A PDE starts with unknown, domain, and data
Objective: Checkpoint HD01.01: without looking back, name the unknown and every object type in A PDE starts with unknown, domain, and data, reproduce the exact finite row, distinguish classical, weak, entropy, or asymptotic claims, state the theorem bridge with all hypotheses, identify the first invalid inference after the mutation, and explain why a scaling label or finite computation cannot replace that missing step.
HD01.01 begins by saying what the unknown is. It may be a scalar u(t,x), a vector U(t,x), a complex amplitude, or a map into a manifold. State the time interval, spatial domain, boundary conditions, initial data, coefficient fields, and target space before displaying an equation. In ordinary language, “A PDE starts with unknown, domain, and data” asks: identify what is being solved for, where it lives, and which initial or boundary values select a problem rather than an isolated formula A symbol is not explained merely because it appears in a familiar formula.
Read every type aloud. The variable t is time, x is position, ξ is frequency, τ is temporal frequency, u_t is a rate of change, ∇u is a spatial slope, Δu is a sum of second spatial derivatives, f(u) is a flux or forcing only after it is declared, and a norm is a rule that measures size in a named function space. Numbers with different units or mathematical roles cannot be combined merely because the arithmetic is legal.
Write the equation as a balance of mechanisms. A transport term moves information, a divergence term records flux through boundaries, a wave operator couples acceleration to spatial curvature, and a dispersive term separates Fourier phases without automatically dissipating energy. The sign convention matters. Moving one term across the equality changes its printed sign but not its mechanism, provided the whole equation and the convention are updated consistently.
Distinguish a classical solution from a weak solution before differentiating. A classical solution has enough pointwise derivatives for every displayed term. A distributional solution is tested against smooth compactly supported functions so derivatives move onto the test function. An entropy solution adds an admissibility inequality because conservation laws can have several weak continuations after characteristics cross. Never transfer pointwise identities to a weak solution without an approximation or chain-rule theorem.
Keep four ledgers. The equation ledger records signs, coefficients, domains, data, and solution class. The propagation ledger records characteristic speeds, cones, group velocities, or Fourier phases. The estimate ledger records norms, constants, derivative counts, time intervals, and every dependence. The theorem ledger records regularity, hyperbolicity, symmetry, admissible exponent, smallness, compactness, or geometric hypotheses. A conclusion moves forward only when its ledger is complete.
Use the exact finite calibration: HD01.01: for the frozen second-order symbol τ²+3τξ+1ξ², the discriminant is 3²−4·1=5. Record the base point, covector ξ, sign convention, and multiplicity before saying hyperbolic, elliptic, or repeated; one frozen row does not classify a variable-coefficient operator everywhere. Retain fractions, radicals, phases, and symbolic powers until the final display. A rounded decimal can test a row at one precision, but it cannot replace an identity, a distributional equality, an infinite-time limit, or an error estimate. Reconstruct the row directly from the definition before invoking a named result.
Characteristics are paths along which a first-order equation simplifies, not decorative curves. Derive the path equation from the coefficient multiplying the spatial derivative, solve it only on an interval where the coefficient field is defined, and track the carried or forced value. When two projected characteristics meet, a smooth scalar solution may cease to exist; the next object can be a weak solution with a shock, not a multivalued classical graph.
For a conservation law, mass balance and entropy selection answer different questions. Rankine–Hugoniot balance fixes the speed required for a discontinuity to conserve the quantity. An entropy condition decides whether that discontinuity is the physically and mathematically selected weak solution. A rarefaction fan, a compressive shock, and a contact discontinuity have different characteristic geometry. Compute the flux derivatives and compare both sides before naming the wave.
Energy estimates convert the equation into control of a norm. Specify which derivatives enter the energy, whether the energy is positive and equivalent to the desired Sobolev norm, which boundary terms vanish, and how coefficient derivatives enter. Grönwall turns a differential inequality into a time-dependent bound; it does not recover derivatives that were lost, remove a blow-up criterion, or make a local solution global.
Finite propagation is a support theorem, whereas decay is a size theorem. A wave cone says where information cannot arrive before a given time. It does not state the amplitude inside the cone. Pointwise decay usually combines energy with geometry, commutators, oscillation, or vector-field inequalities. Keep the support radius, propagation speed, derivative order, and decay weight in separate columns.
Nonlinear iteration needs a complete map-and-space declaration. Define the solution map, the ball on which it acts, the norm controlling the linear solution and Duhamel term, the time or data smallness factor, and the difference estimate. Mapping a ball into itself is not enough; uniqueness requires contraction or another stability mechanism. The resulting lifespan depends on the exact data norm and constants retained in the proof.
A null form is an algebraic cancellation aligned with characteristic directions. Verify it on the full bilinear symbol or tensor, not on one chosen pair of vectors. The cancellation can improve spacetime integrability and weaken dangerous interactions, but global existence still requires dimension, regularity, decay, smallness, and structural hypotheses. Quasilinear equations also change the propagation geometry, so shock formation remains a separate boundary.
Fourier transformation changes spatial differentiation into multiplication by frequency. Declare the transform normalization and sign, solve the resulting ordinary differential equation in time, and then invert. The modulus of a single Fourier multiplier often stays one, while decay in physical space comes from cancellation among many phases. Stationary points, Hessian rank, amplitude support, and endpoint integrability determine which dispersive estimate is actually available.
Mixed spacetime norms encode different questions in time and space. In L_t^qL_x^r, take the spatial r-norm at each time and then the temporal q-norm; changing the order can change the number. An admissible exponent relation depends on equation, dimension, scaling, and sometimes endpoints or derivative loss. State the propagator and interval before using the word Strichartz, smoothing, maximal, or restriction estimate.
Dyadic decomposition is an accounting system for frequency scales. State the cutoff functions, overlap count, homogeneous or inhomogeneous convention, and treatment of zero frequency. Bernstein inequalities exchange integrability or derivatives for frequency powers only on localized pieces. Paraproducts sort high–low, low–high, and comparable-frequency interactions; they do not make every product estimate valid below its scaling or regularity threshold.
Scaling identifies a dimensionless regularity index, not a theorem. Compute how time, space, amplitude, and the chosen norm transform. A subcritical, critical, or supercritical label is always relative to that scaling and norm. Conservation of mass or energy can control a critical quantity for some signs and powers, but focusing equations may possess solitons, thresholds, or blow-up despite formal conservation.
Scattering means that a nonlinear solution approaches a solution of the associated linear equation in a declared norm as time tends to an endpoint, usually infinity. Bounded energy is not the same as scattering, and weak convergence is not strong asymptotic completeness. A concentration–compactness argument first extracts a minimal obstruction under precise compactness modulo symmetries; a rigidity argument must then rule it out using additional dynamics.
Numerical evidence has its own theorem ledger. Record mesh, timestep, boundary closure, numerical flux or spectral filter, conserved totals, stability condition, convergence study, and residual. A scheme may conserve a discrete mass yet select the wrong entropy solution, or resolve low frequencies while aliasing high-frequency dispersion. A finite computation validates only the declared rows until consistency, stability, compactness, and identification are proved.
The bounded bridge for this unit is “a real simple characteristic factorization supplies local propagation directions, while a positive symmetrizer converts the principal system into an energy estimate only under the stated smoothness and uniformity hypotheses” Rewrite it as hypotheses, mechanism, conclusion, and excluded stronger claims. If even one hypothesis is missing, keep the calculation but downgrade the conclusion. The unit mutation is “classify from lower-order terms, omit the covector, allow characteristic multiplicity to change, confuse phase velocity with domain of dependence, or infer global well-posedness from one frozen symbol” Locate the first proof line that uses the removed property instead of saying vaguely that the theorem no longer applies.
Finish with teach-back. Explain A PDE starts with unknown, domain, and data to a learner who has not taken this course: name the unknown and equation, translate every symbol, compute the exact finite row, identify the norm or weak formulation, list every theorem hypothesis, and show one failure mutation. If any phrase relies on an unsupported shortcut, a generic label, or an unnamed estimate, the explanation is incomplete and must be rewritten.
Bounded theorem HD01.01: after the equation, propagation, estimate, and hypothesis ledgers for A PDE starts with unknown, domain, and data pass, this chapter may use only “a real simple characteristic factorization supplies local propagation directions, while a positive symmetrizer converts the principal system into an energy estimate only under the stated smoothness and uniformity hypotheses”; no stronger global existence, uniqueness, decay, entropy selection, endpoint estimate, scattering, blow-up classification, or optimal threshold follows without another theorem.
Proof route HD01.01.1 — scope: declare the unknown, equation, domain, data, coefficient regularity, boundary behavior, solution class, and the exact proposition being proved. Separate existence, uniqueness, stability, regularity, decay, and asymptotics because none is a synonym for another.
Proof route HD01.01.2 — types: expand every derivative, flux, symbol, norm, and pairing used by A PDE starts with unknown, domain, and data. Verify that each operation maps between the stated spaces and that products, traces, or compositions are defined at the retained regularity.
Proof route HD01.01.3 — finite calibration: derive “HD01.01: for the frozen second-order symbol τ²+3τξ+1ξ², the discriminant is 3²−4·1=5. Record the base point, covector ξ, sign convention, and multiplicity before saying hyperbolic, elliptic, or repeated; one frozen row does not classify a variable-coefficient operator everywhere.” from raw declarations twice, once by direct substitution and once by a conservation, characteristic, Fourier, or dimensional check appropriate to the unit. Agreement validates the row, not the infinite theorem.
Proof route HD01.01.4 — mechanism: isolate the term that transports, conserves, disperses, cancels, or controls energy. Perform the integration by parts, characteristic differentiation, symbol factorization, or Duhamel decomposition with all signs and boundary terms visible.
Proof route HD01.01.5 — estimate: place every term in a named norm, state the inequality used, record its exponent range and constant dependence, and close the estimate without silently losing a derivative. If a small factor is needed, identify whether it comes from time, data size, frequency separation, or structure.
Proof route HD01.01.6 — construction: build approximants by smoothing, Galerkin truncation, vanishing viscosity, iteration, frequency cutoff, or a declared finite scheme. Prove uniform bounds in the topology needed for a limit and retain the initial and boundary data.
Proof route HD01.01.7 — passage and identification: state the convergence mode for each sequence, justify nonlinear passage or entropy production, identify the limiting equation, and prove only the bounded bridge “a real simple characteristic factorization supplies local propagation directions, while a positive symmetrizer converts the principal system into an energy estimate only under the stated smoothness and uniformity hypotheses.” Weak convergence alone does not pass an arbitrary nonlinear product.
Proof route HD01.01.8 — independent audit: rebuild the characteristic speed, energy identity, phase relation, or scaling exponent from a second representation and compare intermediate ledgers. Verify that no literature wording, example data, proof order, or visual geometry entered the reconstruction.
Proof route HD01.01.9 — boundary: activate “classify from lower-order terms, omit the covector, allow characteristic multiplicity to change, confuse phase velocity with domain of dependence, or infer global well-posedness from one frozen symbol,” mark the first invalid implication, and retain the strongest surviving statement. Report whether the finite identity, a priori estimate, weak solution, local lifespan, conditional scattering claim, or only numerical evidence remains.
Worked reconstruction HD01.01: starting only from the original raw record “HD01.01: for the frozen second-order symbol τ²+3τξ+1ξ², the discriminant is 3²−4·1=5. Record the base point, covector ξ, sign convention, and multiplicity before saying hyperbolic, elliptic, or repeated; one frozen row does not classify a variable-coefficient operator everywhere.,” explain every symbol, derive A PDE starts with unknown, domain, and data, fill the four ledgers, justify the bounded theorem bridge, repeat the pivotal computation independently, and then activate the failure mutation without importing a source example, dataset, proof layout, or figure.
- Step HD01.01.A: transcribe the raw equation and data without changing signs, normalization, units, domain, or cutoffs.
- Step HD01.01.B: translate every symbol into an object type and state where each derivative, transform, norm, and product is defined.
- Step HD01.01.C: recompute the exact finite ledger “HD01.01: for the frozen second-order symbol τ²+3τξ+1ξ², the discriminant is 3²−4·1=5. Record the base point, covector ξ, sign convention, and multiplicity before saying hyperbolic, elliptic, or repeated; one frozen row does not classify a variable-coefficient operator everywhere.” and preserve all exact arithmetic before approximation.
- Step HD01.01.D: derive the propagation, energy, entropy, phase, or scaling mechanism that connects the row to A PDE starts with unknown, domain, and data.
- Step HD01.01.E: audit every hypothesis of “a real simple characteristic factorization supplies local propagation directions, while a positive symmetrizer converts the principal system into an energy estimate only under the stated smoothness and uniformity hypotheses” and label pass, fail, or not supplied with evidence.
- Step HD01.01.F: repeat the pivotal calculation by an independent route and compare intermediate quantities before conclusions.
- Step HD01.01.G: apply the mutation, stop at the first broken inference, and report the strongest bounded conclusion in plain language.
Result: Answer HD01.01: the retained exact row is “HD01.01: for the frozen second-order symbol τ²+3τξ+1ξ², the discriminant is 3²−4·1=5. Record the base point, covector ξ, sign convention, and multiplicity before saying hyperbolic, elliptic, or repeated; one frozen row does not classify a variable-coefficient operator everywhere.” It supports the typed meaning “identify what is being solved for, where it lives, and which initial or boundary values select a problem rather than an isolated formula” only for the declared equation, data, and finite scale. The theorem step remains conditional on “a real simple characteristic factorization supplies local propagation directions, while a positive symmetrizer converts the principal system into an energy estimate only under the stated smoothness and uniformity hypotheses”; after “classify from lower-order terms, omit the covector, allow characteristic multiplicity to change, confuse phase velocity with domain of dependence, or infer global well-posedness from one frozen symbol,” stop at the first missing hypothesis and retain only the finite identity, audited estimate, or explicitly conditional conclusion.