a system of PDEs

So far we have mostly imagined a single unknown function. But the world is often coupled: the velocity of a fluid, the electric and magnetic fields, the concentrations of several reacting chemicals — these involve several unknowns at once, each evolving in a way that depends on the others. A system of PDEs is several equations for several unknown functions, tied together so that you must solve them all simultaneously. A single equation for a single unknown is, by contrast, called a scalar equation.

Concretely, instead of one u you have a vector of unknowns, say (u, v, w), and a set of equations each relating their partial derivatives. A famous example is the Maxwell equations, four coupled PDEs linking the electric and magnetic fields; another is the Navier-Stokes equations of fluid flow, coupling the three velocity components and the pressure. The coupling is the essential feature: you generally cannot solve one equation in isolation because each unknown appears in several equations. There is also a deep two-way street between systems and scalar equations: a single high-order scalar equation can be rewritten as a first-order system by naming its derivatives as new unknowns (just as a second-order ODE becomes two first-order ODEs), and this trick is the basis of much numerical and theoretical work.

Systems matter because most real physics is a system, and because they raise questions a scalar equation cannot. Classification (hyperbolic, parabolic, elliptic) extends to systems but becomes richer — a first-order system can have several characteristic speeds, one per 'mode', and the right notion is something like symmetric hyperbolicity that guarantees the whole coupled set is well-posed. The unsolved global regularity of the three-dimensional Navier-Stokes system — a Millennium Prize problem — is a vivid reminder that systems can be profoundly harder than any single equation, and that some of the hardest open questions in mathematics live exactly here.