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 may involve initial or boundary values. It is named after Augustin-Louis Cauchy.

Formal statement

For a partial differential equation defined on ${\displaystyle \mathbb {R} ^{n+1}}$ and a smooth manifold ${\displaystyle S\subset \mathbb {R} ^{n+1}}$ of dimension ${\displaystyle n}$ (${\displaystyle S}$ 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$${\displaystyle {\begin{aligned}&{\frac {\partial ^{n_{i}}u_{i}}{\partial t^{n_{i}}}}=F_{i}\left(t,x_{1},\dots ,x_{n},u_{1},\dots ,u_{N},\dots ,{\frac {\partial ^{k}u_{j}}{\partial t^{k_{0}}\partial x_{1}^{k_{1}}\dots \partial x_{n}^{k_{n}}}},\dots \right)\\&{\text{for }}i,j=1,2,\dots ,N;\,k_{0}+k_{1}+\dots +k_{n}=k\leq n_{j};\,k_{0}<n_{j}\end{aligned}}}$$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–Kovalevskaya theorem, named in honor of Cauchy and Sofya Kovalevskaya, states: 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 ${\displaystyle (t^{0},x_{1}^{0},x_{2}^{0},\dots ,x_{n}^{0})}$.

See also

• Cauchy boundary condition
• Cauchy horizon

References

Further reading

• Hille, Einar (1956)[1954]. Some Aspect of Cauchy's Problem Proceedings of 1954 ICM vol III section II (analysis half-hour invited address) p. 1 0 9 ~ 1 6.
• Sigeru Mizohata(溝畑 茂 1965). Lectures on Cauchy Problem. Tata Institute of Fundamental Research.
• Sigeru Mizohata (1985).On the Cauchy Problem. Notes and Reports in Mathematics in Science and Engineering. 3. Academic Press, Inc.. ISBN 9781483269061
• Arendt, Wolfgang; Batty, Charles; Hieber, Matthias; Neubrander, Frank (2001), Vector-valued Laplace Transforms and Cauchy Problems, Birkhauser.

External links

• Cauchy problem at MathWorld.