Cauchy problem

Formal statement

For a partial differential equation defined on Rn+1 and a smooth manifold S ⊂ Rn+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{align}&\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 \end{align} 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).

References

References

1. Hadamard, Jacques. (1923). "Lectures on Cauchy's Problem in Linear Partial Differential Equations". Yale University Press.
2. Petrovsky, I. G.. (1991). "Lectures on Partial Differential Equations". Interscience.