Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
Explicit local Lipschitz bound for the regularized SPH vector field
Statement
Let K be a nonempty compact convex subset of the positive-density domain of Fixed-width regularized conservative SPH particle system. Use ||z||#=max_i{|q_i|,|v_i|,|e_i|}, M=Σ_im_i, V=sup_{z∈K}max_i|v_i|, K₁=||∇W_ε||∞, K₂=||D²W_ε||∞, and a(ρ,e)=P(ρ,e)/ρ². Define A₀=max_{z∈K,1≤i≤N}|a(ρ_i(q),e_i)|, A_ρ=max_{z∈K,1≤i≤N}|∂ρa(ρ_i(q),e_i)|, A_e=max{z∈K,1≤i≤N}|∂_ea(ρ_i(q),e_i)|, and A_Lip=2MK₁A_ρ+A_e. By A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value, these maxima are finite and attained, since K is nonempty compact and the functions involved are continuous. Then the full state vector field F is Lipschitz on K with constant
Facts & Assumptions
Given: A nonempty compact convex positive-density tube K, the fixed finite masses, a C¹ pressure map P, and a C² kernel W_ε.
Fixed-width regularized conservative SPH particle system: ρ_i=Σ_jm_jW_ε(q_i−q_j), A_i=a(ρ_i,e_i), and the q, v and e components of F are the displayed finite sums.
Scaling identities for kernel moments and derivatives: ||∇W_ε||∞=K₁ and ||D²W_ε||∞=K₂ are finite for fixed ε.
The mean value inequality: if is continuous and differentiable on with , then : a C¹ map on a line segment changes by at most the supremum derivative norm times the segment length.
A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value: continuous real-valued functions on a nonempty compact metric space are bounded and attain their maxima and minima.
The chain rule for total derivatives: : derivatives of the density and pressure-coefficient compositions are given by the chain rule.
Sums, scalar multiples, products and quotients: , , , and when : derivatives distribute over finite sums, coordinatewise.
Cauchy-Schwarz with its equality case, the triangle inequality for , the parallelogram law and polarisation: Euclidean inner products satisfy Cauchy-Schwarz.
Proof
Let z,z′∈K, δ=||z−z′||#, and z(s)=(1−s)z+sz′ for 0≤s≤1; convexity keeps this segment in K. Differentiating ρ_i(q(s)) using [F5] and the finite-sum rule [F6] gives Σ_jm_j∇W_ε(q_i(s)−q_j(s))·(q_i′−q_i−q_j′+q_j), whose absolute value is at most 2MK₁δ by [F1], [F2], and Cauchy-Schwarz [F7]. Along the same segment, A_i(s)=a(ρ_i(q(s)),e_i(s)) has derivative bounded by 2MK₁A_ρδ+A_eδ=A_Lipδ, by [F5]. The one-dimensional mean value inequality [F3] on s∈[0,1] yields |A_i−A_i′|≤A_Lipδ.
For the acceleration component, subtract its two finite sums. The coefficient difference contributes at most 2MA_LipK₁δ, and the kernel-gradient difference at most 4MA₀K₂δ because the triangle inequality [F7] gives |(q_i−q_j)−(q_i′−q_j′)|≤2δ and [F3] bounds the change in ∇W_ε by K₂ times its argument change. Thus |v̇_i−v̇_i′|≤M(2A_LipK₁+4A₀K₂)δ.
For ė_i, the coefficient difference contributes at most 2MVA_LipK₁δ; the relative-velocity difference at most 2MA₀K₁δ by the triangle inequality [F7]; and the kernel-gradient difference at most 4MA₀VK₂δ by [F3]. Also |q̇_i−q̇_i′|≤δ. Taking the maximum of the three component bounds gives the stated L.
Depends on
- Fixed-width regularized conservative SPH particle system
- Scaling identities for kernel moments and derivatives
- The mean value inequality: if $f : [a,b] \to \mathbb{R}^m$ is continuous and differentiable on $(a,b)$ with $\lVert f'\rVert_2 \le M$, then $\lVert f(b)-f(a)\rVert_2 \le M(b-a)$
- A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value
- The chain rule for total derivatives: $D(g\circ f)(a)=Dg(f(a))\circ Df(a)$
- Sums, scalar multiples, products and quotients: $(f+g)'(c) = f'(c) + g'(c)$, $(\alpha f)'(c) = \alpha f'(c)$, $(fg)'(c) = f'(c)g(c) + f(c)g'(c)$, and $(f/g)'(c) = \bigl(f'(c)g(c) - f(c)g'(c)\bigr)/g(c)^{2}$ when $g(c) \ne 0$
- Cauchy-Schwarz $\lvert\langle x,y\rangle\rvert \le \lVert x\rVert_2\lVert y\rVert_2$ with its equality case, the triangle inequality for $\lVert\cdot\rVert_2$, the parallelogram law and polarisation
Used by
Dependency tree · two levels
65 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Volker Springel, Smoothed Particle Hydrodynamics in Astrophysics, Annual Review of Astronomy and Astrophysics 48 (2010), author full text (standard reference, not scraped)
- Benjamin Seibold, MIT OpenCourseWare 18.336, Numerical Methods for Partial Differential Equations, Lecture 25: Particle Methods (Spring 2009) (standard reference, not scraped)