Cauchy–Kowalevski theorem

In mathematics, the Cauchy–Kowalevski theorem is the main local existence and uniqueness theorem for analytic partial differential equations associated with Cauchy initial value problems. A special case was proved by harvs|authorlink=Augustin Cauchy|txt=yes|last=Cauchy|first=Augustin|year=1842, and the full result by harvs|txt=yes|authorlink=Sofia Kovalevskaya|first=Sophie |last=Kowalevski|year=1875.

First order Cauchy–Kowalevski theorem

Let "V" and "W" be finite-dimensional vector spaces, with "n" = dim "W". Let "A"₁, ..., "A"_"n"−1 be analytic functions with values in End ("V") and "b"an analytic function with values in "V", defined on some neighbourhood of (0,0) in "V" x "W". Then there is a neighbourhood of 0 in "W" on which the quasilinear Cauchy problem

:f _"x""n" = "A"₁("x","f") "f"_"x"1 + ··· + "A"_"n"−1("x","f") f_"x""n"−1 + "b"("x","f"),

with initial condition

:"f"("x") = 0 on "x"_"n" = 0,

has a unique analytic solution near 0.

The theorem and its proof are valid for analytic functions of either real or complex variables. Lewy's example shows that the theorem is not valid for smooth functions.

Proof by analytic majorization

Both sides of the partial differential equation can be expanded as formal power series and give recurrence relations for the coefficients of the formal power series for "f" that uniquely determine the coefficients. The Taylor series coefficients of the "A"_"i"'s and "b" are majorized in matrix and vector norm by a simple scalar rational analytic function. The corresponding scalar Cauchy problem involving this function instead of the "A"_"i"'s and "b" has an explicit local analytic solution. The absolute values of its coefficients majorize the norms of those of the original problem; so the formal power series solution must convergewhere the scalar solution converges.

Higher-order Cauchy–Kowalevski theorem

If "F" and "f"_"j" are analytic functions near 0, then the non-linear Cauchy problem

:$partial\_t^k\; h\; =\; Fleft(x,t,partial\_t^j,partial\_x^alpha\; h\; ight),$

with initial conditions

:$partial\_t^j\; h(x,0)\; =\; f\_j(x)\; ,$

where "j" < "k" and |"α"| + "j" ≤ "k", has a unique analytic solution near 0.

This follows from the first order problem by considering the derivatives of "h" appearing on the right hand side as components of a vector-valued function.

Example

The heat equation

:$partial\_t\; h\; =\; partial\_x^2\; h\; ,$

with the condition

:$h(0,x)\; =\; \{1over\; 1+x^2\}\; ext\{\; for\; \}t\; =\; 0$

has a unique formal power series solution (expanded around (0,0)). However this formal power series does not converge for any non-zero values of "t", so there are no analytic solutions in a neighborhood of the origin. This shows that the condition |"α"| + "j" ≤ "k" above cannot be dropped. (This example is due to Kowalevski.)

References

*citation|last=Cauchy|first=Augustin|year=1842|journal=Comptes Rend.|volume=15 Reprinted in Oeuvres completes, 1 serie, Tome VII, pages 17–58.
*citation | last= Folland|first= Gerald B. | title= Introduction to Partial Differential Equations| year=1995| publisher= Princeton University Press| id = ISBN0691043612
*citation|id=MR|0717035|first=L.|last= Hörmander|title=The analysis of linear partial differential operators I|series= Grundl. Math. Wissenschaft. |volume= 256 |publisher= Springer |year=1983|ISBN=3-540-12104-8 (linear case)
*citation|last=Kowalevski|first= Sophie |journal = Journal für die reine und angewandte Mathematik|volume=80|year=1875|pages=1–32
url=http://docserver.digizeitschriften.de/digitools/resolveppn.php?PPN=509874|title=Zur Theorie der partiellen Differentialgleichung (The surname given in the paper is "von Kowalevsky", which may be a misprint.)
*springer|id=C/c020920|title=Cauchy–Kovalevskaya theorem|first=A.M.|last= Nakhushev

External links

* [http://planetmath.org/encyclopedia/CauchyKovalevskayaTheorem.html PlanetMath]