Unit 2: The Cauchy Problem, Method of Characteristics & Non-Linear First-Order PDEs

1. Rigorous Formulation of the Cauchy Problem

Consider the quasilinear partial differential equation:

on an open domain $\Omega \subseteq \mathbb{R}^2$. Let $\Gamma \subset \mathbb{R}^3$ be a simple regular $C^1$ space curve parametrized by arc parameter $s \in [a, b]$:

The Cauchy problem consists in determining a $C^1$ surface $z = u(x, y)$ defined on a neighborhood of the projection curve $\gamma_0 = (x_0(s), y_0(s))$ such that:

1. $u(x, y)$ satisfies the PDE on $\Omega$: $P(x, y, u) u_x + Q(x, y, u) u_y = R(x, y, u)$.
2. The surface contains $\Gamma$: $u(x_0(s), y_0(s)) = z_0(s)$ for all $s \in [a, b]$.

2. Characteristic System and Coordinate Transformation

To solve the Cauchy problem, we construct the characteristic curves emanating from each point of $\Gamma$. For each fixed $s$, consider the initial value problem for the autonomous ODE system:

By the Picard-Lindelöf theorem for ODEs, since $P, Q, R$ are $C^1$, this system possesses a unique local solution $(X(s, t), Y(s, t), Z(s, t))$ defined for $(s, t) \in [a, b] \times (-\delta, \delta)$.

3. The Local Existence and Uniqueness Theorem

Theorem 2.1 (Cauchy-Kovalevskaya Local Existence for Quasilinear First-Order PDEs): Let $P, Q, R \in C^1(\mathbb{R}^3)$ and let $\Gamma = (x_0(s), y_0(s), z_0(s))$ be a $C^1$ curve. Suppose that at $s = s_0$:

Then there exists an open neighborhood $U$ of $(x_0(s_0), y_0(s_0))$ in $\mathbb{R}^2$ containing a unique $C^1$ function $z = u(x, y)$ solving the Cauchy problem.

Proof: Consider the planar mapping $\Phi: (s, t) \mapsto (X(s, t), Y(s, t))$. At $t = 0$, the Jacobian matrix of $\Phi$ is:

Its determinant is precisely:

By the Inverse Function Theorem, $\Phi$ is a local $C^1$-diffeomorphism from an open neighborhood $V$ of $(s_0, 0)$ onto an open neighborhood $U$ of $(x_0(s_0), y_0(s_0))$. Thus, we can invert the mapping smoothly:

Now define the candidate solution surface by:

Along the initial curve ($t = 0$), $u(x_0(s), y_0(s)) = Z(s, 0) = z_0(s)$, satisfying the initial condition. To verify that $u$ satisfies the PDE, differentiate $Z(s, t) = u(X(s, t), Y(s, t))$ with respect to $t$:

Since $(X, Y, Z)$ are characteristic trajectories, $X_t = P, Y_t = Q, Z_t = R$:

Hence, $u(x, y)$ is a genuine $C^1$ solution. Uniqueness follows because any integral surface containing $\Gamma$ must contain all characteristic curves issuing from $\Gamma$. $\blacksquare$

1. Wave Steepening in Nonlinear Conservation Laws

Consider the inviscid Burgers equation, the prototype for nonlinear wave propagation and gas dynamics:

subject to initial profile $u(x, 0) = f(x)$.

The characteristic equations are:

Along a characteristic trajectory issuing from $(x_0, 0)$:

1. $u(x, t) = f(x_0)$ is constant along each characteristic.
2. The characteristic curve is a straight line in the $(x, t)$ plane:

Thus, points with higher values of $u$ travel with greater speed $c = u$.

2. Gradient Catastrophe and Shock Formation Time

Differentiating the implicit relation $u = f(x - u t)$ with respect to $x$:

Solving for the spatial gradient $u_x$:

Theorem 2.2 (Gradient Catastrophe / Breaking Time): If $f'(x_0) \ge 0$ for all $x_0 \in \mathbb{R}$ (rarefaction / expansion profile), characteristics diverge forward in time and the classical $C^1$ solution exists for all $t > 0$. If there exists any point $x_0$ where $f'(x_0) < 0$ (compressive profile), the denominator vanishes at a finite time. The earliest such time, called the breaking time (or shock formation time), is given by:

At $t = t_{\text{shock}}$, the gradient $u_x \to -\infty$, characteristics intersect, and the single-valued classical solution ceases to exist.

3. Weak Solutions and the Rankine-Hugoniot Condition

Beyond $t > t_{\text{shock}}$, physics dictates the formation of a shock wave: a propagating jump discontinuity across a curve $x = s(t)$.

Definition 2.1 (Rankine-Hugoniot Jump Condition): For a scalar conservation law $u_t + (f(u))_x = 0$, the propagation speed $\dot{s}(t) = \frac{ds}{dt}$ of a discontinuity separating state $u_L$ on the left and $u_R$ on the right satisfies:

For Burgers' equation $f(u) = \frac{1}{2}u^2$:

The shock speed is the arithmetic mean of the states immediately to its left and right.

1. Contact Elements and the Geometry of Solution Surfaces

Let $F(x, y, z, p, q) = 0$ be a general non-linear first-order PDE. At any point $P_0(x_0, y_0, z_0)$, a planar element through $P_0$ is defined by:

The set $(x_0, y_0, z_0, p, q)$ is called a contact element or tangent element. For a fixed point $(x_0, y_0, z_0)$, the PDE $F(x_0, y_0, z_0, p, q) = 0$ is a relation between $p$ and $q$. Thus, at each point, there is not a single tangent plane (as in linear PDEs), but a one-parameter family of tangent planes:

where $F(x_0, y_0, z_0, p(t), q(t)) = 0$.

2. The Monge Cone

Definition 2.2 (Monge Cone): The envelope of this one-parameter family of tangent planes passing through $(x_0, y_0, z_0)$ is a cone with vertex at $(x_0, y_0, z_0)$, known as the Monge cone. Every integral surface $z = u(x, y)$ passing through $(x_0, y_0, z_0)$ must have a tangent plane that belongs to this family; consequently, the integral surface must be tangent to the Monge cone along a straight-line generator!

To determine the generators of the Monge cone, differentiate the tangent plane equation with respect to parameter $t$:

Differentiating $F(x_0, y_0, z_0, p(t), q(t)) = 0$ with respect to $t$:

Substituting this into the envelope equation:

Along this generator, the increment $dZ$ on the tangent plane is:

Therefore, the direction ratios of the generator of the Monge cone are:

This direction is the characteristic direction for non-linear equations.

1. Compatibility and Charpit's Auxiliary Equations

Paul Charpit (1784) established the universal method for finding a complete integral of an arbitrary non-linear PDE $F(x, y, z, p, q) = 0$. The foundational strategy is to seek a second relation:

such that $F = 0$ and $\Phi = a$ are compatible: that is, solving them simultaneously for $p$ and $q$:

makes the Pfaffian differential form:

an exact (integrable) differential.

By Frobenius' integrability theorem, the integrability condition is:

2. Complete Derivation of Charpit's System

Differentiating both $F(x, y, z, p, q) = 0$ and $\Phi(x, y, z, p, q) = a$ with respect to $x$ (treating $z$ as depending on $x, y$):

Similarly differentiating with respect to $y$:

Eliminating the mixed second derivatives $\frac{\partial p}{\partial x}, \frac{\partial q}{\partial x}, \frac{\partial p}{\partial y}, \frac{\partial q}{\partial y}$ using $\frac{\partial p}{\partial y} = \frac{\partial q}{\partial x}$ leads to the linear first-order PDE for $\Phi$:

Applying Lagrange's auxiliary ODE method to this equation gives Charpit's Auxiliary Equations:

Theorem 2.3 (Charpit's Auxiliary System):

or with signs reversed:

Any non-trivial first integral $\Phi(x, y, z, p, q) = a$ obtained from this system provides the compatible relation. Solving for $p$ and $q$ and integrating $dz = p dx + q dy$ yields the complete integral with two arbitrary constants $a$ and $b$.

1. The Four Classical Standard Forms

When the non-linear equation $F(x, y, z, p, q) = 0$ lacks certain variables, Charpit's system simplifies drastically into four canonical forms:

Standard Form I: Equations Involving Only $p$ and $q$ ($f(p, q) = 0$)

Here $F_x = F_y = F_z = 0$. Charpit's equations give:

Substituting $p = a$ into $f(p, q) = 0$ gives $q = Q(a)$ (constant). Integrating $dz = p dx + q dy = a dx + Q(a) dy$:

This is the complete integral, representing a family of planes.

Standard Form II: Equations Not Involving $x$ and $y$ ($f(z, p, q) = 0$)

Here $F_x = F_y = 0$. Charpit's system gives $\frac{dp}{p F_z} = \frac{dq}{q F_z} \implies \frac{dp}{p} = \frac{dq}{q} \implies q = a p$. Assume $z = Z(u)$ where $u = x + a y$. Then:

Substituting into $f(z, p, q) = 0$ yields an ODE for $Z(u)$:

Standard Form III: Separable Equations ($f(x, p) = g(y, q)$)

Here $x$ and $p$ can be separated from $y$ and $q$. Set each side equal to an arbitrary constant $a$:

Integrating the total differential:

Standard Form IV: Clairaut's Equation ($z = p x + q y + f(p, q)$)

As established in Unit 1, setting $p = a$ and $q = b$ directly yields the complete integral: