14.5 Review of the course

The course has followed a small number of ideas through many equations.

Type decides behaviour. First-order equations and hyperbolic second-order equations carry information along characteristics at finite speed (Sections 1, 2, 4, 5): they conserve energy, keep singularities, and may form shocks if non-linear. Parabolic equations smooth instantly and propagate at infinite speed, and run only forwards in time (Section 6). Elliptic equations describe equilibria, have no characteristics, and are determined by their boundary values (Sections 8, 9). Section 3 showed why: the sign of a discriminant.

The right data make a problem well posed. Cauchy data for hyperbolic equations, initial and boundary data for parabolic, boundary data alone for elliptic; Hadamard’s examples show what goes wrong otherwise.

Explicit solutions. Characteristics and d’Alembert’s formula; spherical means, Kirchhoff and Poisson; the heat kernel; the Poisson kernel and Green’s functions with images; separation of variables with Fourier, Bessel and Legendre expansions (Sections 7, 11); Fourier and Laplace transforms (Section 10).

Estimates without solutions. The maximum principle and energy identities give uniqueness, stability, comparison and decay in any region (Sections 6, 8, 12); the Dirichlet principle and Hilbert-space methods give existence of weak solutions; and the discrete versions of the same estimates decide whether a numerical scheme converges (Section 13).

Non-linearity. Shocks and entropy conditions for conservation laws, fronts and patterns for reaction–diffusion: phenomena with no linear counterpart, studied with the linear tools of the rest of the course.

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