Incompressible flow: foundations and forced equations

A course on incompressible fluid equations, from pressure, energy, vorticity and scaling to weak and strong solutions, residual stresses and forced-flow constructions. It supplies the foundations for studying singularity constructions and their exact force and regularity requirements.

Fifty-four complete lesson drafts, self-checked by GPT-6 Astra at Ultra. No independent review of this version is claimed. The rest of the course is in preparation.

Two hundred and seventy solved exercises accompany complete proofs. The first lesson uses multivariable calculus and integration by parts; the second also assumes basic Lebesgue integration and supplies its Fourier ingredients. The third constructs periodic weak solutions with the full force and mean, proves compactness and the energy inequality, recovers pressure and proves weak–strong uniqueness and stability under changes of force. The fourth gives the complete whole-space construction, spatial-tail bounds and low-frequency pressure, and extends both domains to sums of the two stated force classes. The fifth constructs H1 strong solutions, retains the full original viscosity, force and periods, and proves stability and maximal continuation. The sixth proves the nonendpoint velocity criterion for energy weak solutions, the small L3 endpoint, force and viscosity comparison, and exact boundary maps with retained wall work and periodic surface forces. The seventh constructs critical mild solutions with their full pressure and force, proves small-data global existence in homogeneous spaces, retains arbitrary periodic means, and tests the initial trace and completeness in an original source. The eighth proves vorticity, directional and middle-eigenvalue continuation, finite-sum criteria and a small endpoint tail. Exact strain reconstruction and visible source corrections retain the periodic constant defect, full pressure pairing and compact transition forces.

The ninth constructs Euler solutions for every real s greater than 5/2, proves continuous dependence in the full top norm, and proves vorticity and strain continuation with the original force and periodic mean. Complete logarithmic estimates, source corrections and a compact multiscale example are included.

The tenth proves actual velocity-tail continuation and two finite weighted heat estimates. Original viscosity, the full heat kernel, additive equation errors, cutoff supports and both time endpoints remain explicit. Exact concentration and Fourier examples test the source steps. The unforced whole-space endpoint is supplied in lesson 20.

The eleventh proves a finite moving-annulus energy estimate with all sphere, harmonic, stretching and force contributions. Complete weighted Sobolev and local curl arguments support a repaired radius average, exact physical zoom and two reproducible figures. The following lessons supply pointwise annular regularity; the unforced whole-space endpoint is supplied in lesson 20.

The twelfth proves global nonlinear energy and total speed with the full force and original viscosity. Complete kernel and frequency arguments supply an actual bound for annular boundary displacement. Every tensor interaction is retained.

The thirteenth selects an actual annulus with controlled initial data and both force contributions. Exact heat propagation closes its energy budget; complete local div-curl and Fourier proofs give a local L4-time velocity bound. The localized vorticity equation and force heat potential retain every boundary term.

The fourteenth completes the positive-time annular heat iteration, full Newton velocity and gradient maps, and first vorticity derivative bounds. The actual unforced or compact-force receiving annulus can be chosen with all four maxima arbitrarily small. Every boundary and force term remains.

The fifteenth proves original-viscosity frequency backpropagation with the same amplitude, complete low/high interactions and every exterior kernel term. Exact parameter selection and a convergent lowest-frequency estimate recover the original source hierarchy. Finite backward chains retain their full time and displacement costs.

The sixteenth uses complete total speed and disjoint persistence intervals to prove quantitative backward iteration with the original viscosity, time, frequency and amplitude. It recovers the exact source time separation, improves constants using full intervals, and retains a quantitative set of earlier frequencies. The moving-frame exercise proves the actual Galilean map and its whole-space critical-class restriction.

The seventeenth proves original local vorticity mass, its persistence, and a quantitative regularity interval with velocity, vorticity and first derivatives. The full heat and Newton maps retain every cutoff and boundary term. A definite smaller interval meets the finite Gaussian coefficient bounds and preserves the same mass. Five solved exercises include the exact amplitude-duration optimum, energy thresholds, earlier heat lengths, transported frequency grid and complete off-center Gaussian.

The eighteenth proves the complete outward Gaussian and final-time annular transfer with every original coefficient, time interval and support. Both weighted-transfer cases give actual final-time vorticity and velocity mass. A smaller shell strengthens the bound. Five solved exercises include independent error budgets, longer Gaussian durations, an optimal covering radius, the best initial Gaussian time for the uniform estimate and the full available derivative neighborhood. The following lesson constructs the complete uniform scale family.

The nineteenth constructs original annuli uniformly over the actual scale family. Full energy, heat and Newton coefficients give positive scale-independent velocity mass and the frequency-event bound. Five exercises prove exact overlap, a full Lp curl receiver, quantitative mass improvements, a strictly smaller envelope and a joint radius-and-shell optimization. The next lesson supplies the complete unforced whole-space endpoint.

The twentieth proves the unforced whole-space bounded-L3 endpoint for the original strong and energy weak solutions. It retains the source sixth enstrophy term and the entire original heat field. Full frequency absorption, all-order positive-time smoothing, original pressure and a common H1 restart complete the argument. Smaller envelopes, both exact projection bands and jointly optimized Young weights and preceding time windows strengthen its receiving bounds. Five exercises include an explicit common restart time.

The twenty-first constructs the full periodic residual stress and pressure maps, a sharp inverse-divergence norm and explicit rational Beltrami families. It proves every finite correction term with original viscosity and force, and supplies five solved exercises on pressure freedom, the sharp trace-free positive domain, mixed families, resonant mean correction and minimum finite force cost.

The twenty-second proves the original intermittent covering and norms, complete derivative bounds and a stronger periodic product estimate with the missing amplitude retained. Five solved exercises give exact fourth moments, a sharp boundary resonance, rectangular periods and the full temporal pair correction with pressure and constant modes.

The twenty-third proves the full intermittent velocity, pressure and residual stress with original viscosity and every quadratic term. Explicit periodic kernels give complete oscillatory inverse bounds. Five solved exercises give exact pair matrices, square norms, the temporal correction cost, an optimal frequency split and the entire energy increment.

The twenty-fourth constructs the original mollified equation, complete stress cutoffs and smooth energy amplitudes, including their zeros and every derivative scale. It corrects the source energy-volume factor by an exact comparison. Five solved exercises retain the actual shear pressure, cutoff geometry, arbitrary volume, smoothing loss and every term in the full physical energy error.

The twenty-fifth proves the complete original velocity step, including both small corrections, the full increment and every derivative through order three. It constructs the periodic projection kernel and proves all finite-exponent bounds and the required smooth-input estimate. Five solved exercises give exact projections, optimized kernel bounds, the full projected fourth moment and the original amplitude tail.

The twenty-sixth proves the complete original stress step: every viscous, transport, correction, oscillation and mollification term, the exact pressure projection and all first derivatives. It corrects the printed parameter example and proves a larger admissible range. Five solved exercises give exact stress and curl maps, the complete parameter region, a changing-mean force example and sharper base thresholds.

The twenty-seventh completes the prescribed-energy weak solution: every energy contribution, the original time endpoints, convergence, pressure means, continuous integrable vorticity and exact time locality. Five solved exercises cover sharper base thresholds, Lipschitz energy profiles, all pressure modes, the original heat cancellation and a continuum sharing the same initial velocity.

The twenty-eighth constructs the original vanishing-viscosity limit, retaining the full starting tensor, energy index, coefficient dependence and time intervals. One fixed base gives strong positive-Sobolev and pressure convergence. Five solved exercises test the physical equation and prove explicit rates in the original viscosity.

The twenty-ninth constructs the original radial vortex maps, sharp Green bounds, a complete smooth background and its arbitrarily deep spectral well. It proves the exact eigenfunction dictionary, the integer threshold window and full slow-tail cutoff estimates. Five solved exercises retain the physical pressure, energy, circulation and sign costs. The following lesson supplies the nonreal Rayleigh eigenvalue.

The thirtieth constructs an actual unstable Euler vortex with an integer angular mode. It proves the full oriented boundary integral, localized operator bounds, uniform complex root construction, smooth physical mode and original pressure. Five solved exercises include exact origin coefficients and sharp weighted moment thresholds.

The thirty-first constructs the compact unstable vortex and its full weighted Euler generator. It proves spectral persistence with a definite cutoff radius, the complete Biot-Savart kernel, the closed transport domain and finite algebraic multiplicity. Five solved exercises retain the zero mode, strengthen the weight range and give an explicit ring correction with its full physical volume.

The thirty-second constructs the complete original three-dimensional ring kernel and transfers the actual unstable spectral space. Exact axis and transport maps, every curvature term, the physical pressure and complete Euler and viscous background forces are proved. Five solved exercises strengthen the stretching bound and preserve the whole enclosed algebraic multiplicity.

The thirty-third proves the original physical ring generator and the entire heat-and-dilation domain. Full cylindrical volume, small stretching, compact coupling, precise transport domains and smooth-input bounds receive the actual Euler mode. The exact viscous semigroup retains all coefficients, energy and smoothing terms. Five solved exercises include explicit radius thresholds, a velocity-integrability counterexample and the corrected amplitude and pressure in the curl map.

The thirty-fourth proves the complete original viscous instability and full velocity receiver. Exact domains, semigroups, resolvents and compact spectral transfer retain every viscosity, amplitude and dilation term. Endpoint vorticity integrability, an explicit counterexample family and the entire velocity and pressure maps are proved. Five solved exercises construct the full velocity generator, retain the nonlinear residual and calculate the exact physical tail and heat-semigroup norm.

The thirty-fifth proves the actual velocity spectral growth and complete smoothing at every real Sobolev order, retaining the original viscosity, amplitude and drift. The full solenoidal space and the axisymmetric no-swirl space have their own proved maximal rates. Five solved exercises calculate exact Jordan and real-mode norms, construct the original compact force with its energy identity, and prove the complete nonlinear integral map.

The thirty-sixth constructs the complete original nonlinear correction and distinct suitable Leray flows with one force and zero initial velocity. Every physical coefficient, pressure, time endpoint and energy identity is retained. Five solved exercises give a sharper contraction interval, an explicit second-order correction, a continuum of flows with the same force, both pressure orders and the exact initial concentration.

The thirty-seventh constructs the exact averaged operator, original-viscosity local evolution, energy defect and full force and pressure maps. An explicit Gaussian example and a complete dilation correction test the source formulas. Five solved exercises retain the full frequency relation, optimize the Gaussian rotation, construct the Sobolev cancellation correction and weighted evolution, and prove the original dyadic local flow with both boundary energy fluxes.

The thirty-eighth constructs the original local cascade with its sharp Sobolev maps, complete local evolution and an exact energy-transfer pump. Five solved exercises prove real and complex averaging composition, solve every initial state on the pump plane, compute the full diffusion defect and construct its infinite-dimensional invariant space with a global viscous evolution and complete physical force.

The thirty-ninth proves the complete original local-cascade averaging theorem, including all tensor coefficients, the entire rotation fiber, full Haar and surface measures, a constructed smooth extension, every Fourier multiplier and the complete scale sum. Five solved exercises give exact closing moments, the minimum tensor kernel, the full retraction derivative, a sharp bounded-weight classification for the original pump and quantitative stability of its actual viscous flows.

The fortieth constructs the actual original-viscosity cascade on its maximal interval, with complete band equations, every boundary energy term, the full orthogonal diffusion defect and an exact positive memory kernel. Five solved exercises give the entire heat response and resolvent, a stronger annular-width estimate, the signed scale-energy current and the derivative of the actual flow with respect to viscosity.

The forty-first constructs every original quadratic gate and the complete scheduled dissipative transfer, including its sharp exponent endpoint. Five solved exercises prove the exact gate correspondence, a stronger transfer estimate, the unique damped-output peak, global energy-class continuation and exact terminal regularity. Critical weighted square-sum divergence holds throughout the final stages, with its full physical-time rate.

The forty-second proves the complete original autonomous delayed transfer, with visible product-sign, energy-sign, interval and coefficient-table repairs. Five solved exercises classify every quadratic polynomial integral, retain all amplitude and clock factors and the full inter-scale pump, construct a neighborhood of persistent finite transfer, and give a much sharper trigger-time interval.

The forty-third proves strict positive viscous activation at every scale, exact physical history maps and complete receiver bounds. Five solved exercises retain the full activation-coefficient product, quantify the positive memory series, construct the exact moment maps, prove the strictly positive orthogonal heat remainder and strengthen all receiver bounds with their original viscous decay. All six viscous supporting notes are included; their complete uniform induction proves the original averaged-equation finite endpoint.

The forty-fourth proves the full original viscous rotation corrector and actual first transfer, including every intermediate forcing interval, all upper bands and the corrected receiving-drive range. Five solved exercises determine exact energy positivity, strengthen the complete velocity transfer, retain all physical Sobolev factors, solve the weighted-tail comparison and compute the exact scalar forcing asymptotic.

The forty-fifth proves the full original-viscosity uniform cascade and its finite maximal endpoint, retaining every physical history, incoming pump, weighted tail and corrected checkpoint. Five solved exercises strengthen amplitude and lifetime bounds, prove Sobolev growth throughout every interval, retain the complete band sums, and establish matching critical growth and its exact time-integrability threshold.

The forty-sixth derives the complete original exterior and interior cylindrical profile maps, including dynamic clocks, signed swirl, curvature, the regular axis, moving period, source elliptic operator and periodic pressure. Five solved exercises retain the original time endpoint, characterize the axial pressure obstruction, prove the exact flux-energy decomposition, construct a full smooth swirl example and strengthen the weighted stream estimate using both boundary values.

The forty-seventh reconstructs the original fields from all six profile residuals, proves the complete sinh-coordinate and sampling-measure maps, and calculates explicit finite-network derivative bounds. Five solved exercises retain the physical pressure and time, test incompatible predictions, compare exact measures, specify a six-layer network and strengthen grid control using all pure second residual derivatives and both quadratic cell-width contributions.

The forty-eighth proves the original anchored Burgers branch, its complete regularity and physical characteristic map. Explicit smooth approximants show the exact scope of residual-derivative losses, with full parameter bounds for the finite network. Five solved exercises give the cubic inverse, original characteristics, all derivative contributions, matching residual-decay coefficients and the losses of a completely specified network.

The forty-ninth proves the original CCF mode defects, signed symmetries and full Hilbert-transform error bounds. Five solved exercises calculate the anchor maps, a specified velocity and both mode defects, the sharp joint error coefficient, every original nonuniform-grid weight and a complete transform budget with its global cutoff terms.

The fiftieth proves the original physical clocks, full weighted transport and exact analytic corrections in the Chen–Hou construction. The complete frozen model identifies the precise derivative threshold and the different behavior of the full Sobolev norm. Five solved exercises calculate a logarithmic clock, a complete annular weight, a nonzero correction tail, every Gaussian derivative norm and the full three-component correction with its gradient defect.

The fifty-first proves the full original two-point transport and reflected-domain correction, the singular-cell repair budget and a rigorous enclosure of the horizontal constant below2.549. Five solved exercises retain both coupled components, an actual mesh region, the rectangle edge in a transport partner, the cell-to-global comparison and four exact cutoff-array lookups with their receiving costs.

The fifty-second certifies all3896 original horizontal upper entries, proves the sharp anisotropy map and reconstructs the full physical K2 transport formulas. It evaluates both singular constants and the horizontal residual exactly, restores the edge-kernel factor and strengthens the height-dependent edge bound. Five solved exercises carry these results through their original physical receiving coefficients.

The fifty-third proves the complete outer and residual-edge K2 bounds, exact horizontal partner maps and the full vertical velocity increments. Every cutoff band, local sign and residual mass remains. A stronger middle bound recovers the source estimate after explicit corrections to intermediate formulas. Five solved exercises quantify measure concentration, test exact physical velocity differences and strengthen whole parameter cells.

The fifty-fourth certifies both complete saved vertical derivative bounds, retaining every original cost and singular endpoint. It identifies and repairs a local extra-area error, reconstructs the weighted velocity integral and proves the actual-exponent selector despite its false all-exponent generalization. Five solved exercises integrate the full residual, quantify midpoint improvement, store an exact correction and enlarge the actual-exponent cell.

Later lessons develop the full residual-stress iterations, forced singularity constructions and compact viscous flows. The series connects the earlier human results, the Alpöge–Buckmaster programme, OpenAI's proposed construction and subsequent mathematical derivations.

Offline reader and editable sources · Editable sources only