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.[1] A Cauchy problem can be an initial value problem or a boundary value problem (for this case see also Cauchy boundary condition). 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 ${\displaystyle u_{1},\dots ,u_{N))$ of the differential equation with respect to the independent variables ${\displaystyle t,x_{1},\dots ,x_{n))$ that satisfies[2] {\displaystyle {\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 ${\displaystyle t=t_{0))$,

${\displaystyle {\frac {\partial ^{k}u_{i)){\partial t^{k))}=\phi _{i}^{(k)}(x_{1},\dots ,x_{n})\quad {\text{for ))k=0,1,2,\dots ,n_{i}-1}$

where ${\displaystyle \phi _{i}^{(k)}(x_{1},\dots ,x_{n})}$ are given functions defined on the surface ${\displaystyle S}$ (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 ${\displaystyle F_{i))$ are analytic in some neighborhood of the point ${\displaystyle (t^{0},x_{1}^{0},x_{2}^{0},\dots ,\phi _{j,k_{0},k_{1},\dots ,k_{n))^{0},\dots )}$, and if all the functions ${\displaystyle \phi _{j}^{(k)))$ are analytic in some neighborhood of the point ${\displaystyle (x_{1}^{0},x_{2}^{0},\dots ,x_{n}^{0})}$, then the Cauchy problem has a unique analytic solution in some neighborhood of the point ${\displaystyle (t^{0},x_{1}^{0},x_{2}^{0},\dots ,x_{n}^{0})}$.

## References

1. ^ Hadamard, Jacques (1923). Lectures on Cauchy's Problem in Linear Partial Differential Equations. New Haven: Yale University Press. pp. 4–5. OCLC 1880147.
2. ^ Petrovsky, I. G. (1991) [1954]. Lectures on Partial Differential Equations. Translated by Shenitzer, A. (Dover ed.). New York: Interscience. ISBN 0-486-66902-5.

3.^ Hille,Einar (1956)[1954]. Some Aspect of Cauchy's Problem Proceedings of '5 4 ICM vol III section II (analysis half-hour invited address) p.1 0 9 ~ 1 6 .

4.^ Sigeru Mizohata(溝畑 茂 1965). Lectures on Cauchy Problem. Tata Institute of Fundamental Research.

5.^ Sigeru Mizohata (1985).On the Cauchy Problem. Notes and Reports in Mathematics in Science and Engineering. 3. Academic Press, Inc.. ISBN 9781483269061


6.^Arendt, Wolfgang; Batty, Charles; Hieber, Matthias; Neubrander, Frank (2001), Vector-valued Laplace Transforms and Cauchy Problems, Birkhauser.