Quasilinear Characteristics and Cauchy Kovalevskaya

Let $n\ge 1$ and $V\subseteq {\mathbb{R}}^{n-1}$ be open, let $\gamma :V\to {\mathbb{R}}^n$ and $\varphi :V\to \mathbb{R}$ be $C^1$, with $\mathrm{rank}D\gamma (y)=n-1$ for every $y\in V$, and let $a:O\to {\mathbb{R}}^n$ and $b:O\to \mathbb{R}$ be smooth on an open set $O\subseteq {\mathbb{R}}^n\times \mathbb{R}$. Assume that $(\gamma (y),\varphi (y))\in O$ for every $y\in V$. The Cauchy problem for the quasilinear equation is

$a(x,u(x))\cdot Du(x)=b(x,u(x)),u(\gamma (y))=\varphi (y)\mathrm{.}$

It is semilinear when $a=a(x)$ is independent of $u$. At $y_0\in V$, a classical local solution is a $C^1$ function $u$ on an open neighbourhood $\mathrm{\Omega }$ of $\gamma (y_0)$ such that $(x,u(x))\in O$ and the PDE holds for every $x\in \mathrm{\Omega }$, and such that there is a neighbourhood $W\subseteq V$ of $y_0$ with $\gamma (W)\subseteq \mathrm{\Omega }$ and $u(\gamma (y))=\varphi (y)$ for every $y\in W$. This specializes the first-order classification in Linear, semilinear, quasilinear, and fully nonlinear partial differential equations; the word noncharacteristic will be tested by the rank calculation below, rather than by importing the space-time transport convention of Noncharacteristic Cauchy surfaces for first-order transport.

For the problem in Semilinear and quasilinear first-order Cauchy problems on a parametrised hypersurface, its augmented characteristic strip is a map $(X,Z)(s,y)$ satisfying

$\dot{X}=a(X,Z),\dot{Z}=b(X,Z),(X,Z)(0,y)=(\gamma (y),\varphi (y))\mathrm{.}$

Here $s$ is the characteristic parameter and $y\in V$ labels the initial point. The system concerns only $(X,Z)$; no differential equation for a putative gradient is included in this definition.

At every $y_0\in V$, the system in The augmented characteristic system for a quasilinear first-order PDE has, after shrinking to $\mathrm{\mid }s\mathrm{\mid }<ϵ$ and a neighbourhood $W$ of $y_0$, a unique common solution $(X,Z)$ for $y\in W$. It is $C^1$ in $(s,y)$.

Let $u\in C^1(\mathrm{\Omega })$ solve $a(x,u)\cdot Du=b(x,u)$. If $X$ solves $\dot{X}=a(X,u(X))$ while it stays in $\mathrm{\Omega }$, then $Z:=u(X)$ satisfies $\dot{Z}=b(X,Z)$. Thus $(X,Z)$ solves the augmented characteristic ODE system.

If a $C^1$ solution has $u(\gamma (y))=\varphi (y)$ and $p_0(y):=Du(\gamma (y))$, then

$p_0(y)D\gamma (y)=D\varphi (y),p_0(y)\cdot a(\gamma (y),\varphi (y))=b(\gamma (y),\varphi (y))\mathrm{.}$

The first condition is tangential compatibility. The rank condition for the projected strip is a separate hypothesis; it is not derived from this identity.

For the local strip $(X,Z)$, the derivative of $\mathrm{\Psi }(s,y):=X(s,y)$ at $s=0$ is

$D_{(s,y)}\mathrm{\Psi }(0,y)=[a(\gamma (y),\varphi (y)),D\gamma (y)]\mathrm{.}$

Consequently $D_{(s,y)}\mathrm{\Psi }(0,y)$ is invertible exactly when this $n\times n$ matrix has rank $n$.

At $y_0\in V$, suppose $[a(\gamma (y_0),\varphi (y_0)),D\gamma (y_0)]$ has rank $n$. Then, after shrinking the characteristic strip, $\mathrm{\Psi }(s,y)=X(s,y)$ is a $C^1$ diffeomorphism onto an open set $U$, and

$u(x):=Z({\mathrm{\Psi }}^{-1}(x))$

is the unique $C^1$ function obtained by this inverse-projection construction. It attains $u(\gamma (y))=\varphi (y)$; the next lemma verifies its PDE.

The function $u=Z\circ {\mathrm{\Psi }}^{-1}$ constructed in Local quasilinear characteristic graph construction satisfies $a(x,u(x))\cdot Du(x)=b(x,u(x))$ on its local domain.

For a characteristic strip, a crossing or caustic is a point or time at which the projected map $(s,y)\mapsto X(s,y)$ loses local rank or local one-to-one graphing. It is a failure of the projection needed to define $u=Z\circ X^{-1}$, not a claim that the lifted ODE $(X,Z)$ has reached its maximal lifespan. For a forward-time family, a first crossing time is an infimum in $[0,\mathrm{\infty }]$, with value $\mathrm{\infty }$ when the relevant set is empty.

Let $u\in C^2(\mathrm{\Omega })$ solve $u_t+u{u}_x=0$ on an open $\mathrm{\Omega }\subseteq {\mathbb{R}}^2$. Let $I$ be an open interval containing zero and $X\in C^1(I)$ satisfy $(t,X(t))\in \mathrm{\Omega }$ and $\dot{X}=u(t,X)$. Then $q(t):=u_x(t,X(t))$ satisfies

$\dot{q}=-q^2,q(t)=\frac{q(0)}{1+tq(0)}$

for every $t\in I$; in particular $1+tq(0)$ cannot vanish in $I$.

For $u_t+u{u}_x=0$ with $C^1$ datum $u(0,\xi )=u_0(\xi )$, every classical characteristic has

$u(t,X(t,\xi ))=u_0(\xi ),X(t,\xi )=\xi +t{u}_0(\xi )\mathrm{.}$

Thus $X_{\xi }=1+t{u}_0^{\mathrm{\prime }}(\xi )$ and its first forward crossing time is

$T_{\ast }:=\inf \{t\ge 0:\mathrm{\exists }\xi ,1+t{u}_0^{\mathrm{\prime }}(\xi )=0\}\in [0,\mathrm{\infty }],$

where the infimum of the empty set is $\mathrm{\infty }$.

If $u_0^{\mathrm{\prime }}(\xi )\ge 0$ for all $\xi$, then $1+t{u}_0^{\mathrm{\prime }}(\xi )\ge 1$ for $t\ge 0$. Hence the Burgers projection has no forward crossing. This says nothing about any unrelated global lifespan condition.

Let $u,v\in C^1$ solve the same quasilinear Cauchy problem. On a connected region where their common characteristic projection from the data is a diffeomorphism, $u=v$.

Let $F:O\subseteq {\mathbb{R}}^n\times \mathbb{R}\times {\mathbb{R}}^n\to \mathbb{R}$. A general first-order equation has the form $F(x,u,Du)=0$; it is fully nonlinear when its dependence on the highest-order variable $p=Du$ is not affine after $(x,u)$ is fixed. A local complete integral on open sets $U,A\subseteq {\mathbb{R}}^n$ is a $C^2$ family $S:U\times A\to \mathbb{R}$ such that $(x,S(x;\alpha ),D_{x}S(x;\alpha ))\in O,F(x,S(x;\alpha ),D_{x}S(x;\alpha ))=0,\det D_{x\alpha }^{2}S(x;\alpha )\mathrm{\ne }0$ for every $(x,\alpha )\in U\times A$. Thus the $n$ parameters enter essentially rather than merely labelling repeated copies of one solution. A stationary envelope is a value $u(x)=S(x;\alpha (x))$ selected by $S_{\alpha }(x;\alpha (x))=0$. This definition asserts neither global representation nor differentiability of an envelope.

If $S_{\alpha }(x_0;{\alpha }_0)=0$ and $S_{\alpha \alpha }(x_0;{\alpha }_0)$ is invertible, then locally $S_{\alpha }(x;\alpha (x))=0$ defines a $C^1$ parameter $\alpha (x)$. The envelope $u=S(x;\alpha (x))$ has $Du=S_x(x;\alpha (x))$ and satisfies $F(x,u,Du)=0$.

For $F\in C^2(O)$ on $O\subseteq {\mathbb{R}}^n\times \mathbb{R}\times {\mathbb{R}}^n$, the Lagrange–Charpit system is

$\dot{X}=F_p,\dot{Z}=P\cdot F_p,\dot{P}=-F_x-P{F}_z,$

with all derivatives evaluated at $(X,Z,P)$. The alternative contact normalization $\dot{Z}=P\cdot F_p-F$ agrees with this one only along $F=0$; it is not silently substituted off that constraint.

Along a Charpit curve,

$\frac{d}{ds}F(X,Z,P)=0.$

Consequently an initial point in $F=0$ remains in $F=0$ while the curve exists.

Let $(X,Z,P)(s,y)$ be a $C^1$ Charpit strip lying in $F=0$. If $Z_y(0,y)=P(0,y)\cdot X_y(0,y)$, then

$Z_y(s,y)=P(s,y)\cdot X_y(s,y)$

while the strip exists.

If $u\in C^2$ solves $F(x,u,Du)=0$, put $P=Du(X)$ and let $\dot{X}=F_p(X,u(X),Du(X))$. Then

$\dot{P}=-F_x-P{F}_z,$

so the momentum equation is forced by differentiating the PDE.

Let $O\subseteq {\mathbb{R}}^n\times \mathbb{R}\times {\mathbb{R}}^n$ be open and $F\in C^2(O)$. Let $V\subseteq {\mathbb{R}}^{n-1}$ be open, let $y_0\in V$, and let $\gamma :V\to {\mathbb{R}}^n$, $\varphi :V\to \mathbb{R}$, and $p_0:V\to {\mathbb{R}}^n$ be $C^1$ maps such that $(\gamma (y),\varphi (y),p_0(y))\in O$, $F(\gamma (y),\varphi (y),p_0(y))=0$, and $D\varphi (y)=p_0(y)D\gamma (y)$ for every $y\in V$. Suppose also that $\mathrm{rank}[F_p(\gamma (y_0),\varphi (y_0),p_0(y_0)),D\gamma (y_0)]=n\mathrm{.}$ Then the Charpit strip through $(\gamma ,\varphi ,p_0)$ projects locally to a classical graph $u$ satisfying $F(x,u,Du)=0$. It is unique while that projection is locally invertible among graphs obtained by inverse-projecting this fixed Charpit strip.

A crossing makes the characteristic projection fail to define a single-valued classical graph. It does not by itself choose a continuation: entropy conditions for conservation laws and viscosity inequalities for Hamilton–Jacobi equations are extra selection principles. No such weak-solution theorem is claimed here.

Cauchy–Kovalevskaya on a noncharacteristic analytic hypersurface is proved in this library by analytic flattening, normal-form reduction and coefficient-majorant arguments. Its conclusion is local existence and uniqueness of an analytic solution germ for analytic Cauchy data on a noncharacteristic analytic hypersurface. For an implicit nonlinear equation it fixes a compatible jet and its selected analytic branch. These hypotheses do not assert an extension to arbitrary smooth coefficients or data.