frac{\partial u_i}{\partial x_n}=\sum_{k=1}^{n-1} \sum_{j=1}^N a_{i j}^k(\mathbf{p}) \frac{\partial u_j}{\partial x_k}+b_i(\mathbf{p}), \quad i=1, \ldots, N,$$
where $\mathbf{p}$ stands for the vector $\left(x_1, \ldots, x_{n-1}, u_1, \ldots, u_N\right)$, and the functions $a_{i j}^k$ and $b_i$ are assumed analytic. The initial conditions are
$$u_i=0 \quad \text { on } x_n=0, \quad i=1, \ldots, N .$$
A general noncharacteristic initial-value problem for a system of PDEs can always be reduced to the form $(2.45),(2.46)$; below we shall discuss the reduction algorithm in detail. We shall start the section by reviewing some basic facts about real analytic functions.

## 数学代写|偏微分方程代考Partial Differential Equations代写|Real Analytic Functions

Analytic functions are functions which can be represented locally by power series. We shall use the multi-index notation introduced in the previous section and write the power series of a function of $n$ variables in the form
$$f(\mathbf{x})=\sum_\alpha c_\alpha \mathbf{x}^\alpha$$
where $\alpha=\left(\alpha_1, \ldots, \alpha_n\right)$ is a multi-index and $\mathbf{x}^\alpha$ has the meaning introduced in equation (2.6).
We note the following facts about power series:

Suppose that (2.47) converges absolutely for $\mathbf{x}=\mathbf{y}$, where all components of $\mathbf{y}$ are different from zero. Then it converges absolutely in the domain $D=\left{\mathbf{x} \in \mathbb{R}^n|| x_i|<| y_i \mid, i=1, \ldots, n\right}$ and it converges uniformly absolutely in any compact subset of $D$.
In $D$, the power series (2.47) can be differentiated term by term. We shall obtain an estimate for the derivatives. Let $\left|x_i\right| \leq q\left|y_i\right|$ for $i=1, \ldots, n$, where $0 \leq q<1$. We compute
$$D^\beta f(\mathbf{x})=\sum_{\alpha \geq \beta} c_\alpha D^\beta \mathbf{x}^\alpha=\sum_{\alpha \geq \beta} c_\alpha \frac{\alpha !}{(\alpha-\beta) !} \mathbf{x}^{\alpha-\beta},$$
and hence
\begin{aligned} \left|D^\beta f(\mathbf{x})\right| & \leq \sum_{\alpha \geq \beta} \frac{\alpha !}{(\alpha-\beta) !}\left|c_\alpha\right| q^{|\alpha-\beta|}\left|\mathbf{y}^{\alpha-\beta}\right| \ & \leq \frac{1}{\left|\mathbf{y}^\beta\right|} \sup \alpha\left(\left|c\alpha\right|\left|\mathbf{y}^\alpha\right|\right) \sum_{\alpha \geq \beta} \frac{\alpha !}{(\alpha-\beta) !} q^{|\alpha-\beta|} . \end{aligned}
We have (see Problem 2.7)
$$\sum_{\alpha \geq \beta} \frac{\alpha !}{(\alpha-\beta) !} q^{|\alpha-\beta|}=\frac{\beta !}{(1-q)^{n+|\beta|}},$$
and with
$$M=(1-q)^{-n} \sup \alpha\left(\left|c\alpha\right|\left|\mathbf{y}^\alpha\right|\right), \quad r=(1-q) \min _i\left|y_i\right|,$$
we finally obtain
$$\left|D^\beta f(\mathbf{x})\right| \leq M|\beta| ! r^{-|\beta|} .$$
We have
$$c_\alpha=\frac{1}{\alpha !} D^\alpha f(0)$$

# 偏微分方程代写

## 数学代写|偏微分方程代考Partial Differential Equations代写|The Cauchy-Kovalevskaya Theorem

$$\frac{\partial u_i}{\partial x_n}=\sum_{k=1}^{n-1} \sum_{j=1}^N a_{i j}^k(\mathbf{p}) \frac{\partial u_j}{\partial x_k}+b_i(\mathbf{p}), \quad i=1, \ldots, N,$$

$$u_i=0 \quad \text { on } x_n=0, \quad i=1, \ldots, N .$$

## 数学代写|偏微分方程代考Partial Differential Equations代写|Real Analytic Functions

$$f(\mathbf{x})=\sum_\alpha c_\alpha \mathbf{x}^\alpha$$

$$D^\beta f(\mathbf{x})=\sum_{\alpha \geq \beta} c_\alpha D^\beta \mathbf{x}^\alpha=\sum_{\alpha \geq \beta} c_\alpha \frac{\alpha !}{(\alpha-\beta) !} \mathbf{x}^{\alpha-\beta},$$

\begin{aligned} \left|D^\beta f(\mathbf{x})\right| & \leq \sum_{\alpha \geq \beta} \frac{\alpha !}{(\alpha-\beta) !}\left|c_\alpha\right| q^{|\alpha-\beta|}\left|\mathbf{y}^{\alpha-\beta}\right| \ & \leq \frac{1}{\left|\mathbf{y}^\beta\right|} \sup \alpha\left(\left|c\alpha\right|\left|\mathbf{y}^\alpha\right|\right) \sum_{\alpha \geq \beta} \frac{\alpha !}{(\alpha-\beta) !} q^{|\alpha-\beta|} . \end{aligned}

$$\sum_{\alpha \geq \beta} \frac{\alpha !}{(\alpha-\beta) !} q^{|\alpha-\beta|}=\frac{\beta !}{(1-q)^{n+|\beta|}},$$

$$M=(1-q)^{-n} \sup \alpha\left(\left|c\alpha\right|\left|\mathbf{y}^\alpha\right|\right), \quad r=(1-q) \min i\left|y_i\right|,$$ 我们最终得到 $$\left|D^\beta f(\mathbf{x})\right| \leq M|\beta| ! r^{-|\beta|} .$$ 我们有 $$c\alpha=\frac{1}{\alpha !} D^\alpha f(0)$$

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