Partial Differential Equations¶

Job: organize the frontier areas of modern PDE theory at postdoc depth — elliptic regularity (Schauder, De Giorgi–Nash–Moser, Calderón–Zygmund, fully nonlinear Krylov–Safonov / Evans–Krylov), nonlinear dispersive equations (Strichartz, NLS well-posedness and scattering, energy-critical Kenig–Merle), Navier–Stokes regularity (Leray–Hopf, Caffarelli–Kohn–Nirenberg, Koch–Tataru), and the calculus of variations (direct method, quasiconvexity, Γ-convergence, Mumford–Shah) — with sharp hypotheses and the open problems at each frontier.

Core¶

PDE theory separates into four structural regimes — elliptic (steady states), dispersive (wave-spreading nonlinear), parabolic fluid (Navier–Stokes, the Millennium problem), and variational (minimizers and free boundaries) — each with distinct function spaces, a priori estimates, and open problems. The foundations page (Foundations) gives the classification, weak-solution framework, and energy-method sketch; this subdomain develops the deep regularity theory, the dispersive nonlinear machinery, the fluid regularity problem, and the variational regularity / Γ-convergence in full.

Layer	Content
Elliptic	Schauder (\(C^{k,\alpha}\)); De Giorgi–Nash–Moser (measurable coefficients \(\Rightarrow C^\alpha\)); Calderón–Zygmund \(L^p\); Krylov–Safonov / Evans–Krylov (fully nonlinear)
Dispersive	Strichartz (Keel–Tao); NLS LWP / GWP / scattering; Kenig–Merle energy-critical; Duyckaerts–Kenig–Merle soliton resolution
Navier–Stokes	Leray–Hopf weak; energy inequality; CKN partial regularity; Koch–Tataru \(BMO^{-1}\); ESS blow-up criterion
Variations	direct method; De Giorgi minimizer regularity; Morrey quasiconvexity; Γ-convergence; Mumford–Shah / SBV

Reading map¶

Page	Focus
Foundations	classification; weak / distributional / viscosity solutions; energy methods sketch; maximum principles
Elliptic Theory	Schauder; De Giorgi–Nash–Moser; Calderón–Zygmund \(L^p\); Krylov–Safonov / Evans–Krylov fully nonlinear
Dispersive & Wave	dispersive / Strichartz (Keel–Tao); NLS LWP/GWP/scattering; Kenig–Merle energy-critical
Navier–Stokes	Leray–Hopf; energy inequality; CKN partial regularity; Koch–Tataru; the Millennium problem
Calculus of Variations	direct method; De Giorgi regularity; quasiconvexity; Γ-convergence; Mumford–Shah / SBV

Dependency order¶

Functional analysis: Banach/Hilbert, duality, weak compactness, Lax–Milgram — the function-space framework.
Fourier analysis: Fourier restriction, oscillatory integrals — the Strichartz machinery.
Microlocal analysis: pseudodifferential operators, wavefront sets, elliptic regularity / propagation of singularities.
Geometric measure theory: currents, varifolds, finite-perimeter sets — the variational / minimal-surface regularity.
Dynamical systems: the long-time behavior / scattering connection.
SPDE: stochastic perturbations; the deterministic PDE theory is the \(\nu\to\infty\) / zero-noise reference.
Foundations \(\to\) elliptic \(\to\) dispersive / Navier–Stokes \(\to\) variations.

Interfaces¶

Area	Contact
Functional analysis	Sobolev, Lax–Milgram, weak compactness, semigroups
Fourier analysis	Strichartz via restriction; Calderón–Zygmund singular integrals
Microlocal analysis	\(\Psi\)DO, elliptic regularity, propagation of singularities
Geometric measure theory	minimal surfaces, finite-perimeter sets, SBV, Mumford–Shah
Dynamical systems	long-time behavior, attractors, scattering as asymptotic dynamics
SPDE	stochastic forcing; regularity structures; the deterministic PDE as reference
Mathematical physics	Navier–Stokes / Euler fluids; NLS in quantum mechanics; wave maps
Complex geometry / Kähler	Monge–Ampère / Calabi–Yau as fully nonlinear elliptic (Caffarelli)

Notation (paper-standard)¶

\(W^{k,p}(\Omega)\), \(H^s(\Omega)=W^{s,2}\) Sobolev spaces; \(C^{k,\alpha}(\overline\Omega)\) Hölder spaces
\(L=-\mathrm{div}(A(x)\nabla)\) divergence-form; \(L=a^{ij}(x)\partial_{ij}\) non-divergence form; uniform ellipticity \(\lambda|\xi|^2\le a^{ij}\xi_i\xi_j\le\Lambda|\xi|^2\)
Schauder estimate \(\|u\|_{C^{k+2,\alpha}}\le C(\|f\|_{C^{k,\alpha}}+\|u\|_{C^0})\); Calderón–Zygmund \(\|D^2 u\|_{L^p}\le C\|\Delta u\|_{L^p}\)
Strichartz pair \((q,r)\) admissible: \(\frac{2}{q}+\frac{n}{r}=\frac{n}{2}\) (Schrödinger); dispersion \(\|e^{it\Delta}u_0\|_{L^\infty}\le|t|^{-n/2}\|u_0\|_{L^1}\)
NLS \(i\partial_t u+\Delta u=\pm|u|^{p-1}u\); critical exponents \(p_{mass}=1+4/n\), \(p_{energy}=1+4/(n-2)\)
Navier–Stokes: \(u\in L^\infty_t L^2_x\cap L^2_t H^1_x\) (Leray–Hopf); \(\Sigma\) singular set; \(\dim_{\mathcal{H}(\Sigma)}\le 1\) (CKN)
\(F[u]=\int L(x,u,\nabla u)\) variational functional; quasiconvex / polyconvex / rank-one convex
\(\Psi^m\) pseudodifferential operators of order \(m\); \(\Gamma\)-convergence \(F_j\xrightarrow{\Gamma}F\)

See it¶

The four frontiers, one open problem each. (1) Elliptic: the fully nonlinear regularity — the Evans–Krylov \(C^{2,\alpha}\) estimate for convex solutions of \(\det D^2u=f\) is established, but the optimal regularity for non-convex fully nonlinear equations (e.g. general uniformly elliptic \(F(D^2u)=0\) with \(F\) concave) remains partly open (the Nadirashvili–Vlăduţ counterexamples to \(C^{2,\alpha}\) for non-concave \(F\)). (2) Dispersive: the soliton resolution conjecture (Duyckaerts–Kenig–Merle) — for energy-critical wave / NLS, do all finite-energy solutions decompose as a sum of a soliton (ground state) and a dispersive linear part as \(t\to\infty\)? Proven in special cases (radial, 3D wave), open in general. (3) Navier–Stokes: the Millennium regularity problem — for \(u_0\in C_c^\infty(\mathbb{R}^3)\), does a global smooth solution exist? (Leray–Hopf weak solutions exist globally; smooth local solutions exist; the gap is global strong regularity.) (4) Variations: the Mumford–Shah regularity — the jump set of the minimizer is known to be \(C^{1,\alpha}\) outside a singular set of dimension \(\le n-3\) (Ambrosio–Fusco–Pallara), but the structure of the full singular set and the optimal regularity remain partly open. Each frontier has a sharp open problem; the four together define the modern PDE landscape.

Check¶

State the precise sense in which De Giorgi–Nash–Moser 'rescued' elliptic theory from the smooth-coefficient hypothesis, and identify the regularity it gives.

Before De Giorgi–Nash–Moser, the Schauder theory required \(C^{k,\alpha}\) coefficients to obtain \(C^{k+2,\alpha}\) solutions — the regularity of the solution was tied to the regularity of the coefficients. Hilbert's 19th problem asked whether minimizers of regular variational functionals are smooth; the obstacle was that the Euler–Lagrange equation has measurable coefficients (the gradient of the solution enters the coefficient \(A(x)=D^2 L(\nabla u)\)), so Schauder does not apply. De Giorgi (divergence form) and Nash–Moser (non-divergence) independently proved that uniformly elliptic equations with merely bounded measurable coefficients have solutions that are \(C^\alpha\) (Hölder) — a regularity independent of the coefficient smoothness. This resolved Hilbert 19th (combined with the higher-regularity bootstrap) and detached elliptic regularity from the coefficient hypothesis; the \(C^\alpha\) is the sharp regularity for measurable coefficients (counterexamples show no better holds in general). The rescue: regularity is intrinsic to ellipticity, not to coefficient smoothness.

Why is the Navier–Stokes energy inequality (not equality) the key subtlety, and where does the equality fail?

For smooth solutions, multiplying \(\partial_t u+(u\cdot\nabla)u-\nu\Delta u=-\nabla p\) by \(u\) and integrating gives \(\frac{d}{dt}\frac12\|u\|_{L^2}^2=-\nu\|\nabla u\|_{L^2}^2\) (equality) — the convective term \((u\cdot\nabla)u\) vanishes after integration by parts (it is skew-symmetric: \(\int u\cdot(u\cdot\nabla)u=0\) for divergence-free \(u\)), and the pressure term vanishes. For Leray–Hopf weak solutions obtained as Galerkin limits, the convective term passes to the limit only as an inequality — the weak limit of \((u^N\cdot\nabla)u^N\) is bounded below by the limit solution's contribution, and one obtains \(\frac{d}{dt}\frac12\|u\|_{L^2}^2\le-\nu\|\nabla u\|_{L^2}^2\). The failure of equality is the energy defect (possible dissipation in singular regions); proving equality for all weak solutions would imply regularity (no energy defect \(\Rightarrow\) smooth). Hence the inequality vs equality distinction is exactly the regularity question — see Navier–Stokes.

Pitfalls¶

Schauder needs \(C^{k,\alpha}\) coefficients; De Giorgi–Nash–Moser is the measurable-coefficient substitute giving only \(C^\alpha\): do not conflate the two regimes.
Strichartz endpoint is sharp (Keel–Tao); the naive endpoint \((q,r)=(\infty,2)\) fails for Schrödinger.
Energy-critical scattering needs the ground-state threshold (Kenig–Merle); below the threshold is a theorem, above it blow-up occurs — the threshold is essential, not a technicality.
The Navier–Stokes regularity problem is open — do not claim smoothness of Leray–Hopf solutions.
The energy inequality (not equality) for weak solutions is the key subtlety; confusing the two is the standard error.
Quasiconvexity is not convexity (the vectorial case is genuinely different); rank-one convex \(\subsetneq\) quasiconvex \(\subsetneq\) polyconvex \(\subsetneq\) convex.
Γ-convergence is a variational convergence, not a topological one — it does not preserve PDE structure, only minimizers.

Refs¶

Gilbarg–Trudinger, Elliptic Partial Differential Equations of Second Order; Evans, PDE; Ladyzhenskaya–Ural'tseva
Tao, Nonlinear Dispersive Equations; Kenig, Lectures on the energy-critical wave equation; Sogge, Hangzhou lectures
Lemarié-Rieusset, The Navier–Stokes Problem in the 21st Century; Seregin, Lecture Notes on Regularity Theory
Giusti, Direct Methods in the Calculus of Variations; Ambrosio–Fusco–Pallara, Functions of Bounded Variation and Free Discontinuity Problems

Next¶

Foundations · Elliptic Theory · Dispersive & Wave · Navier–Stokes · Calculus of Variations · Functional analysis · Fourier analysis · Microlocal analysis · Geometric measure theory