In the theory of hyperbolic partial differential equations, specifically first-order equations of the form $F(\nabla u, u, x) = 0$, classical solutions $u \in C^1(\Omega)$ often fail to exist globally. This breakdown occurs when the characteristic curves intersect or focus, causing the coordinate transformation from parameters to physical space to become singular. The **Envelope of Shocks** (or caustic) is the geometric locus in the domain $\Omega$ where this singularity occurs. ## Motivation Nonlinear transport equations propagate information along characteristic curves. Unlike in the linear case, the geometry of these curves often depends on the solution value $u$. This can cause the curves to converge. The envelope marks the precise [boundary](/page/Boundary) in the domain where the "[method of characteristics](/page/Method%20of%20Characteristics)" fails. At these points, the mapping from the boundary parametrization to the domain ceases to be invertible. Physically, this typically corresponds to the gradient $\nabla u$ becoming infinite (gradient catastrophe), signaling the onset of a discontinuous (weak) solution or shock wave. ## Formal Definition To define the envelope rigorously on a potentially curved boundary $\Gamma$, we must work within local coordinate charts. This allows us to identify the abstract derivative of the flow map with a concrete Jacobian matrix of numbers. [definition:EnvelopeOfCharacteristics] Let $(V, \Phi)$ be a local coordinate chart for the boundary $\Gamma$ around a point $\tilde{c}$, where $V' = \Phi(\Gamma \cap V) \subset \mathbb{R}^{n-1}$ is the flattened parameter space with coordinates $y' = (y_1, \dots, y_{n-1})$. Let $\Psi: I \times V' \to \Omega$ be the **Local Characteristic Flow Map** defined by $\Psi(s, y') = X(s, y')$, where $X(s, y')$ is the spatial component of the solution to the characteristic ODE system. The **Jacobian Matrix** $D\Psi(s, y')$ is the $n \times n$ matrix identified by the column vectors corresponding to the characteristic velocity and the wavefront tangent vectors: \begin{align*} D\Psi(s, y') = \begin{pmatrix} \vline & \vline & & \vline \\ \frac{\partial X}{\partial s} & \frac{\partial X}{\partial y_1} & \cdots & \frac{\partial X}{\partial y_{n-1}} \\ \vline & \vline & & \vline \end{pmatrix} \end{align*} The **Envelope of Shocks** (relative to this chart) is the set of points $x \in \Omega$ where this matrix is singular: \begin{align*} \det(D\Psi(s, y')) = 0 \end{align*} The full envelope is the union of such [sets](/page/Set) over an atlas covering $\Gamma$. [/definition] > **Note: Why do we work locally?** > The concept of the envelope is geometric and global, but the calculation of a **determinant** requires representing a [linear map](/page/Linear%20Map) as a matrix. To form a matrix, we must choose a basis for the tangent space. On a general curved manifold $\Gamma$ (like a sphere), there is no single global coordinate system (basis) that works everywhere without singularity. > > By restricting ourselves to a local chart $(V, \Phi)$, we secure a valid local basis $(\frac{\partial}{\partial y_1}, \dots, \frac{\partial}{\partial y_{n-1}})$. This allows us to compute the numerical value of $\det(D\Psi)$ unambiguously. The condition $\det(D\Psi)=0$ is invariant under coordinate changes, so checking it in local charts is sufficient to define the envelope globally. ## Examples [example:BurgersSquared] Consider the conservation law problem on $\Omega = \{ (x_1, x_2) \in \mathbb{R}^2 : x_1 > 0 \}$ (where $x_1$ represents time $t$ and $x_2$ represents space): \begin{align*} \frac{\partial u}{\partial x_1} + u^2 \frac{\partial u}{\partial x_2} = 0, \quad u(0, x_2) = x_2. \end{align*} Here, the [function](/page/Function) $F: \mathbb{R}^2 \times \mathbb{R} \times \mathbb{R}^2 \to \mathbb{R}$ is defined by: \begin{align*} F(a, b, c) = a_1 + b^2 a_2 \end{align*} **(1) The ODE System** We compute the necessary partial [derivatives](/page/Derivative) of $F$: \begin{align*} \frac{\partial F}{\partial a_1} = 1, \quad \frac{\partial F}{\partial a_2} = b^2, \quad \frac{\partial F}{\partial b} = 2b a_2, \quad \frac{\partial F}{\partial c_i} = 0. \end{align*} The characteristic system for the trajectory $T(s) = (G(s), U(s), X(s))$ becomes: \begin{align*} \dot{X}_1 &= 1 \\ \dot{X}_2 &= U^2 \\ \dot{U} &= G_1(1) + G_2(U^2) = a \cdot \nabla_a F = 0 \quad (\text{since } F=0 \text{ implies } a \cdot \nabla_a F = 0 \text{ for homogeneous parts}) \end{align*} **(2) The Flow Map $\Psi$** We parametrize the boundary $\Gamma$ (the $x_2$-axis at $x_1=0$) using the identity chart, so $y' \in \mathbb{R}$ corresponds to the initial position $x_2$. The initial conditions at $s=0$ are: \begin{align*} X(0, y') &= (0, y') \\ U(0, y') &= g(X(0, y')) = y' \end{align*} Solving the ODEs: * $X_1(s, y') = s$ * $U(s, y') = y'$ (constant along characteristic) * $X_2(s, y') = X_2(0) + \int_0^s U^2 d\tau = y' + (y')^2 s$ Thus, the Characteristic Flow Map is: \begin{align*} \Psi(s, y') = \begin{pmatrix} s \\ y' + s(y')^2 \end{pmatrix} \end{align*} **(3) The Envelope** We compute the Jacobian matrix $D\Psi$ with respect to the parameters $(s, y')$: \begin{align*} D\Psi = \begin{pmatrix} \frac{\partial X_1}{\partial s} & \frac{\partial X_1}{\partial y'} \\ \frac{\partial X_2}{\partial s} & \frac{\partial X_2}{\partial y'} \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ (y')^2 & 1 + 2s y' \end{pmatrix} \end{align*} The determinant is: \begin{align*} \det(D\Psi) = 1(1 + 2s y') - 0 = 1 + 2s y' \end{align*} The envelope occurs where $\det(D\Psi) = 0$: \begin{align*} 1 + 2s y' = 0 \implies y' = -\frac{1}{2s} \end{align*} Substituting this critical parameter back into the map for $X_2$: \begin{align*} X_2 &= \left(-\frac{1}{2s}\right) + s \left(-\frac{1}{2s}\right)^2 = -\frac{1}{2s} + \frac{1}{4s} = -\frac{1}{4s}. \end{align*} Since $X_1 = s$, the curve in physical space is the hyperbola $x_2 = -\frac{1}{4x_1}$. [/example] ## Key Results [theorem:GradientBlowup] Let $u$ be a solution constructed via the Method of Characteristics. If the Jacobian determinant $\det(D\Psi)$ approaches zero at a point $x_0$ (and the transported values remain bounded), then $|\nabla u(x_0)| \to \infty$. [/theorem] [proof] Recall that the gradient $\nabla u$ at a point $x = \Psi(s, y')$ is related to the derivative of the transported value $U$ by the Chain Rule. Differentiating $u(\Psi(s, y')) = U(s, y')$ with respect to the parameters $(s, y')$ yields the linear system: \begin{align*} (D\Psi)^T \nabla u = \begin{pmatrix} \dot{U} \\ \nabla_{y'} U \end{pmatrix} \end{align*} To solve for $\nabla u$, we must invert the transpose of the Jacobian matrix $(D\Psi)^T$. By Cramer's rule, the inverse matrix contains the term $1/\det(D\Psi)$. Thus, as the determinant vanishes, the components of $\nabla u$ become unbounded, provided the vector on the right-hand side is not orthogonal to the null space of $(D\Psi)^T$. [/proof] ## Problems [problem] Consider the inviscid Burgers' equation with inverted data: \begin{align*} \frac{\partial u}{\partial x_1} + u \frac{\partial u}{\partial x_2} = 0, \quad u(0, x_2) = -x_2. \end{align*} Using the formal notation $F(a, b, c) = a_1 + b a_2$, construct the flow map $\Psi$ and find the time $x_1$ where the Jacobian determinant first vanishes. [/problem] [solution] **1. Setup:** Here $F = a_1 + b a_2$. $\frac{\partial F}{\partial a_1} = 1$, $\frac{\partial F}{\partial a_2} = b$. The spatial ODEs are: \begin{align*} \dot{X}_1 = 1, \quad \dot{X}_2 = U. \end{align*} Since $F=0$ along the solution, we have $\dot{U} = 0$. **2. Flow Map:** Parametrize the boundary by $y'$. $X(0, y') = (0, y')$, $U(0, y') = -y'$. Solutions: \begin{align*} X_1(s, y') &= s \\ U(s, y') &= -y' \\ X_2(s, y') &= y' + \int_0^s (-y') d\tau = y'(1 - s) \end{align*} The map is $\Psi(s, y') = (s, y'(1-s))$. **3. Jacobian Analysis:** \begin{align*} D\Psi = \begin{pmatrix} 1 & 0 \\ -y' & 1 - s \end{pmatrix} \end{align*} Determinant: \begin{align*} \det(D\Psi) = 1(1-s) = 1 - s. \end{align*} The determinant vanishes when $1 - s = 0$, or $s = 1$. Since $X_1 = s$, the envelope (and the first shock) forms at time $x_1 = 1$. [/solution] ## References - Evans, L. C. (2010). *Partial Differential Equations*. American Mathematical Society. - John, F. (1982). *Partial Differential Equations*. Springer-Verlag.