Navier–Stokes: new knowl collection

14 new knowls.

Foundations

• Binomial coefficient
• Ceiling function
• Finite product
• Finite set
• Finite sum
• First-order logic and inference rules
• Floor function
• Mathematical axiom
• Mathematical proof
• Mathematical proposition
• Mathematical theorem
• Nearest integer
• Rising factorial
• Telescoping sum

37 new knowls.

Linear algebra

• Block matrix
• Cross product in three-dimensional Euclidean space
• Dual basis
• Elementary row operation
• Hadamard product
• Inverse matrix
• Left inverse of a linear map
• Moving frame for a family of subspaces
• Orthogonal reflection across a hyperplane
• Outer product
• Quadratic map over the real numbers
• Right inverse of a linear map
• Unit vector

Real and multivariable analysis

• A sufficient criterion for smoothness at a cylindrical axis
• Cartesian coordinates
• Completing the square
• Cosine function
• Curl of a three-dimensional vector field
• Cylindrical coordinates and their orthonormal frame
• Divergence of a Euclidean vector field
• Divergence-free field
• Euclidean Laplacian
• Even and odd parity of a function
• Exponential function
• Logarithmic radial derivative
• Matrix-valued tensor field in Euclidean coordinates
• Natural logarithm
• Nonnegative square root
• Product rule for derivatives
• Quotient rule
• Real power of a positive number
• Row divergence of a matrix field
• Sine function
• Stokes streamfunction for an axisymmetric meridional field
• Vector field on a Euclidean open set
• Vector potential for a divergence-free field

Topology

• Euclidean annulus

22 new knowls.

Function spaces and operators

• Absolutely convergent series in a Banach space
• Bounded bilinear map between normed spaces
• Continuous Banach-valued functions on a compact space
• Integral operator with a measurable kernel

Measure and integration

• Bochner integral
• Complete measure
• Completeness of Lebesgue spaces
• Completion of a measure space
• Differentiation under an integral sign
• Hausdorff dimension
• Hausdorff measure in a metric space
• Integrable majorant for a family
• Interpolation between two Lebesgue norms
• Locally integrable function
• Mixed Lebesgue norm
• Parabolic Hausdorff measure
• Product sigma-algebra
• Sigma-algebra generated by a family
• Sigma-finite measure
• Strongly measurable Banach-valued function

Real and multivariable analysis

• Sum of nonnegative extended real terms

Topology

• Parabolic metric on space-time

37 new knowls.

Complex analysis

• Cauchy estimate for holomorphic derivatives
• Common holomorphic neighborhood for a parameter family
• Holomorphic dependence of a parameter integral
• Holomorphic logarithm of a nonvanishing function
• Locally uniform limit theorem for holomorphic functions
• Nonvanishing on a neighborhood of a compact set
• Power of a holomorphic function using a chosen logarithm

Function spaces and operators

• A mixed-derivative estimate after radial inversion
• Contraction criterion on a closed norm ball
• Holomorphic realization of weighted analytic coefficients
• Operator inversion from summable bounds on powers
• Product estimate in a two-index analytic coefficient space
• Radial operator bounds in an analytic coefficient norm
• Two-index analytic coefficient space

Linear algebra

• Matrix exponential

Measure and integration

• Ordered integration simplex

Ordinary differential equations

• Continuation criterion for a finite-dimensional ODE
• Continuous-dependence estimate for ODEs
• Differential equations
• Duhamel formula for a linear ODE
• Fundamental matrix and principal propagator
• Gronwall inequality
• Holomorphic dependence of local ODE solutions
• Initial-value problem for an ODE
• Integrating factor for a scalar linear ODE
• Linear ordinary differential system
• Ordinary differential equation
• Peano–Baker series
• Picard iteration
• Picard–Lindelöf local existence and uniqueness theorem
• Regular inverse of Y d2/dY2 plus nu d/dY
• Regular singular point of a second-order linear ODE
• Scalar comparison barriers for an ODE
• Smooth dependence of ODE solutions on parameters

Real and multivariable analysis

• Radial averaging operator from the origin
• Real-analytic function

Topology

• Locally Lipschitz map

21 new knowls.

Real and multivariable analysis

• Borel extension of a prescribed smooth jet
• Cartesian jet of a function
• Compact derivative seminorm
• Compactly supported function
• Derivative estimates for scaled cutoffs
• Flat exponential
• Flat factors absorb polynomial derivative growth
• Flat function along a set
• Hadamard division lemma in one normal variable
• Localization through a vector potential
• Multi-index Leibniz rule
• Normalized primitive of an exponential flat factor
• Smooth cutoff function
• Smooth extension of a function
• Smooth zero extension across a flat boundary
• Squared partition of unity

Topology

• Compact exhaustion of an open Euclidean set
• Positive margin around a compact subset of an open set

Analysis

• Locally finite sum of smooth functions
• Lp estimate for sums with bounded overlap
• Smooth convergence of a series from derivative bounds

50 new knowls.

Fluid dynamics

• Advective transport time scale
• Affine velocity field
• Angular velocity about an axis
• Angular-velocity equation in axisymmetric flow
• Axial velocity
• Axisymmetric Navier–Stokes component equations
• Axisymmetric vector field
• Centripetal acceleration for circular motion
• Conservative incompressible momentum equation
• External force in a normalized fluid equation
• Fluid dynamics
• Fluid velocity field
• Incompressible Euler equations
• Incompressible Navier–Stokes equations
• Incompressible velocity field
• Kinetic energy of an incompressible velocity field
• Material derivative
• Meridional velocity
• Momentum flux tensor in an incompressible fluid
• Parallel shear flow
• Particle trajectory of a velocity field
• Pressure in incompressible flow
• Quadratic advection of a velocity field
• Radial velocity in cylindrical coordinates
• Rate-of-strain tensor
• Reynolds number
• Rotating flow about a fixed axis
• Specific angular momentum about an axis
• Swirl component of a cylindrical velocity
• Transport operator along a vector field
• Viscosity in the incompressible Newtonian model
• Viscous diffusion time scale
• Vortex line
• Vorticity equation for incompressible flow
• Vorticity of a velocity field

Linear algebra

• Symmetric matrix
• Transpose of a matrix

Partial differential equations

• Cauchy problem for an evolution PDE
• Classical solution of a PDE
• Conservation law in divergence form
• Finite-time blowup in a specified norm
• Initial datum for an evolution equation
• Linear partial differential operator
• Local well-posedness of an evolution problem
• Maximal existence time in a solution class
• Partial differential equation
• Partial differential equations
• Principal part of a differential operator
• Transport equation

Real and multivariable analysis

• Limit superior of a function at an endpoint

38 new knowls.

Asymptotic estimates

• Absorbing logarithmic losses into a power
• Asymptotic equivalence
• Asymptotic expansion
• Asymptotic scale
• Big-O notation
• Comparable positive functions
• Concentration scale
• Finite truncation of an expansion
• Formal expansion
• Hierarchy of parameter choices
• Leading order
• Leading term of an asymptotic expansion
• Little-o notation
• Perturbation
• Power law
• Quadratic self-interaction
• Remainder estimate
• Uniform parameter estimate

Fluid dynamics

• Linearized Navier–Stokes operator
• Navier–Stokes momentum residual
• Rescaling unit viscosity to positive viscosity
• Scaling symmetry of Navier–Stokes

Function spaces and operators

• Derivative loss in an estimate

Partial differential equations

• Differential polynomial
• Self-similar profile
• Similarity variables

Real and multivariable analysis

• Anisotropic dilation
• Chain rule in implicit anisotropic coordinates
• Envelope-controlled amplitude
• Flat logarithmic interval weight
• Implicit anisotropic similarity coordinates
• Linearization of a nonlinear map
• Normalized dyadic chart
• Norms of an anisotropically concentrating profile
• Product bounds for weighted coefficients and amplitudes
• Scaling of weighted radial moments
• Smooth dyadic partition of a positive scale
• Weighted coefficient class

35 new knowls.

Fluid dynamics

• Angular average in cylindrical components
• Curl completion of a transverse oscillatory wave
• Phase transport defect
• Shearing wave on an affine flow
• Viscous damping of a Fourier mode

Fourier and harmonic analysis

• Carrier frequency parameter
• Decay of smooth periodic Fourier coefficients
• Fourier character on the standard torus
• Fourier harmonic
• Gaussian bound from a decreasing net growth rate
• Local wavevector
• Mean of a product of periodic Fourier sums
• Orthogonality of periodic Fourier characters
• Oscillatory modulation
• Oscillatory phase
• Oscillatory pulse
• Periodic antiderivative
• Periodic Fourier coefficient
• Periodic Fourier multiplier
• Periodic Fourier series
• Pulse envelope
• Smooth periodic Fourier reconstruction
• Transverse polarization of a vector wave
• Wave amplitude
• Wavelength
• Wavenumber
• Zero Fourier mode

Measure and integration

• Projection onto zero-mean functions
• Zero-mean function

Ordinary differential equations

• Duhamel estimate with a common envelope
• Factoring scalar damping from a matrix evolution
• Instability of a linear equilibrium
• Linear evolution under a moving normal constraint

Real and multivariable analysis

• Evaluation along an auxiliary map
• Periodic function

35 new knowls.

Asymptotic estimates

• Logarithmic selection of an iteration level

Foundations

• Descent of a function through a surjection
• Strict partial order

Fourier and harmonic analysis

• Diophantine direction
• Directional inverse with a Diophantine bound
• Fourier criterion for descent under a torus covering
• Nearly resonant Fourier mode
• Normalized torus average
• Resonant mode of a directional derivative
• Small divisor
• Torus coverings preserve normalized averages

Linear algebra

• Full-rank Euclidean lattice
• Fundamental domain of a Euclidean lattice
• Index of an integer sublattice

Real and multivariable analysis

• Chain rule for iterated torus variables
• Irrational number
• Quadratic algebraic conjugate
• Quadratic irrational
• Quadratic irrational lower bound

Topology

• Common torus for iterated integer coverings
• Deck transformation
• Flat torus
• Integer-matrix torus covering
• Local lifts of cells under a torus covering
• Separated centers under finitely many torus covers

Analysis

• Vanishing products from separated auxiliary supports

Discrete structures

• Countable graph
• Directed acyclic graph
• Directed cycle
• Directed graph
• Directed path
• Greedy coloring of a countable bounded-degree graph
• Proper graph coloring
• Support intersection graph
• Vertex degree

29 new knowls.

Convex geometry

• Conical combination
• Convex cone
• Planar cone defined by a quadratic inequality
• Positive span
• Two-generator cone test
• Uniform directional margin inside a cone

Fluid dynamics

• Radial-tangential velocity covariance

Function spaces and operators

• Small solution of an invertible linear equation with quadratic error

Linear algebra

• Linearized inverse for squared amplitudes
• Moment pairing matrix
• Positive realization by squared amplitudes
• Vandermonde matrix

Measure and integration

• Averaged second-moment matrix

Partial differential equations

• Solvability compatibility condition

Probability and moments

• Covariance matrix
• Derivatives of the log moment generating function
• Exponential tilt
• Variance of a normalized exponential weight

Real and multivariable analysis

• Circle reparametrization by a positive density
• Compactly supported weighted radial primitive
• Cumulative weighted integral
• Inverse loss when two moment exponents coalesce
• Invertibility of moments on ordered bump supports
• Matching prescribed integral moments
• Positive powers of flat weighted profiles
• Preservation of exterior primitives by moment matching
• Projection removing one weighted moment
• Weighted radial moment
• Zero count for distinct real powers

20 new knowls.

Fluid dynamics

• Cylindrical swirl diffusion operator
• Integrability of the subtracted swirl heat tail
• Integral profile for radial swirl heat flow
• Pure swirl solution from a radial heat flow

Measure and integration

• Reference-subtracted integral

Ordinary differential equations

• Differential equation for the Gamma average

Partial differential equations

• Diffusion in a fixed direction
• Duhamel formula for the heat equation
• Euclidean heat kernel
• Heat equation
• Heat semigroup
• Heat smoothing estimate
• Solution of the heat equation by Gaussian convolution

Real and multivariable analysis

• Gamma average of an algebraic factor
• Gamma function on positive real arguments
• Gaussian integral
• Radial Laplacian
• Smooth nonanalytic endpoint of a Gamma average
• The constant pi
• Tricomi U for positive real arguments

27 new knowls.

Fluid dynamics

• Canonical pressure from an integrable tensor

Fourier and harmonic analysis

• Calderón–Zygmund kernel
• Calderón–Zygmund operator
• Euclidean L2 Fourier multiplier
• Kernel cancellation in a multiplication commutator
• Kernel of a double Riesz transform in three dimensions
• Lp boundedness of Calderón–Zygmund operators
• Principal-value singular integral
• Riesz transform
• Scaled cutoff commutator estimate

Function spaces and operators

• Derivative bound between Fourier Sobolev spaces
• Dirac delta distribution
• Fourier Sobolev space on Euclidean space
• Integrable functions in negative Sobolev spaces
• Local Sobolev space
• Multiplication operator
• Smooth compact-support approximation in Euclidean Sobolev space
• Sobolev embedding into bounded continuous derivatives
• Sobolev estimate with a scaled cutoff
• Three-dimensional Sobolev inequality

Linear algebra

• Kronecker delta

Partial differential equations

• A harmonic global Sobolev distribution vanishes
• Harmonic distribution
• Newtonian fundamental solution in three dimensions
• Newtonian potential in three dimensions
• Poisson equation

Real and multivariable analysis

• Mollifier

22 new knowls.

Convex geometry

• Absorbing a subquadratic power

Fluid dynamics

• Changing viscosity by spatial rescaling
• Classical uniqueness from difference energy
• Cutoff pressure-flux estimate for a velocity difference
• Delayed parabolic rescaling at a fixed terminal time
• Difference equation for two incompressible flows
• Energy bound from a time-integrable L2 force
• Global kinetic-energy identity
• Identifying pressure gradients without a pressure growth assumption
• Integration cancellation for divergence-free transport and pressure
• Kinetic-energy inequality
• Local kinetic-energy balance for smooth incompressible flow
• Localized difference-energy identity
• Periodization of separated incompressible flows
• Relative kinetic energy
• Uniformly bounded kinetic energy
• Uniqueness against a smooth compactly supported reference flow
• Viscous dissipation in the incompressible energy balance

Fourier and harmonic analysis

• Periodization of a compactly supported smooth function

Linear algebra

• Frobenius norm

Partial differential equations

• Supremum growth obstructs continuous Sobolev continuation

Real and multivariable analysis

• Unbounded values approaching a point prevent continuous extension

17 new knowls.

Asymptotic estimates

• Asymptotic summation by shrinking cutoffs
• Diagonal choice of shrinking scales
• Exponential decay in the square of a logarithm
• Infinite-order decay with derivatives

Foundations

• Recursion on the natural numbers

Fourier and harmonic analysis

• Rapid decay of a shifted torus integral

Function spaces and operators

• Derivative requirements on a finite computation graph
• Residual update with a fixed linear inverse

Partial differential equations

• Difference estimate for a differential polynomial
• Flat residual from asymptotic summation
• Localized characteristic primitive with an explicit defect
• Shifted characteristic primitive on a torus

Real and multivariable analysis

• Asymptotic summation preserving divergence freedom
• Derivatives of a cutoff composed with a controlled scale
• One-sided jet limits from uniform derivative bounds
• Smooth extension from bounded endpoint jets
• Weighted shifted radial primitive

29 new knowls.

Differential geometry

• The h-principle

Fluid dynamics

• Active scalar equation
• Averaged Navier–Stokes model
• Caffarelli–Kohn–Nirenberg partial regularity theorem
• Endpoint L-infinity-in-time L3 regularity criterion
• Euler subsolution with prescribed energy
• Hypodissipative Navier–Stokes equations
• Incompressible porous media equation
• Leray projection on Euclidean space
• Leray–Hopf solution
• Local energy inequality
• Navier–Stokes energy class
• Reynolds averaging stress
• Reynolds defect system
• Singular set of a weak velocity
• Suitable weak solution
• Weak Navier–Stokes solution

Fourier and harmonic analysis

• Fractional Laplacian on Euclidean space
• Order-zero Fourier multiplier

Function spaces and operators

• Weakly continuous path

Partial differential equations

• Backward uniqueness
• Convex integration
• Differential inclusion
• First-order differential relation
• Formal and holonomic solutions
• Nonuniqueness for an initial-value problem
• Partial regularity
• Regularity criterion
• Weak formulation of a differential equation

Real and multivariable analysis

• Geometric series — added while connecting the complete paper to the batch collection.