 Cauchy problem - Wikiwand

# Cauchy problem

A Cauchy problem in mathematics asks for the solution of a partial differential equation that satisfies certain conditions that are given on a hypersurface in the domain. A Cauchy problem can be an initial value problem or a boundary value problem (for this case see also Cauchy boundary condition) or it can be either of them. It is named after Augustin Louis Cauchy.

## Formal statement

For a partial differential equation defined on Rn+1 and a smooth manifold SRn+1 of dimension n (S is called the Cauchy surface), the Cauchy problem consists of finding the unknown functions $u_{1},\dots ,u_{N)$ of the differential equation with respect to the independent variables $t,x_{1},\dots ,x_{n)$ that satisfies

{\begin{aligned}&{\frac {\partial ^{n_{i))u_{i)){\partial t^{n_{i))))=F_{i}\left(t,x_{1},\dots ,x_{n},u_{1},\dots ,u_{N},\dots ,{\frac {\partial ^{k}u_{j)){\partial t^{k_{0))\partial x_{1}^{k_{1))\dots \partial x_{n}^{k_{n)))),\dots \right)\\&{\text{for ))i,j=1,2,\dots ,N;\,k_{0}+k_{1}+\dots +k_{n}=k\leq n_{j};\,k_{0} subject to the condition, for some value $t=t_{0)$ ,

${\frac {\partial ^{k}u_{i)){\partial t^{k))}=\phi _{i}^{(k)}(x_{1},\dots ,x_{n})\quad {\text{for ))k=0,1,2,\dots ,n_{i}-1$ where $\phi _{i}^{(k)}(x_{1},\dots ,x_{n})$ are given functions defined on the surface $S$ (collectively known as the Cauchy data of the problem). The derivative of order zero means that the function itself is specified.

## Cauchy–Kowalevski theorem

The Cauchy–Kowalevski theorem states that If all the functions $F_{i)$ are analytic in some neighborhood of the point $(t^{0},x_{1}^{0},x_{2}^{0},\dots ,\phi _{j,k_{0},k_{1},\dots ,k_{n))^{0},\dots )$ , and if all the functions $\phi _{j}^{(k))$ are analytic in some neighborhood of the point $(x_{1}^{0},x_{2}^{0},\dots ,x_{n}^{0})$ , then the Cauchy problem has a unique analytic solution in some neighborhood of the point $(t^{0},x_{1}^{0},x_{2}^{0},\dots ,x_{n}^{0})$ .