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 ${\displaystyle u_1,\dots ,u_N}$ of the differential equation with respect to the independent variables ${\displaystyle t,x_1,\dots ,x_n}$ that satisfies $${\displaystyle \begin{array}{cc}& \frac{{\mathrm{\partial }}^{n_i}{u}_i}{\mathrm{\partial }{t}^{n_i}}=F_i\left(t,x_1,\dots ,x_n,u_1,\dots ,u_N,\dots ,\frac{{\mathrm{\partial }}^k{u}_j}{\mathrm{\partial }{t}^{k_0}\mathrm{\partial }{x}_1^{k_1}\dots \mathrm{\partial }{x}_n^{k_n}},\dots \right)\\ & \text{for }i,j=1,2,\dots ,N;k_0+k_1+\cdots +k_n=k\le n_j;k_0<n_j\end{array}}$$ subject to the condition, for some value ${\displaystyle t=t_0}$,

$${\displaystyle \frac{{\mathrm{\partial }}^k{u}_i}{\mathrm{\partial }{t}^k}={\varphi }_i^{(k)}(x_1,\dots ,x_n)\text{for }k=0,1,2,\dots ,n_i-1}$$

where ${\displaystyle {\varphi }_i^{(k)}(x_1,\dots ,x_n)}$ 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.