6 1. INTRODUCTION

1.2.2. Systems of partial diﬀerential equations.

a. Linear systems.

1. Equilibrium equations of linear elasticity

μΔu + (λ + μ)D(div u) = 0.

2. Evolution equations of linear elasticity

utt − μΔu − (λ + μ)D(div u) = 0.

3. Maxwell’s equations

⎧

⎨

⎩

Et = curl B

Bt = − curl E

div B = div E = 0.

b. Nonlinear systems.

1. System of conservation laws

ut + div F(u) = 0.

2. Reaction-diﬀusion system

ut − Δu = f(u).

3. Euler’s equations for incompressible, inviscid flow

ut + u · Du = −Dp

div u = 0.

4. Navier–Stokes equations for incompressible, viscous flow

ut + u · Du − Δu = −Dp

div u = 0.

See Zwillinger [Zw] for a much more extensive listing of interesting PDE.

1.3. STRATEGIES FOR STUDYING PDE

As explained in §1.1 our goal is the discovery of ways to solve partial diﬀer-

ential equations of various sorts, but—as should now be clear in view of the