Partial Differential Equations and Characteristics

Let $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$ be open, let $m\ge 1$, and let

$F:\mathrm{\Omega }\times {\mathbb{R}}^{N(m,n)}\to \mathbb{R}$

be a function of the point $x$ and of the jet coordinates $z_{\alpha }(\mathrm{\mid }\alpha \mathrm{\mid }\le m)$, where $N(m,n)$ is the number of multi-indices of length at most $m$. Assume that $F$ depends nontrivially on at least one jet coordinate. A scalar partial differential equation of order at most $m$ is an equation

$F(x,(D^{\alpha }u(x){)}_{\mathrm{\mid }\alpha \mathrm{\mid }\le m})=0$

for an unknown scalar field $u:\mathrm{\Omega }\to \mathbb{R}$.

Its order is the largest $\mathrm{\mid }\alpha \mathrm{\mid }$ for which $F$ depends nontrivially on $z_{\alpha }$. A classical solution of an equation of order at most $m$ is a function $u\in C^m(\mathrm{\Omega })$ satisfying the displayed equation at every $x\in \mathrm{\Omega }$.

If $\mathrm{\Sigma }\subseteq \mathrm{\Omega }$ is a codimension-one surface, prescribing values of $u$ and possibly of some derivatives on $\mathrm{\Sigma }$ is Cauchy data. If $\mathrm{\partial }\mathrm{\Omega }$ is used instead, the prescribed values are boundary data.

Consider an order-$m$ scalar PDE on $\mathrm{\Omega }$.

It is linear when it has the form

${\sum }_{\mathrm{\mid }\alpha \mathrm{\mid }\le m}{a}_{\alpha }(x)D^{\alpha }u(x)=f(x),$

so every jet variable enters affinely and with coefficients depending only on $x$.

It is semilinear when the top-order derivatives enter linearly with coefficients depending only on $x$, while lower-order terms may depend nonlinearly on $(u,Du,\dots ,D^{m-1}u)$.

It is quasilinear when the top-order derivatives still enter linearly, but their coefficients may depend on $(x,u,Du,\dots ,D^{m-1}u)$.

It is fully nonlinear when the dependence on the order-$m$ jet is not affine. Thus semilinear and quasilinear equations are affine in the top-order variables after the lower jet is fixed, whereas fully nonlinear equations are not. The classification is by dependence on the highest-order jet, not by how the equation happens to be typeset.

Let

$Lu={\sum }_{\mathrm{\mid }\alpha \mathrm{\mid }\le m}{a}_{\alpha }(x)D^{\alpha }u$

be a linear scalar differential operator of order $m$ on an open $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$.

Its principal part is the homogeneous order-$m$ operator

$L_{m}u:={\sum }_{\mathrm{\mid }\alpha \mathrm{\mid }=m}{a}_{\alpha }(x)D^{\alpha }u\mathrm{.}$

Its principal symbol is the homogeneous polynomial in the covector $\xi \in ({\mathbb{R}}^n{)}^{\ast }$ given by

$p_m(x,\xi ):={\sum }_{\mathrm{\mid }\alpha \mathrm{\mid }=m}{a}_{\alpha }(x){\xi }^{\alpha }\mathrm{.}$

If the equation is quasilinear, first freeze the lower jet $(u,Du,\dots ,D^{m-1}u)$ at the point under discussion; the resulting linear operator in the highest derivatives has a principal part and principal symbol defined by the same formula.

Let

$L={\sum }_{\mathrm{\mid }\alpha \mathrm{\mid }\le m}{a}_{\alpha }(x)D_x^{\alpha }$

be a linear scalar differential operator of order $m$, let $x=\mathrm{\Phi }(y)$ be a smooth local coordinate change with smooth inverse, and let $\tilde{L}$ be the operator in the $y$-coordinates defined by $\tilde{L}(v)=L(v\circ {\mathrm{\Phi }}^{-1})\circ \mathrm{\Phi }$. Then

${\tilde{p}}_m(y,\eta )=p_m(\mathrm{\Phi }(y),D\mathrm{\Phi }(y{)}^{-T}\eta )$

for every covector $\eta$, and every derivative of $\mathrm{\Phi }$ of order at least $2$ contributes only to lower-order terms of $\tilde{L}$.

Let $p_m(x,\xi )$ be the principal symbol of an order-$m$ scalar equation.

A nonzero covector $\xi \in T_x^{\ast }\mathrm{\Omega }$ is characteristic at $x$ when

$p_m(x,\xi )=0.$

If $\mathrm{\Sigma }\subseteq \mathrm{\Omega }$ is a $C^1$ hypersurface locally written as $\mathrm{\Sigma }\cap U=\{\varphi =0\}$ with $d\varphi \mathrm{\ne }0$ on $\mathrm{\Sigma }\cap U$, then $\mathrm{\Sigma }$ is characteristic at $x\in \mathrm{\Sigma }$ when its conormal $d\varphi (x)$ is characteristic, and noncharacteristic at $x$ otherwise.

Accordingly, Cauchy data prescribed on $\mathrm{\Sigma }$ are noncharacteristic near $x$ when the defining conormal is nowhere characteristic there.

Let $\mathrm{\Sigma }\cap U=\{\varphi =0\}=\{\psi =0\}$ be a $C^1$ hypersurface with $d\varphi \mathrm{\ne }0$ and $d\psi \mathrm{\ne }0$ on $\mathrm{\Sigma }\cap U$. If $p_m$ is the principal symbol of an order-$m$ scalar operator, then

$p_m(x,d\varphi (x))=0⟺p_m(x,d\psi (x))=0$

for every $x\in \mathrm{\Sigma }\cap U$.

For a real scalar second-order principal symbol

$p_2(x,\xi )={\sum }_{i,j=1}^n{a}_{ij}(x){\xi }_i{\xi }_j,$

assume $a_{ij}=a_{ji}$.

The symbol is elliptic at $x$ when $p_2(x,\xi )\mathrm{\ne }0$ for every $\xi \mathrm{\ne }0$; equivalently, the quadratic form has one definite sign.

Fix a nonzero covector $\tau$. The symbol is strictly hyperbolic at $x$ relative to $\tau$ when $p_2(x,\tau )\mathrm{\ne }0$ and, for every $\zeta$ not proportional to $\tau$, the polynomial $\lambda \mapsto p_2(x,\zeta +\lambda \tau )$ has two distinct real roots. The first condition says that $\tau$ is noncharacteristic and prevents the root condition from becoming vacuous in one dimension.

A space-time operator of the form

$u_t-{\sum }_{i,j=1}^n{a}_{ij}(x,t)u_{x_i{x}_j}+\text{lower-order terms}$

is parabolic at $(x,t)$ when the spatial quadratic form $\sum a_{ij}(x,t){\xi }_i{\xi }_j$ is positive semidefinite and nonzero, and, in the standard time covector, the equation contains a first-order time derivative rather than a second-order one. The nonzero condition ensures that the operator still has a second-order spatial principal part at the point.

Fix a point $x_0$ of a scalar second-order operator whose principal part is

${\sum }_{i,j=1}^n{a}_{ij}(x_0){\mathrm{\partial }}_{x_i{x}_j},a_{ij}(x_0)=a_{ji}(x_0)\mathrm{.}$

Then there is a linear change of coordinates near $x_0$ in which the frozen principal quadratic form becomes

${\xi }_1^2+\cdots +{\xi }_p^2-{\xi }_{p+1}^2-\cdots -{\xi }_{p+q}^2,$

with the remaining $r=n-p-q$ directions absent. The triple $(p,q,r)$ depends only on the quadratic form, not on the chosen coordinates.

For a second-order equation in two variables with principal part

$A(x,y)u_{xx}+2B(x,y)u_{xy}+C(x,y)u_{yy},$

the discriminant is

$\mathrm{\Delta }(x,y):=B(x,y{)}^2-A(x,y)C(x,y)\mathrm{.}$

At a point where $(A,B,C)\mathrm{\ne }(0,0,0)$, the principal part is called elliptic when $\mathrm{\Delta }<0$, parabolic when $\mathrm{\Delta }=0$, and hyperbolic when $\mathrm{\Delta }>0$. If $A=B=C=0$ at the point, the second-order principal part vanishes there and is degenerate rather than parabolic.

Here “parabolic” names the nonzero rank-one case in the pointwise classification of a binary second-order principal form. This is distinct from the heat-type space-time convention, which requires a first-order time derivative and a nonzero semidefinite spatial principal form.

For a scalar second-order equation in two variables, the sign of the discriminant $\mathrm{\Delta }=B^2-AC$ is unchanged by a smooth local coordinate change. Moreover, the characteristic directions are exactly the tangent directions whose normal covectors are characteristic, so they are also coordinate invariant.

Let

$A{u}_{xx}+2B{u}_{xy}+C{u}_{yy}+\text{lower-order terms}=0$

have constant real coefficients in its principal part.

If $B^2-AC<0$, an invertible linear change of variables and multiplication by a nonzero scalar reduce the principal part to $u_{\xi \xi }+u_{\eta \eta }$.

If $B^2-AC>0$, such a change reduces it to $u_{\xi \xi }-u_{\eta \eta }$.

If $B^2-AC=0$ but $(A,B,C)\mathrm{\ne }(0,0,0)$, such a change reduces it to $u_{\xi \xi }$.

The names on this page combine related but differently formulated pointwise notions: ellipticity and strict hyperbolicity are conditions on real scalar second-order principal symbols, whereas the stated parabolic condition is for a space-time operator with a first-order time derivative and a semidefinite spatial quadratic form. These notions do not by themselves give a global taxonomy of equations that change type from point to point, systems with matrix-valued principal symbols, higher-order equations, or fully nonlinear equations whose leading behavior is not a single quadratic form.

Let $\mathrm{\Omega }\subseteq {\mathbb{R}}^n\times \mathbb{R}$ be open and let $a:\mathrm{\Omega }\to {\mathbb{R}}^n$, $c:\mathrm{\Omega }\to \mathbb{R}$, and $f:\mathrm{\Omega }\to \mathbb{R}$. The scalar first-order equation

$u_t(x,t)+a(x,t)\cdot {\mathrm{\nabla }}_{x}u(x,t)+c(x,t)u(x,t)=f(x,t)$

is a linear transport equation.

Fix $(x_0,t_0)\in \mathrm{\Omega }$. A characteristic through $(x_0,t_0)$ is a solution $X(\cdot )$ of the ODE

$X^{\mathrm{\prime }}(s)=a(X(s),s),X(t_0)=x_0,$

on an interval containing $t_0$. The corresponding space-time curve is $s\mapsto (X(s),s)$. If these initial-value problems have unique solutions for varying initial data, write $X(s;t_0,x_0)$ for the solution through $(x_0,t_0)$. The resulting maps $x_0\mapsto X(s;t_0,x_0)$, on the domains where they are defined, form the characteristic flow of the transport field. Without uniqueness there are characteristics, but no single-valued characteristic flow.

Let $u\in C^1(\mathrm{\Omega })$ satisfy

$u_t+a\cdot {\mathrm{\nabla }}_{x}u+cu=f,$

and let $X$ be a characteristic solving $X^{\mathrm{\prime }}(s)=a(X(s),s)$. Then

$\frac{d}{ds}u(X(s),s)+c(X(s),s)u(X(s),s)=f(X(s),s)\mathrm{.}$

Let $a$ be $C^1$ on an open neighborhood of the graph of a characteristic $X(\cdot ;{\xi }_0)$ solving

$X^{\mathrm{\prime }}(t)=a(X(t),t),X(t_0;{\xi }_0)={\xi }_0,$

on a compact interval $I$ containing $t_0$. Then, after shrinking to a neighborhood $U_0$ of ${\xi }_0$, every initial point $\xi \in U_0$ determines a characteristic $X(\cdot ;\xi )$ on the same interval $I$, the map $(t,\xi )\mapsto X(t;\xi )$ is continuous on $I\times U_0$, and for each $t\in I$ the map $\xi \mapsto X(t;\xi )$ is $C^1$. Its Jacobian matrix $Y(t,\xi )=D_{\xi }X(t;\xi )$ satisfies

$Y^{\mathrm{\prime }}(t,\xi )=D_{x}a(X(t,\xi ),t)Y(t,\xi ),Y(t_0,\xi )=I_n\mathrm{.}$

Assume $a$ is regular enough that for each $(x,t)$ in a region $U$ the unique characteristic $s\mapsto (X(s;t,x),s)$ is defined from $s=t$ to $s=0$ and its whole segment remains in $U$, and assume these characteristics have the usual flow consistency. If $u\in C^1(U)$ solves

$u_t+a\cdot {\mathrm{\nabla }}_{x}u=0,u(x,0)=u_0(x),$

then

$u(x,t)=u_0(X(0;t,x))\mathrm{.}$

Hence there is at most one classical solution on $U$. Conversely, any $C^1$ function on $U$ satisfying this formula is that unique classical solution.

Let $U\subseteq {\mathbb{R}}^n\times \mathbb{R}$ be a region. Assume $a\in C^1(U;{\mathbb{R}}^n)$ and $c,f\in C(U)$, and assume that for each $(x,t)\in U$ there is a unique characteristic $X(s;t,x)$ on the closed interval with endpoints $0$ and $t$, its space-time graph remains in $U$, and the characteristic family has the usual flow consistency. If $u\in C^1(U)$ solves

$u_t+a\cdot {\mathrm{\nabla }}_{x}u+cu=f,u(x,0)=u_0(x),$

then

$u(x,t)=e^{-{\int }_0^{t}c(X(\tau ;t,x),\tau )d\tau }{u}_0(X(0;t,x))+{\int }_0^t{e}^{-{\int }_s^{t}c(X(\tau ;t,x),\tau )d\tau }f(X(s;t,x),s)ds\mathrm{.}$

Conversely, any $C^1$ function satisfying this formula solves the transport equation on $U$.

Let $\mathrm{\Omega }\subseteq {\mathbb{R}}^n\times \mathbb{R}$ be the domain of the transport equation, and let $\sigma :U\subseteq {\mathbb{R}}^n\to \mathrm{\Omega }$ be a $C^1$ parametrized hypersurface, written

$\sigma (\eta )=(\gamma (\eta ),\tau (\eta ))\mathrm{.}$

For the transport equation $u_t+a(x,t)\cdot {\mathrm{\nabla }}_{x}u+c(x,t)u=f(x,t)$, let the space-time transport vector be

$B(x,t):=(a(x,t),1)\in {\mathbb{R}}^{n+1}\mathrm{.}$

The surface $\sigma$ is noncharacteristic at ${\eta }_0$ when the $(n+1)\times (n+1)$ matrix whose columns are the $n$ tangent vectors ${\mathrm{\partial }}_{{\eta }_1}\sigma ({\eta }_0),\dots ,{\mathrm{\partial }}_{{\eta }_n}\sigma ({\eta }_0)$ and the vector $B(\sigma ({\eta }_0))$ has nonzero determinant. Equivalently, $B(\sigma ({\eta }_0))$ is not tangent to the data surface.

A prescribed datum for the transport equation is a function $g:U\to \mathbb{R}$, interpreted as the Cauchy condition $u(\gamma (\eta ),\tau (\eta ))=g(\eta )$.

Let $a,c,f$ be $C^1$ near a point $(x_{\ast },t_{\ast })$, and let $\sigma :U\subseteq {\mathbb{R}}^n\to {\mathbb{R}}^n\times \mathbb{R}$ be a $C^1$ parametrized hypersurface written $\sigma (\eta )=(\gamma (\eta ),\tau (\eta ))$, with datum $g:U\to \mathbb{R}$ of class $C^1$. Assume $\sigma ({\eta }_{\ast })=(x_{\ast },t_{\ast })$. If $\sigma$ is noncharacteristic at ${\eta }_{\ast }$, then there are neighborhoods $V$ of ${\eta }_{\ast }$ and $W$ of $(x_{\ast },t_{\ast })$ and a unique $u\in C^1(W)$ such that

$u_t+a\cdot {\mathrm{\nabla }}_{x}u+cu=f\text{on }W,$

and

$u(\gamma (\eta ),\tau (\eta ))=g(\eta )(\eta \in V)\mathrm{.}$