3.5: Theorem of Cauchy-Kovalevskaya
( \newcommand{\kernel}{\mathrm{null}\,}\)
Consider the quasilinear system of first order (3.3.1) of Section 3.3. Assume an initial manifolds S is given by χ(x)=0, ∇χ≠0, and suppose that χ is not characteristic. Then, see Section 3.3, the system (3.3.1) can be written as
uxn=n−1∑i=1ai(x,u)uxi+b(x,u)u(x1,…,xn−1,0)=f(x1,…,xn−1)
Here is u=(u1,…,um)T, b=(b1,…,bn)T and ai are (m×m)-matrices.
We assume that ai, b and f are in C∞ with respect to their arguments. From (3.5.1) and (3.5.2) it follows that we can calculate formally all derivatives Dαu in a neighborhood of the plane {x: xn=0}, in particular in a neighborhood of 0∈R. Thus we have a formal power series of u(x) at x=0:
$$u(x)\sim \sum\frac{1}{\alpha !}D^\alpha u(0) x^\alpha.\]
For notations and definitions used here and in the following see the appendix to this section.
Then, as usually, two questions arise:
- Does the power series converge in a neighborhood of 0∈R?
- Is a convergent power series a solution of the initial value problem (3.5.1), (3.5.2)?
Remark. Quite different to this power series method is the method of asymptotic expansions. Here one is interested in a good approximation of an unknown solution of an equation by a finite sum ∑Ni=0ϕi(x) of functions ϕi. In general, the infinite sum
∑∞i=0ϕi(x) does not converge, in contrast to the power series method of this section. See [15] for some asymptotic formulas in capillarity.
Theorem 3.1. (Cauchy-Kovalevskaya). There is a neighborhood of 0∈R such there is a real analytic solution of the initial value problem (3.5.1), (3.5.2). This solution is unique in the class of real analytic functions.
Proof. The proof is taken from F. John \cite{John}. We introduce u−f as the new solution for which we are looking at and we add a new coordinate u⋆ to the solution vector by setting u⋆(x)=xn. Then
$$u^\star_{x_n}=1,\ u^\star_{x_k}=0,\ k=1,\ldots,n-1,\ u^\star(x_1,\ldots,x_{n-1},0)=0\]
and the extended system (3.5.1), (3.5.2) is
$$
\left(u1,xn⋮um,xnu⋆xn\right)=
\sum_{i=1}^{n-1}\left(ai000\right)
\left(u1,xi⋮um,xiu⋆xi\right)+
\left(b1⋮bm1\right),
\]
where the associated initial condition is u(x1,…,xn−1,0)=0.
The new u is u=(u1,…,um)T, the new ai are ai(x1,…,xn−1,u1,…,um,u⋆) and the new b is b=(x1,…,xn−1,u1,…,um,u⋆)T.
Thus we are led to an initial value problem of the type
uj,xn=n−1∑i=1N∑k=1aijk(z)uk,xi+bj(z), j=1,…,Nuj(x)=0 if xn=0,
where j=1,…,N and z=(x1,…,xn−1,u1,…,uN).
The point here is that aijk and bj are independent of xn. This fact simplifies the proof of the theorem.
From (3.5.8) and (3.5.9) we can calculate formally all Dβuj. Then we have formal power series for uj:
$$u_j(x)\sim \sum_\alpha c_\alpha^{(j)}x^\alpha,\]
where
$$c_\alpha^{(j)}=\frac{1}{\alpha!}D^\alpha u_j(0).\]
We will show that these power series are (absolutely) convergent in a neighborhood of 0∈R, i.e., they are real analytic functions, see the appendix for the definition of real analytic functions. Inserting these functions into the left and into the right hand side of (3.5.8) we obtain on the right and on the left hand side real analytic functions. This follows since compositions of real analytic functions are real analytic again, see Proposition A7 of the appendix to this section. The resulting power series on the left and on the right have the same coefficients caused by the calculation of the derivatives Dαuj(0) from (3.5.8). It follows that uj(x), j=1,…,n, defined by its formal power series are solutions of the initial value problem (3.5.8), (3.5.9).
Set
$$d=\left(\frac{\partial}{\partial z_1},\ldots,\frac{\partial}{\partial z_{N+n-1}}\right)\]
Lemma A. Assume u∈C∞ in a neighborhood of 0∈R. Then
Dαuj(0)=Pα(dβaijk(0),dγbj(0)),
where |β|, |γ|≤|α| and Pα are polynomials in the indicated arguments with non-negative integers as coefficients which are independent of ai and of b.
Proof. It follows from equation (3.5.8) that
DnDαuj(0)=Pα(dβaijk(0),dγbj(0),Dδuk(0)).
Here is Dn=∂/∂xn and α, β, γ, δ satisfy the inequalities
$$|\beta|,\ |\gamma|\le|\alpha|,\ \ |\delta|\le|\alpha|+1,\]
and, which is essential in the proof, the last coordinates in the multi-indices α=(α1,…,αn), δ=(δ1,…,δn) satisfy δn≤αn since the right hand side of (3.5.8) is independent of xn.
Moreover, it follows from (3.5.8) that the polynomials Pα have integers as coefficients. The initial condition (3.5.9) implies
Dαuj(0)=0,
where α=(α1,…,αn−1,0), that is, αn=0. Then, the proof is by induction with respect to αn. The induction starts with αn=0, then we replace $D^\delta u_k(0)$ in the right hand side of (???) by (???), that is by zero. Then it follows from (???) that
$$D^\alpha u_j(0)=P_\alpha (d^\beta a_{jk}^i(0),d^\gamma b_j(0),D^\delta u_k(0)),\]
where α=(α1,…,αn−1,1).
◻
Definition. Let f=(f1,…,fm), F=(F1,…,Fm), fi=fi(x), Fi=Fi(x), and f, F∈C∞. We say f is majorized by F if
|Dαfk(0)|≤DαFk(0), k=1,…,m
for all α. We write f<<F, if f is majorized by F.
Definition. The initial value problem
Uj,xn=n−1∑i=1N∑k=1Aijk(z)Uk,xi+Bj(z)Uj(x)=0if xn=0,
j=1,…,N, Aijk, Bj real analytic, is called majorizing problem to (3.5.8), (3.5.9) if
$$
a_{jk}^i<<A_{jk}^i\ \ \mbox{and}\ b_j<<B_j.
\]
Lemma B. The formal power series
∑α1α!Dαuj(0)xα,
where Dαuj(0) are defined in Lemma A, is convergent in a neighborhood of 0∈R if there exists a majorizing problem which has a real analytic solution U in x=0, and
|Dαuj(0)|≤DαUj(0).
Proof. It follows from Lemma A and from the assumption of Lemma B that
|Dαuj(0)|≤Pα(|dβaijk(0)|,|dγbj(0)|)≤Pα(|dβAijk(0)|,|dγBj(0)|)≡DαUj(0).
The formal power series
∑α1α!Dαuj(0)xα,
is convergent since
∑α1α!|Dαuj(0)xα|≤∑α1α!DαUj(0)|xα|.
The right hand side is convergent in a neighborhood of x∈R by assumption.
◻
Lemma C. There is a majorising problem which has a real analytic solution.
Proof. Since aiij(z), bj(z) are real analytic in a neighborhood of z=0 it follows from Proposition A5 of the appendix to this section that there are positive constants M and r such that all these functions are majorized by
Mrr−z1−…−zN+n−1.
Thus a majorizing problem is
Uj,xn=Mrr−x1−…−xn−1−U1−…−UN(1+n−1∑i=1N∑k=1Uk,xi)Uj(x)=0 if xn=0,
j=1,…,N.
The solution of this problem is
Uj(x1,…,xn−1,xn)=V(x1+…+xn−1,xn), j=1,…,N,
where V(s,t), s=x1+…+xn−1, t=xn, is the solution of the Cauchy initial value problem
Vt=Mrr−s−NV(1+N(n−1)Vs),V(s,0)=0.
which has the solution, see an exercise,
V(s,t)=1Nn(r−s−√(r−s)2−2nMNrt).
This function is real analytic in (s,t) at (0,0). It follows that Uj(x) are also real analytic functions. Thus the Cauchy-Kovalevskaya theorem is shown.
◻
Example 3.5.1: Ordinary differential equations
Consider the initial value problem
y′(x)=f(x,y(x))y(x0)=y0,
where x0∈R1 and y0∈R are given. Assume f(x,y) is real analytic in a neighborhood of (x0,y0)∈R1×R. Then it follows from the above theorem that there exists an analytic solution y(x) of the initial value problem in a neighborhood of x0. This solution is unique in the class of analytic functions according to the theorem of Cauchy-Kovalevskaya. From the Picard-Lindel\"of theorem it follows that this analytic solution is unique even in the class of C1-functions.
Example 3.5.2: Partial differential equations of second order
Consider the boundary value problem for two variables
uyy=f(x,y,u,ux,uy,uxx,uxy)u(x,0)=ϕ(x)uy(x,0)=ψ(x).
We assume that ϕ, ψ are analytic in a neighborhood of x=0 and that f is real analytic in a neighbourhood of
(0,0,ϕ(0),ϕ′(0),ψ(0),ψ′(0)).
There exists a real analytic solution in a neighborhood of 0∈R2 of the above initial value problem.
In particular, there is a real analytic solution in a neighborhood of 0∈R2 of the initial value problem
△u=1u(x,0)=0uy(x,0)=0.
The proof follows by writing the above problem as a system. Set p=ux, q=uy, r=uxx,
s=uxy, t=uyy, then
t=f(x,y,u,p,q,r,s).
Set U=(u,p,q,r,s,t)T, b=(q,0,t,0,0,fy+fuq+fqt)T and
A=(00000000100000000000001000000100fp0frfs).
Then the rewritten differential equation is the system
Uy=AUx+b with the initial condition
U(x,0)=(ϕ(x),ϕ′(x),ψ(x),ϕ″(x),ψ′(x),f0(x)),
where f0(x)=f(x,0,ϕ(x),ϕ′(x),ψ(x),ϕ″(x),ψ′(x)).
Contributors and Attributions
Integrated by Justin Marshall.