Cauchy–Kowalevski theorem

The Cauchy - Kovalevskaya, named after Augustin- Louis Cauchy and Sofya Kovalevskaya, is a set of the mathematical theory of partial differential equations. He ensures the existence and uniqueness of solutions of such an equation, more precisely the so-called Cauchy problem, under suitable Analytizitätsvoraussetzungen.

The Cauchy problem

First, a special form of the Cauchy problem is considered. For this purpose let a function in variables that are written because of the special role of the last variable. The -th derivative with respect to is denoted by, for a multi- index is a derivative with respect to the first variable.

Given now a natural number, functions and variables in a function. The Cauchy problem asks in this situation for a function in the variables that satisfies the following conditions:

In a neighborhood of 0, where the variables of adjacent and above all possible multi- indices of length and natural numbers running with. The arity of has just been chosen so that this is possible. Equation (1 ) is then a condition on the derivative of th by which depends on the right side only of a lower order - derivatives. By ( 2) are the derivations of smaller order for the so-called boundary or initial values ​​prescribed. They call and also the data of the Cauchy problem, is called the order of the problem. Note to the fact that all occurring derivatives have an order less equal, and on the left a derivative of order actually occurs. Any function that satisfies the above equations, is called a solution of the Cauchy problem.

Wording of the sentence

The Cauchy - Kovalevskaya says:

Are and the functions in the above formulation of the Cauchy problem analytically, so there is a neighborhood of the origin a unique analytical solution of the Cauchy problem.

More general formulation

In a more general formulation one considers functions in variables to excel without one of these variables in particular. It is set at a normal field point from a sufficiently smooth hypersurface. The normal derivative in direction will be denoted by.

Now functions and a function with points are given. In general Cauchy problem one asks for functions with

In a neighborhood of.

In this form it is generally not a properly posed problem and you can no existence and uniqueness results expected, also not when, and which are assumed to be analytic. You need to the additional condition that you can resolve (1 ) to a maximum dissipation. But then you can transform the above-described specific formulation of the Cauchy problem, the present situation by a suitable coordinate transformation. I basically want to do so that the analyticity of the functions is preserved, and that imaged on the hypersurface and the point to 0. One then speaks of a so-called non- characteristic Cauchy problem. Casually can also express the set of Cauchy- Kovalevskaya so that has a non - characteristic analytic Cauchy problem locally, ie in a neighborhood of, a unique analytical solution.

Comments

For a positive number, the Cauchy problem

Obviously the solution

How to easily recalculate. If we let go now, the Cauchy data converge uniformly to 0 The solution, however, oscillates faster and does not converge for. This goes back to J. Hadamard example shows that the solution of the Cauchy problem not depends continuously on the data of the Cauchy problem.

Next, the question arises whether one can weaken the Analytizitätsvoraussetzung to " infinitely differentiable " in the set of Cauchy- Kovalevskaya. The example 1957 found by Lewy is a surprisingly simple example of a Cauchy problem with infinitely differentiable data, which has no solution.