Quasilinear partial differential equation

A quasilinear partial differential equation is linear in its highest-order derivatives, while its coefficients may depend on the independent variables, the unknown function, and lower-order derivatives.

For a $C^1$ flux $F$ and a weak solution of $\mathrm{div}F(Du)=0$, discrete integration by parts and the fundamental theorem of calculus along a line segment show that $w_h={\delta }_{\ell ,-h}u$ solves $\mathrm{div}(A_{h}D{w}_h)=0$, with $p_{-}=Du(x-h{e}_{\ell })$ and $p_{+}=Du(x)$. Coercive quadratic-form bounds and operator norm bounds for $DF$ on all segments between these gradients pass to $A_h$. This produces a divergence-form elliptic operator without initially assuming second partial derivatives of $u$.

The p-Laplacian is the nonlinear divergence-form operatorIt is the Euler-Lagrange operator of the p-energy. For $p>2$ it is degenerately elliptic where $Du=0$.

For a scalar differential operator of order $k$, the principal symbol replaces each order-$k$ derivative ${\partial }^{\alpha }$ by ${\xi }^{\alpha }$ and discards lower-order terms. For a quasilinear partial differential equation, the coefficients are evaluated at the prescribed lower-order data.

A hypersurface $\Sigma =\{\varphi =0\}$ is characteristic for a differential equation at $x\in \Sigma$ when its principal symbol vanishes on the conormal $d\varphi (x)$:It is non-characteristic where this quantity is nonzero.

For a scalar second-order partial differential equation in two variables, a tangent direction along which Cauchy data fail to determine the highest transverse derivative through the equation. For principal part $A{u}_{xx}+2B{u}_{xy}+C{u}_{yy}$, a tangent $(dx,dy)$ is characteristic exactly when $A(dy{)}^2-2Bdxdy+C(dx{)}^2=0$. Equivalently its conormal annihilates the principal symbol. A hyperbolic partial differential equation has two distinct real characteristic directions at each point.

For principal part $A{u}_{xx}+2B{u}_{xy}+C{u}_{yy}$, the displayed quadratic relation determines the characteristic directions. Curves tangent to them are characteristic curves. Lower-order terms do not enter this relation, but they may be singular where the principal coefficients alone suggest an otherwise ordinary classification. In particular, classifying a principal part on a singular line does not extend the full differential equation onto that line.

A characteristic plane is a planar characteristic hypersurface. For $u_{zz}-m^2{u}_{xx}=0$, the planes $x\pm mz=\text{constant}$ carry the two travelling-wave families.

A non-characteristic hypersurface satisfies $p(x,d\varphi (x))\ne 0$. This lets the equation solve for the highest derivative normal to the hypersurface.

A characteristic coordinate is a function constant along one family of characteristics. For a second-order equation in two variables, two independent characteristic coordinates reduce the principal part to a mixed derivative.

For a hyperbolic partial differential equation in two variables, two independent characteristic coordinates remove both pure second partial derivatives, leaving a nonzero mixed second partial derivative. A nonzero Jacobian determinant is essential: two characteristic invariants that become dependent do not give a valid coordinate transformation.

In $x,y>0$, the change of variables in the title reducesto the one-dimensional wave equation $U_{ss}-U_{rr}=0$. Indeed $u_{xx}=U_{rr}/(4x)-U_r/(4x^{3/2})$ and $u_{yy}=y{U}_{ss}/4+U_s/(4\sqrt{y})$; the first-derivative terms cancel exactly against the right-hand side. The independent characteristic coordinates $\xi =r+s$, $\eta =r-s$ then give $V_{\xi \eta }=0$ and $u=g(\xi )+h(\eta )$. Two characteristic families alone generally reduce only the principal part; the cancellation of lower-order terms is the extra feature that gives this simple exact solution.

The change of variables $X=\rho e^{-t}$, $S=e^{-t}>0$ turnsinto $w_{SS}-w_{XX}=0$. Indeed, with $D={\partial }_t+\rho {\partial }_{\rho }$, the original operator is $D^2+D-{\partial }_{\rho }^2$, while $D=-S{\partial }_S$ and ${\partial }_{\rho }=S{\partial }_X$. Its characteristic curves are $\rho =-1+C{e}^t$ and $\rho =1+C{e}^t$, including the characteristic lines $\rho =\pm 1$. General smooth solutions have the form $F(e^{-t}(\rho +1))+G(e^{-t}(\rho -1))$. Zero Cauchy data on $-1<\rho <1$ at $t=0$ force vanishing throughout that strip for $t\ge 0$, but not for all negative time: $u=(e^{-t}(\rho +1)-2{)}_{+}^3$ is a global $C^2$ counterexample. Thus the characteristic barriers describe a forward domain of dependence, not a barrier in both time directions.

A scalar second-order equationis hyperbolic where $B^2-AC>0$. Equivalently, its principal symbol has two distinct real characteristic directions.

This partial differential equation has discriminant $y$, so it is a hyperbolic partial differential equation above $y=0$ and an elliptic partial differential equation below it. In its hyperbolic region the displayed characteristic coordinates have Jacobian determinant $18\sqrt{y}$. The chain rule givesThe first derivative terms arise from differentiating the coordinate coefficients and cannot be discarded. The coordinate change degenerates at the line where the two characteristic directions merge.

On $x>0$, the linear partial differential equation $u_{xx}-x^2{u}_{yy}-u_x/x=0$ has characteristic curves $y\pm x^2/2=\text{constant}$. The change of variables $\xi =x^2-2y$, $\eta =x^2+2y$ has Jacobian determinant $8x\ne 0$. The chain rule givesThus its canonical form of a second-order partial differential equation is $U_{\xi \eta }=0$, and its general twice differentiable solution is $u=f(x^2-2y)+g(x^2+2y)$. The first derivative term exactly cancels the derivatives of the coordinate coefficients.

More generally, for positive differentiable $a(x)$, choose $s(x)$ with $s^{\prime }(x)=a(x)$. Then the same cancellation givesThe characteristic coordinates are $s\pm y$. The quadratic example is $a(x)=x$, $s=x^2/2$, with both coordinates multiplied by two.

For the Euler-type operator $-x^2{\partial }_x^2-x{\partial }_x+{\partial }_y^2$, set $z=\ln |x|$ on either half-plane $x\ne 0$. The chain rule gives $x{\partial }_x={\partial }_z$ and $x^2{\partial }_x^2+x{\partial }_x={\partial }_z^2$. The operator becomes ${\partial }_y^2-{\partial }_z^2$. Its characteristic coordinates $\xi =y+z$, $\eta =y-z$ reduce it further to $4{\partial }_{\xi }{\partial }_{\eta }$. Thus the constant-forcing equation $-x^2{u}_{xx}-x{u}_x+u_{yy}=c$ has solutions $u=(c/4)\xi \eta +g(\xi )+h(\eta )$, with arbitrary sufficiently differentiable functions $g,h$. The logarithmic change of variables cannot by itself describe smooth passage across $x=0$.

For a one-dimensional first-order system $v_t+A(v,x,t)v_x=s$, hyperbolicity means that the coefficient matrix has real eigenvalues and a complete set of eigenvectors. Distinct real eigenvalues give strict hyperbolicity and separate characteristic curves. Along a curve of speed $\lambda$, a left eigenvector $\ell A=\lambda \ell$ gives the compatibility equation $\ell dv/dt=\ell s$.

A first-order system $U_t+A(U)U_x=C(U)$ is strictly hyperbolic at a state when $A$ has distinct real eigenvalues. These separate its characteristic families and permit independent compatibility equations from left eigenvectors. At a repeated eigenvalue strict hyperbolicity fails; if the matrix is also defective, even hyperbolicity in the complete-eigenvector sense fails. For $A=\left(\begin{array}{cc}u& 2\\ 2v& u\end{array}\right)$ the eigenvalues $u\pm 2\sqrt{v}$ are distinct precisely when $v>0$.

For a one-dimensional wave equation whose principal symbol is ${\xi }_t^2-c(y{)}^2{\xi }_y^2$, a travel-time coordinate isIts two families of characteristic curves are $s(y)-t=\text{constant}$ and $s(y)+t=\text{constant}$ wherever the wave speed $c$ is positive.

For $c(y)=1-y^2$ in the strip $|y|<1$, the travel-time coordinate is $s(y)=\mathrm{artanh}y$. The characteristic curves areequivalently $y=\tanh (C\mp x)$. Outside the strip the same separated ordinary differential equation gives hyperbolic cotangent branches, and the equilibrium curves $y=\pm 1$ are also characteristic.