Many partial differential equations are written with derivatives that classical solutions simply do not possess. Even when a classical solution exists, proving existence by direct differentiation is often unrealistic: the natural a priori estimates live at the level of integrals, not pointwise derivatives. The modern resolution is to redefine what it means \to “solve” a […]

A striking feature of many elliptic and parabolic equations is that their solutions are constrained by the boundary data and the forcing in a one-sided, order-preserving way. This is not a minor technical convenience; it is a structural statement about diffusion-type operators. Maximum principles formalize it: under appropriate hypotheses, a solution cannot attain an interior […]

Fourier methods are often introduced as a clever way to solve PDE on simple domains, but their importance goes deeper. They provide a direct mechanism for diagonalizing linear translation-invariant operators, turning PDE into decoupled ODEs in time or in one variable. They also reveal how smoothing and dispersion emerge from the spectrum of the operator. […]

Building examples in PDE is not about hunting for exotic formulas. It is about controlling which features of the equation are active. A good example isolates a mechanism: propagation, smoothing, boundary influence, loss of regularity, blow‑up, finite‑speed effects, dispersion, or the failure of an estimate outside its hypotheses. This recipe is meant to be used. […]

Partial differential equations reward precision and punish assumptions that you did not pay for. Many mistakes in PDE are not “careless algebra.” They are category mistakes: mixing pointwise and weak meanings, confusing the role of boundary conditions, or applying an estimate outside the regime where it is valid. This article collects common failure modes and […]

Parabolic equations are the place where PDE technique becomes concrete fast: they are time‑directed, they smooth rough data, and they reward careful bookkeeping. If you can prove the right statement for the heat equation and its close relatives, you have learned a proof pattern that reappears across nonlinear diffusion, fluid models with dissipation, and many […]