Loading field map
A partial differential equation (PDE) is an equation that relates a function of several variables to its own partial derivatives. Where an ordinary differential equation describes a system evolving along a single independent variable—typically time—a PDE describes a quantity that varies across a spatial domain, or across both space and time. The unknown is not a number or a curve but a field: a temperature distribution, a fluid velocity, an electromagnetic potential, a probability density. PDEs are the language in which most of classical and modern physics, and much of geometry and finance, is written.
The subject is defined less by a single method than by a central difficulty: the space of possible functions is infinite-dimensional, and the equations impose constraints that are local, relating the value of a function at a point to its infinitesimal behavior nearby. The central questions are whether a given equation has a solution, whether that solution is unique, how it depends on the data supplied, and what qualitative features it exhibits—smoothness, singularities, oscillations, decay. The field is organized around the different types of equations, because the type determines which questions are meaningful and which tools apply.
The modern theory begins with a classification that is algebraic in origin but profoundly physical in consequence. A second-order linear PDE in two variables can be written in the form
\[ a u{xx} + 2b u{xy} + c u_{yy} + \text{(lower-order terms)} = f, \]
and its character is determined by the sign of the discriminant \(b^2 - ac\). If the discriminant is negative, the equation is elliptic; if zero, parabolic; if positive, hyperbolic. The names come from the conic sections, but the distinction is dynamical.
Elliptic equations, of which Laplace's equation \(\Delta u = 0\) and Poisson's equation \(\Delta u = f\) are the prototypes, describe equilibrium. The solution at any point is a weighted average of its surroundings; there are no preferred directions, no propagation, no time. The physical meaning is that of a steady state: the temperature distribution in a heated body after it has settled, the gravitational potential around a mass, the electrostatic field in a region of fixed charges. The mathematical signature of ellipticity is that solutions are smooth wherever the equation and the data are smooth, and that the problem is well-posed only when boundary data are specified on a closed curve or surface. The Dirichlet problem—find a function with prescribed values on the boundary of a domain—is the canonical elliptic question.
Parabolic equations, with the heat equation \(u_t = \Delta u\) as the prototype, describe diffusion. They have a distinguished time direction, but information travels at infinite speed in a smoothed, dissipative manner. The heat equation takes an initial temperature distribution and spreads it out; irregularities are immediately smoothed, and the solution becomes analytic for positive time even if the initial data were merely continuous. The canonical problem is the initial-value problem, possibly with boundary conditions: given the state at time zero, find the state at all later times.
Hyperbolic equations, with the wave equation \(u_{tt} = \Delta u\) as the prototype, describe propagation. Information travels along characteristic curves or surfaces at finite speed, and the equation preserves singularities rather than smoothing them. A sharp initial disturbance in a wave equation remains a sharp disturbance, traveling along characteristics. The canonical problem is again the initial-value problem, but now the data must include both the initial value and the initial time derivative, and the solution at a point depends only on the data in its past light cone.
This trichotomy is not a mere taxonomy. It determines which boundary conditions make a problem well-posed, which numerical methods will be stable, and what regularity the solutions possess. A problem that is well-posed for one type is typically ill-posed for another: prescribing both value and normal derivative on a closed boundary, natural for the wave equation, is impossible for Laplace's equation; prescribing only the initial value, natural for the heat equation, is insufficient for the wave equation.
The classification extends to systems and to variable-coefficient equations, where the type can change from point to point. The Euler–Tricomi equation, which models transonic flow, is elliptic in one region and hyperbolic in another, and the line where the type changes is itself a central object of study. For nonlinear equations, the type is not fixed in advance but depends on the solution, which is one reason nonlinear PDEs are so much harder.
The subject began in the eighteenth century with specific physical problems and specific attempts to solve them. The vibrating string, the cooling of a solid body, the motion of a pendulum with a moving support, the gravitational potential of a nonspherical body—each produced an equation and a struggle to find a formula. The early workers—Daniel Bernoulli, d'Alembert, Euler, Lagrange, Laplace, Fourier—developed a collection of techniques that remain the core of the applied mathematician's toolkit.
The method of characteristics solves first-order equations and some second-order hyperbolic equations by reducing them to ordinary differential equations along special curves. The method of separation of variables seeks solutions of the form \(u(x,t) = X(x)T(t)\), reducing the PDE to a pair of ordinary differential equations linked by a constant. The boundary conditions then force the constant to take discrete values, producing an infinite sequence of eigenfunctions, and the general solution is assembled as a superposition. Fourier's treatment of the heat equation, which required representing an arbitrary initial function as a sum of sines and cosines, gave rise to the theory of Fourier series and forced a reexamination of what it means for a function to be represented by an infinite series.
The method of Green's functions expresses the solution to a linear PDE with a source term as an integral of the source against a fundamental solution—the response to a point source. The Green's function for Laplace's equation in three dimensions is the Newtonian potential \(1/4\pi r\); for the heat equation it is the Gaussian kernel; for the wave equation it is a distribution supported on the light cone. These explicit formulas are not merely computational conveniences. They encode the qualitative structure of the solution: the smoothing of the heat kernel, the finite-speed propagation of the wave kernel, the nonlocality of the elliptic Green's function.
The classical period also produced the variational approach. Many elliptic equations arise as the Euler–Lagrange equations of an energy functional: the minimizer of the Dirichlet integral \(\int |\nabla u|^2\) satisfies Laplace's equation. This observation, due to Dirichlet and Riemann, opened the way to proving existence by minimizing an energy, though the rigorous implementation had to wait for the twentieth century. The variational viewpoint remains central, both as a method of proof and as a way of understanding what a PDE "means": the solution is the state that minimizes a cost, subject to constraints.
These classical techniques were not superseded; they were absorbed. The explicit formulas they produced are still the first recourse for understanding a new linear equation, and they serve as the testing ground for more general theories. But they have limits. Separation of variables works only for equations with special symmetries. Green's functions can be written explicitly only for constant-coefficient equations on simple domains. The variational method requires the energy to be bounded below and the space of admissible functions to be compact in a suitable sense. The modern theory grew out of the attempt to extend these successes to equations that resist explicit solution.
The decisive conceptual shift of the twentieth century was the move from classical solutions to weak solutions. A classical solution is a function with enough derivatives to make sense of the equation pointwise. But many physical problems have solutions that are not smooth: shock waves in gas dynamics, interfaces in materials, concentration of vorticity in fluids. The equation, written pointwise, does not make sense at the singularity. The weak formulation multiplies the equation by a smooth test function, integrates by parts to move the derivatives onto the test function, and requires the resulting identity to hold for all test functions. The solution need not be differentiable at all; it need only be integrable.
This move has a price. Weak solutions are not unique in general, and the equation alone does not select the physically correct one. The entropy conditions of hyperbolic conservation laws, which rule out nonphysical shocks, are the most famous example: the weak formulation must be supplemented by an additional inequality that encodes the irreversibility of the underlying process. The theory of weak solutions thus splits into two tasks: proving existence (often by compactness arguments that produce a solution in a large space) and proving uniqueness or selecting the admissible solution (often by additional conditions that are not part of the original equation).
The natural setting for weak solutions is the theory of Sobolev spaces, which measure not only the size of a function but the size of its derivatives in an integrated sense. A function belongs to the Sobolev space \(W^{k,p}\) if it and its derivatives up to order \(k\) lie in \(L^p\). These spaces are the correct arena for the variational method: the Dirichlet integral is finite exactly for functions in \(W^{1,2}\), and the space of functions with finite energy is a Hilbert space, complete under its natural norm. The existence of minimizers follows from the direct method of the calculus of variations: take a minimizing sequence, extract a convergent subsequence using compactness, and show that the limit is a minimizer.
The theory of linear elliptic equations reached a mature form in the mid-twentieth century. The Schauder estimates show that solutions of elliptic equations with Hölder continuous coefficients are themselves Hölder continuous, with estimates that depend only on the ellipticity constants and the domain. The \(L^p\) theory, developed by Calderón and Zygmund, extends these estimates to Lebesgue spaces. The result is a nearly complete picture: for a wide class of linear elliptic equations, existence, uniqueness, and regularity are understood, and the solution inherits exactly the regularity that the data permit.
For parabolic and hyperbolic equations, the theory is organized around semigroup theory, which treats the time evolution as a dynamical system on a function space. The heat equation generates an analytic semigroup: the solution at time \(t\) is obtained from the initial data by applying an operator that is the exponential of the Laplacian. The wave equation generates a group, reflecting its time-reversibility. This viewpoint unifies linear evolution equations and provides a framework for nonlinear perturbations, though the nonlinear theory is far less complete.
The linear theory is, in a sense, finished. The nonlinear theory is not, and it is where most of the subject's current energy lies. Nonlinear PDEs arise whenever the physical law depends on the state of the system itself: the speed of a wave depends on its amplitude, the viscosity of a fluid depends on its temperature, the curvature of a surface depends on its shape. The classification into elliptic, parabolic, and hyperbolic still applies locally, but the type can change as the solution evolves, and the superposition principle—the bedrock of the linear theory—fails entirely.
The field is organized less by a single method than by a set of regimes, each with its own questions and tools. For semilinear equations, where the highest-order terms are linear and the nonlinearity appears only in lower-order terms, the linear theory often provides a starting point, and the question is whether the nonlinearity can cause blow-up—the formation of a singularity in finite time. The critical exponent problem asks, for equations like \(u_t = \Delta u + u^p\), which powers \(p\) allow global existence and which force blow-up. The answer depends on the dimension, and the borderline case is often the hardest.
For quasilinear equations, where the nonlinearity appears in the highest-order derivatives, the situation is more delicate. The equations of fluid mechanics—the Euler and Navier–Stokes equations—are of this type, and the question of whether the three-dimensional Navier–Stokes equations have smooth global solutions for all smooth initial data remains open, one of the Clay Millennium Prize Problems. The difficulty is not a lack of ideas but a lack of estimates: the nonlinear term involves a derivative of the velocity, and the available a priori bounds do not rule out the possibility that the solution develops a singularity.
For fully nonlinear equations, where the highest-order derivatives appear nonlinearly, the theory is even more recent. The Monge–Ampère equation, which governs the curvature of a surface and the optimal transport of mass, is the prototype. The theory of viscosity solutions, developed in the 1980s, provides a notion of weak solution for a large class of fully nonlinear equations, based on the comparison principle rather than on integration by parts. A viscosity solution is a function that cannot be touched from above or below by a smooth solution of the equation; this definition is robust enough to guarantee uniqueness and stability, even when classical solutions do not exist.
The study of conservation laws—equations of the form \(u_t + \nabla \cdot F(u) = 0\), which express the conservation of some quantity—is a field in its own right. The solutions develop shocks, and the theory of shock waves, initiated by Hopf and Lax and developed by many others, provides a framework for understanding how discontinuities form, how they propagate, and which weak solutions are admissible. The entropy condition selects the physically correct solution, and the theory of scalar conservation laws is essentially complete, while systems of conservation laws remain only partially understood.
PDEs are not only a tool for physics; they are also a tool for geometry, and geometry in turn provides PDEs. The most famous example is the Yamabe problem: given a Riemannian manifold, find a conformal change of metric that makes the scalar curvature constant. This reduces to a semilinear elliptic equation, and its solution, completed in the 1980s, was a landmark in geometric analysis. The Calabi conjecture, which asks for a Kähler metric with prescribed Ricci curvature, reduces to a complex Monge–Ampère equation; its resolution by Yau was a central achievement of late-twentieth-century mathematics.
The minimal surface equation—the condition that a surface have zero mean curvature—is a quasilinear elliptic equation, and the study of its solutions, including the existence of singularities, has been a major theme. The harmonic map equations, which describe maps between manifolds that minimize an energy, are a system of semilinear elliptic equations, and their regularity theory has been a source of deep results and open problems.
These geometric applications have driven the development of regularity theory: the study of how smooth solutions must be, given the equation they satisfy. The regularity theory for elliptic equations in high dimensions is subtle because the Sobolev embedding theorems deteriorate as the dimension grows, and the natural energy estimates no longer imply continuity. The theory of partial regularity—showing that solutions are smooth except on a small set—has been a major theme, with applications to minimal surfaces, harmonic maps, and the Navier–Stokes equations.
The current subject is characterized by a productive tension between the abstract and the concrete. On one side are the general theories—weak solutions, semigroup methods, viscosity solutions, microlocal analysis—that apply to broad classes of equations. On the other side are the specific equations of physics and geometry, each with its own structure and its own open problems. The field is not a single research programme but a federation of them, united by common tools and a common history.
Several themes dominate the present. The free boundary problems—where the boundary of the domain is itself unknown and must be found as part of the solution—arise in melting ice, in fluid flow through porous media, in the growth of tumors, and in optimal stopping. The dispersive equations, such as the nonlinear Schrödinger and Korteweg–de Vries equations, describe waves that spread out over time; their theory combines harmonic analysis, dynamical systems, and probability. The stochastic PDEs, where the equation contains a random forcing term, model phenomena from turbulence to financial markets; their theory requires a careful redefinition of what a solution means, because the forcing is too irregular for classical or even weak formulations.
The numerical analysis of PDEs has grown into a discipline of its own, but it remains in constant dialogue with the analytic theory. The stability of a numerical method is often a discrete analogue of an analytic estimate, and the convergence of a numerical scheme is often proved by the same compactness arguments that prove existence. The finite element method, the finite volume method, and the spectral methods are each suited to different types of equations, and the choice of method is guided by the same classification that organizes the analytic theory.
The field's open problems are not merely technical. The Navier–Stokes regularity problem, the question of whether solutions of the three-dimensional Euler equations develop singularities, the uniqueness of weak solutions for conservation laws in several dimensions, the regularity of solutions to the Monge–Ampère equation in high dimensions—each of these is a question about the limits of our understanding of a physical process. The subject advances not by a single breakthrough but by the slow accumulation of estimates, counterexamples, and new notions of solution, each of which sharpens the boundary between what is understood and what is not.