Cauchy-Kovalevskaya theorem

The Cauchy-Kowalewskaja theorem , named after Augustin-Louis Cauchy and Sofja Kowalewskaja , is a theorem from the mathematical theory of partial differential equations . It ensures the existence and uniqueness of solutions to such an equation, more precisely the so-called Cauchy problem , under suitable analytical conditions .

The Cauchy problem

First, a special form of the Cauchy problem is considered. To do this, be a function in variables that are also written because of the special role of the last variable . The -th derivative according to is denoted by, for a multi-index is a derivative according to the first variables. ${\displaystyle u}$ ${\displaystyle n}$ ${\displaystyle x_{1},\ldots ,x_{n-1},t}$ ${\displaystyle j}$ ${\displaystyle t}$ ${\displaystyle \partial _{t}^{j}}$ ${\displaystyle \alpha =(\alpha _{1},\ldots ,\alpha _{n-1})}$ ${\displaystyle \partial _{x}^{\alpha }=\partial _{x_{1}}^{\alpha _{1}}\ldots \partial _{x_{n-1}}^{\alpha _{n-1}}}$ ${\displaystyle n-1}$

Let a natural number , functions for and a function in variables be given. In this situation, the Cauchy problem asks for a function in the variables that meets the following conditions: ${\displaystyle k}$ ${\displaystyle x\mapsto \varphi _{j}(x)=\varphi _{j}(x_{1},\ldots ,x_{n-1})}$ ${\displaystyle j<k}$ ${\displaystyle G}$ ${\displaystyle n-1+{\tbinom {k+n-1}{k}}}$ ${\displaystyle u}$ ${\displaystyle x_{1},\ldots ,x_{n-1},t}$

(1) ${\displaystyle \partial _{t}^{k}u(x,t)=G(x,t,(\partial _{x}^{\alpha }\partial _{t}^{j}u(x,t))_{|\alpha |+j\leq k,j<k})}$

(2) for ${\displaystyle \partial _{t}^{j}u(x,0)=\varphi _{j}(x)}$ ${\displaystyle j=0,\ldots ,k-1}$

in a neighborhood of 0. The variables of running alongside and over all possible multiindices the length and integers with . The arity of was just chosen so that this is possible. Equation (1) is then a condition on the -th derivative of to , which on the right-hand side only depends on -derivatives of smaller order. (2) specifies the lower order derivatives for the so-called boundary or initial values. The data of the Cauchy problem is called and that is called order of the problem. Please note that all derivatives that occur have an order less than or equal and that a derivative of the order actually occurs on the left-hand side . Every function that satisfies the above equations is called a solution to the Cauchy problem. ${\displaystyle G}$ ${\displaystyle x}$ ${\displaystyle t}$ ${\displaystyle \alpha }$ ${\displaystyle n-1}$ ${\displaystyle j=0,\ldots ,k-1}$ ${\displaystyle |\alpha |+j\leq k}$ ${\displaystyle G}$ ${\displaystyle k}$ ${\displaystyle u}$ ${\displaystyle t}$ ${\displaystyle t}$ ${\displaystyle t}$ ${\displaystyle t=0}$ ${\displaystyle G}$ ${\displaystyle \varphi _{j}}$ ${\displaystyle k}$ ${\displaystyle k}$ ${\displaystyle k}$ ${\displaystyle u}$

Formulation of the sentence

Cauchy-Kovalevskaya's theorem says:

If and the functions in the above formulation of the Cauchy problem are analytical, there is a clear analytical solution to the Cauchy problem in a neighborhood of the zero point. ${\displaystyle G}$ ${\displaystyle \varphi _{j}}$

More general wording

In a more general formulation, one considers functions in variables without particularly marking one of these variables. A point from a sufficiently smooth hypersurface with a normal field is given. The normal derivative toward going with designated. ${\displaystyle n}$ ${\displaystyle x_{1},\ldots ,x_{n}}$ ${\displaystyle x^{0}=(x_{1}^{0},\ldots ,x_{n}^{0})}$ ${\displaystyle S}$ ${\displaystyle \nu }$ ${\displaystyle \nu }$ ${\displaystyle \partial _{\nu }}$

Now let functions and a function with places be given. In the general Cauchy problem one asks about functions with ${\displaystyle \varphi _{j}}$ ${\displaystyle F}$ ${\displaystyle n+{\tbinom {k+n-1}{k}}}$ ${\displaystyle u}$

(1) ${\displaystyle F(x,(\partial ^{\alpha }u(x))_{|\alpha |\leq k})=0}$

(2) on ${\displaystyle \partial _{\nu }^{j}u=\varphi _{j}}$ ${\displaystyle S}$

in a neighborhood of . ${\displaystyle x^{0}\in S}$

In this form, it is not generally a properly posed problem and you can no existence and uniqueness statements expected, even not if , and are assumed to be analytical. For this one needs the additional requirement that one can solve (1) for a highest derivative. But then one can transform the present situation by means of a suitable coordinate transformation to the more specific formulation of the Cauchy problem described above. This can then be done in such a way that the analyticity of the functions is retained and that the hypersurface and the point are mapped to 0. One then speaks of a so-called non-characteristic Cauchy problem. The initial data must be given on a hypersurface that is not a characteristic of the partial differential equation (or is tangential to a characteristic). The Cauchy-Kowalewskaja theorem can also be expressed loosely in such a way that a non-characteristic analytic Cauchy problem has a locally unambiguous analytic solution, i.e. in a neighborhood of . ${\displaystyle S}$ ${\displaystyle F}$ ${\displaystyle \varphi _{j}}$ ${\displaystyle S}$ ${\displaystyle x_{n}=0}$ ${\displaystyle x^{0}}$ ${\displaystyle x^{0}}$

Remarks

For a positive number the Cauchy problem has ${\displaystyle k}$

(1) ${\displaystyle \partial _{t}^{2}u=-\partial _{x}^{2}u}$

(2) ${\displaystyle u(x,0)=0,\quad \partial _{t}u(x,0)=ke^{-k/2}\sin kx}$

apparently the solution

${\displaystyle u(x,t)\,=\,e^{-k/2}\sin(kx)\sinh(kt)}$,

how to easily do the math. If you let go, the Cauchy data converge evenly to 0. The solution, on the other hand, oscillates faster and faster and does not converge for . This example, which goes back to J. Hadamard , shows that the solution of the Cauchy problem does not depend continuously on the data of the Cauchy problem. ${\displaystyle k\to \infty }$ ${\displaystyle k\to \infty }$

Furthermore, the question arises whether in Cauchy-Kowalewskaja's theorem one can weaken the analyticity requirement to “as often differentiable as desired”. The example found by Lewy in 1957 is a surprisingly simple example of a Cauchy problem with any number of differentiable data that has no solution.

literature

• Sophie von Kowalevsky: On the theory of partial differential equations. Reimer, Berlin 1874 (dissertation by Sofja Wassiljewna Kowalewskaja, University of Göttingen; published under the spelling of her name then customary in Germany), digitized version .

Individual evidence

1. ↑ Gerald B. Folland: Introduction to Partial Differential Equations. Princeton University Press, 1976, typesetting (1.25)
2. ^ H. Lewy: An example of a smooth linear partial equation without solution. In: Annals of Mathematics. Vol. 66 (1957), pp. 155-158.