Homework 6: Hyperbolic Equations

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Exercise 1. Burger's Equations

a.- Find an infinite number of conserved quantities for Burger's equations, \(u_t = u * u_x\).

b.- Let the initial data for Burger's equation be: \(u_0(x) = heaviside(x)*heaviside(1-x)\). Draw the characteristic surfaces for that data.

c.- Use the equation and integration by parts to get an estimate of the form:

\[ \frac{d \mathcal{E}}{dt} \leq \max{|u_x|} \mathcal{E} \]

where

\[ \mathcal{E} := \int [u^2 + u_x^2 + u_{xx}^2]\;dx \]

Exercise 2. Propagation Cones

a.- Find a covector \(n_a\) so that the following system is hyperbolic:

\[\begin{aligned} l_1^a \nabla_a \phi_1 &= F_1(\phi_1,\phi_2,\phi_3) \\ l_2^a \nabla_a \phi_2 &= F_2(\phi_1,\phi_2,\phi_3) \\ l_3^a \nabla_a \phi_3 &= F_3(\phi_1,\phi_2,\phi_3) \end{aligned}\]

where the three vectors are linearly independent. What that condition on \(n_a\) geometrically means? Find the Cones and Propagation Cones. Understand geometrically both constructions.

b.- How many cones and co-cones can you construct by multiplying some of the equations by \(-1\)?

c.- Find the cones and co-cones for the fluid equations:

\[\begin{aligned} \rho_t &= -(\rho u)_x \;\;\;\;\; \text{particle number conservation}\\ (\rho u)_t &= -(\frac{1}{2}\rho u^2 + p)_x\;\;\;\; \text{momentum conservation} \end{aligned}\]

where \(p = p(\rho)\) is the pressure, \(\rho\) the density and \(u\) the fluid velocity.

Exercise 3. Electromagnetism

Consider the evolutionary Maxwell equations:

\[\begin{aligned} \frac{\partial \vec{E}}{\partial t} &= \nabla \wedge \vec{B} \\ \frac{\partial \vec{B}}{\partial t} &= -\nabla \wedge \vec{E} \end{aligned}\]

a.- Find the cones and co-cones.

b.- Show that:

\[ \mathcal{E} := \int [\vec{E}\cdot \vec{E} + \vec{B} \cdot \vec{B}]\; d^3x \]

is conserved.

The covariant treatment, including constraints, can be see from the example given in page 34 of Geroch article: https://arxiv.org/abs/gr-qc/9602055. There a generic hyperbolization is found for Electromagnetism in terms of the Faraday tensor \(F_{ab}\).