Editing Openai/6935580c-34e4-8001-9136-7bbdf1499790 (section)

=== Assistant: Below I give precise statements, short proofs Fourier / kernel based and textbook references showing that i the heat se… ===
Below I give precise statements, short proofs (Fourier / kernel based) and textbook references showing that (i) the heat semigroup eνtΔe^{\nu t\Delta}eνtΔ is a bounded, strongly continuous (and smoothing) family of operators on the common norms used for Navier–Stokes, and (ii) the Leray projector PPP is a bounded linear operator on those same spaces. I restrict attention to the usual Banach/Sobolev spaces most used in the Picard approach: Lp(R3)L^p(\mathbb{R}^3)Lp(R3) for 1≤p≤∞1\le p\le\infty1≤p≤∞, Sobolev spaces Hs(R3)H^s(\mathbb{R}^3)Hs(R3) (equivalently Ws,2W^{s,2}Ws,2), and Hölder / Besov-type spaces where noted. I give references after each short proof.

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For each 1≤p≤∞1\le p\le\infty1≤p≤∞ and each t>0t>0t>0,

Gtf=Kt∗f,Kt(x)=(4πνt)−3/2e−∣x∣2/(4νt),G_t f = K_t * f, \qquad K_t(x) = (4\pi\nu t)^{-3/2} e^{-|x|^2/(4\nu t)},Gtf=Kt∗f,Kt(x)=(4πνt)−3/2e−∣x∣2/(4νt),
and

∥Gtf∥Lp≤∥f∥Lp(in fact ∥Gt∥Lp→Lp=1).\|G_t f\|_{L^p} \le \|f\|_{L^p}\qquad(\text{in fact }\|G_t\|_{L^p\to L^p}=1).∥Gtf∥Lp≤∥f∥Lp(in fact ∥Gt∥Lp→Lp=1).
Proof (sketch). The heat kernel KtK_tKt is nonnegative and has unit mass ∫Kt=1\int K_t=1∫Kt=1. By Young’s convolution inequality,

∥Gtf∥Lp=∥Kt∗f∥Lp≤∥Kt∥L1∥f∥Lp=∥f∥Lp.\|G_t f\|_{L^p} = \|K_t * f\|_{L^p} \le \|K_t\|_{L^1}\|f\|_{L^p} = \|f\|_{L^p}.∥Gtf∥Lp=∥Kt∗f∥Lp≤∥Kt∥L1∥f∥Lp=∥f∥Lp.
For p=∞p=\inftyp=∞ the same follows from ∥Kt∥L1=1\|K_t\|_{L^1}=1∥Kt∥L1=1.
(Strict contractivity may hold on some spaces; the important fact is boundedness with norm ≤1\le1≤1.)

References. Standard: Evans, Partial Differential Equations, §2.3; Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations; also Henry, Geometric Theory of Semilinear Parabolic Equations.

===== For any s∈Rs\in\mathbb{R}s∈R and any r≥0r\ge0r≥0, =====

∥Gtf∥Hs+r≤C t−r/2 ∥f∥Hs,t>0,\|G_t f\|_{H^{s+r}} \le C\, t^{-r/2}\,\|f\|_{H^s},\qquad t>0,∥Gtf∥Hs+r≤Ct−r/2∥f∥Hs,t>0,
where C=C(ν,r)C=C(\nu,r)C=C(ν,r). Equivalently, for integer k≥0k\ge0k≥0,

∥∇kGtf∥L2≤C t−k/2 ∥f∥L2.\|\nabla^k G_t f\|_{L^2} \le C\, t^{-k/2}\,\|f\|_{L^2}.∥∇kGtf∥L2≤Ct−k/2∥f∥L2.
Proof (Fourier). Using the Fourier transform Gtf^(ξ)=e−νt∣ξ∣2f^(ξ)\widehat{G_t f}(\xi)=e^{-\nu t|\xi|^2}\widehat f(\xi)Gtf(ξ)=e−νt∣ξ∣2f(ξ),

∥Gtf∥Hs+r2=∫R3(1+∣ξ∣2)s+re−2νt∣ξ∣2∣f^(ξ)∣2 dξ.\|G_t f\|_{H^{s+r}}^2 = \int_{\mathbb{R}^3} (1+|\xi|^2)^{s+r} e^{-2\nu t|\xi|^2} |\widehat f(\xi)|^2\,d\xi.∥Gtf∥Hs+r2=∫R3(1+∣ξ∣2)s+re−2νt∣ξ∣2∣f(ξ)∣2dξ.
Factor (1+∣ξ∣2)re−2νt∣ξ∣2≤C t−r(1+|\xi|^2)^{r} e^{-2\nu t|\xi|^2} \le C\, t^{-r}(1+∣ξ∣2)re−2νt∣ξ∣2≤Ct−r (supremum over ξ\xiξ is O(t−r)O(t^{-r})O(t−r)). Thus the integral is ≤Ct−r∥f∥Hs2\le C t^{-r}\|f\|_{H^s}^2≤Ct−r∥f∥Hs2. Take square root to get the stated estimate.

References. Standard Fourier-semigroup calculation; see Pazy (semigroup theory), Evans §2.3, and Temam, Navier–Stokes Equations.

===== GtG_tGt is a strongly continuous semigroup on LpL^pLp and on HsH^sHs: for each fff fixed, =====

lim⁡t↓0∥Gtf−f∥X=0,X=Lp or Hs.\lim_{t\downarrow 0} \|G_t f - f\|_{X} = 0,\qquad X=L^p\ \text{or}\ H^s.t↓0lim∥Gtf−f∥X=0,X=Lp or Hs.
Proof (sketch). For LpL^pLp this follows from the approximation-identity property of KtK_tKt as t→0t\to0t→0. For HsH^sHs, use the Fourier representation: (1+∣ξ∣2)s/2(e−νt∣ξ∣2−1)f^(ξ)→0(1+|\xi|^2)^{s/2}(e^{-\nu t|\xi|^2}-1)\widehat f(\xi)\to0(1+∣ξ∣2)s/2(e−νt∣ξ∣2−1)f(ξ)→0 pointwise and dominated convergence given (1+∣ξ∣2)s∣f^∣2∈L1(1+|\xi|^2)^{s}|\widehat f|^2\in L^1(1+∣ξ∣2)s∣f∣2∈L1. Thus strong continuity holds.

References. Pazy, Semigroups; see also functional-analytic texts and Evans.

===== For 1≤p≤q≤∞1\le p\le q\le\infty1≤p≤q≤∞, =====

∥Gtf∥Lq≤Ct−32(1p−1q)∥f∥Lp,\|G_t f\|_{L^q} \le C t^{-\frac{3}{2}(\frac{1}{p}-\frac{1}{q})} \|f\|_{L^p},∥Gtf∥Lq≤Ct−23(p1−q1)∥f∥Lp,
with the usual Gaussian scaling. This provides short-time control near t=0t=0t=0.

Proof. Estimate KtK_tKt in LrL^rLr norms and apply Young inequality. Standard.

References. Beckner/Young/heat kernel estimates; see Stein, Harmonic Analysis or Davies, Heat Kernels and Spectral Theory.

==== Recall the matrix-valued Fourier multiplier for PPP: ====

Pf^(ξ)=(I−ξ⊗ξ∣ξ∣2)f^(ξ)=:P(ξ)f^(ξ).\widehat{P f}(\xi) = \Big(I - \frac{\xi\otimes\xi}{|\xi|^2}\Big)\widehat f(\xi) =:\mathbb{P}(\xi)\widehat f(\xi).Pf(ξ)=(I−∣ξ∣2ξ⊗ξ)f(ξ)=:P(ξ)f(ξ).
Each component of P(ξ)\mathbb{P}(\xi)P(ξ) is a homogeneous rational function of degree 000 in ξ\xiξ (symbol of order 000).

===== The Leray projector PPP extends to a bounded linear operator on Lp(R3;R3)L^p(\mathbb{R}^3;\mathbb{R}^3)Lp(R3;R3) for every 1<p<∞1<p<\infty1<p<∞. More precisely, the components of PPP are combinations of singular integral operators (Riesz transforms) and the identity; hence Calderón–Zygmund theory applies. =====

Proof (sketch). Write I−ξ⊗ξ/∣ξ∣2=I−ξiξj∣ξ∣2I - \xi\otimes\xi/|\xi|^2 = I - \frac{\xi_i\xi_j}{|\xi|^2}I−ξ⊗ξ/∣ξ∣2=I−∣ξ∣2ξiξj. The scalar multipliers ξk/∣ξ∣\xi_k/|\xi|ξk/∣ξ∣ correspond to Riesz transforms Rk\mathcal{R}_kRk whose kernels are homogeneous of degree −3-3−3 and satisfy Calderón–Zygmund kernel bounds. The entries of P\mathbb{P}P are linear combinations of products of Riesz transforms (order-zero multipliers). By Calderón–Zygmund theory (or the Mikhlin multiplier theorem), these are bounded on LpL^pLp for 1<p<∞1<p<\infty1<p<∞.

References. Stein, Singular Integrals and Differentiability Properties of Functions (Riesz transforms & CZ theory); Grafakos, Classical Fourier Analysis (Mikhlin theorem, multipliers); also Temam or Constantin–Foias for Navier–Stokes.

===== PPP is bounded on Hs(R3)H^s(\mathbb{R}^3)Hs(R3) for every real sss; i.e. =====

∥Pf∥Hs≤Cs∥f∥Hs.\|P f\|_{H^s} \le C_s \|f\|_{H^s}.∥Pf∥Hs≤Cs∥f∥Hs.
Proof (Fourier / multiplier). PPP is a Fourier multiplier with symbol P(ξ)\mathbb{P}(\xi)P(ξ) that is smooth away from ξ=0\xi=0ξ=0 and homogeneous of degree 000. For an HsH^sHs-norm,

∥Pf∥Hs2=∫(1+∣ξ∣2)s∣P(ξ)f^(ξ)∣2 dξ≤sup⁡ξ∥P(ξ)∥2(1+∣ξ∣2)s∣f^(ξ)∣2 dξ=C∥f∥Hs2,\|P f\|_{H^s}^2 = \int (1+|\xi|^2)^s |\mathbb{P}(\xi)\widehat f(\xi)|^2\,d\xi
\le \sup_\xi \|\mathbb{P}(\xi)\|^2 (1+|\xi|^2)^s |\widehat f(\xi)|^2\,d\xi
= C \|f\|_{H^s}^2,∥Pf∥Hs2=∫(1+∣ξ∣2)s∣P(ξ)f(ξ)∣2dξ≤ξsup∥P(ξ)∥2(1+∣ξ∣2)s∣f(ξ)∣2dξ=C∥f∥Hs2,
since P(ξ)\mathbb{P}(\xi)P(ξ) is bounded (matrix entries are bounded by 1). More refined multiplier theorems (Mikhlin) give boundedness on Ws,pW^{s,p}Ws,p and Besov spaces.

References. Mikhlin multiplier theorem (Grafakos), Stein; for direct PDE context see Temam, Navier–Stokes Equations or Lions, Quelques méthodes.

===== By Littlewood–Paley theory and multiplier theorems, PPP is bounded on Besov spaces Bp,qsB^s_{p,q}Bp,qs and Hölder spaces CαC^\alphaCα (appropriate ranges) — essentially because PPP is an order-zero Fourier multiplier satisfying the standard symbol estimates. =====

References. Bahouri–Chemin–Danchin, Fourier Analysis and Nonlinear Partial Differential Equations; Triebel, Theory of Function Spaces.

==== From the facts above we get the exact properties needed in the mild/Picard construction: ====
* eνtΔe^{\nu t\Delta}eνtΔ is bounded X→XX\to XX→X for X=LpX=L^pX=Lp or X=HsX=H^sX=Hs and is strongly continuous there. (Statements 1 and 3.)
* eνtΔe^{\nu t\Delta}eνtΔ is smoothing: it maps XXX to spaces with higher regularity with the explicit gain estimate ∥etΔf∥Hs+r≤Ct−r/2∥f∥Hs\|e^{t\Delta}f\|_{H^{s+r}}\le C t^{-r/2}\|f\|_{H^s}∥etΔf∥Hs+r≤Ct−r/2∥f∥Hs. (Statement 2.)
* PPP is a bounded linear operator on XXX for the ranges above. (Statements 5–7.)
* Consequently the bilinear map B(u,v)=P∇⋅(u⊗v)B(u,v)=P\nabla\cdot(u\otimes v)B(u,v)=P∇⋅(u⊗v) satisfies the algebra estimates used in Picard: - If X=HsX=H^sX=Hs with s>3/2s>3/2s>3/2 then HsH^sHs is an algebra and pointwise product maps Hs×Hs→HsH^s\times H^s\to H^sHs×Hs→Hs; ∇⋅\nabla\cdot∇⋅ maps Hs→Hs−1H^s\to H^{s-1}Hs→Hs−1 and PPP maps Hs−1→Hs−1H^{s-1}\to H^{s-1}Hs−1→Hs−1. Hence ∥B(u,v)∥Hs−1≤C∥u∥Hs∥v∥Hs.\|B(u,v)\|_{H^{s-1}} \le C \|u\|_{H^s}\|v\|_{H^s}.∥B(u,v)∥Hs−1≤C∥u∥Hs∥v∥Hs. - If X=LpX=L^pX=Lp with ppp large and suitable Sobolev embeddings hold, analogous estimates can be arranged (use Besov or Bony paraproduct calculus for critical spaces).

References for the Picard setup.
* Temam, Navier–Stokes Equations (local existence in Sobolev spaces);
* Constantin & Foias, Navier–Stokes Equations (functional setting and semigroup methods);
* Pazy, Semigroups of Linear Operators (abstract semigroup-based local theory);
* Bahouri–Chemin–Danchin for critical spaces.

==== - L. C. Evans, Partial Differential Equations, 2nd ed., Graduate Studies in Mathematics, AMS — heat kernel facts, semigroup basics. ====
* A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer — abstract semigroup theory and strong continuity.
* H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer — Sobolev embeddings and algebra properties.
* P. G. Lemarié-Rieusset, Recent developments in the Navier–Stokes problem, CRC Press — comprehensive Navier–Stokes functional analysis.
* P. Constantin and C. Foias, Navier–Stokes Equations, University of Chicago Press — classical functional setting, Leray projector, semigroup methods.
* E. M. Stein, Singular Integrals and Differentiability Properties of Functions — Calderón–Zygmund theorems and Riesz transforms.
* L. Grafakos, Classical Fourier Analysis — Mikhlin multipliers and detailed proofs for Fourier multipliers on LpL^pLp and Sobolev scales.
* H. Bahouri, J.-Y. Chemin, R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations — for Besov and paraproduct approaches.

==== I can write fully detailed, line-by-line proofs for any of the following (pick one): ====
# The HsH^sHs-smoothing estimate ∥etΔf∥Hs+r≤Ct−r/2∥f∥Hs\|e^{t\Delta} f\|_{H^{s+r}} \le C t^{-r/2}\|f\|_{H^s}∥etΔf∥Hs+r≤Ct−r/2∥f∥Hs (Fourier proof with explicit constants).
# Boundedness of the Leray projector on LpL^pLp via Calderón–Zygmund/Riesz-transform decomposition (construct kernels and verify CZ estimates).
# Verification that for X=HsX=H^sX=Hs with s>3/2s>3/2s>3/2 the bilinear map B(u,v)=P∇⋅(u⊗v)B(u,v)=P\nabla\cdot(u\otimes v)B(u,v)=P∇⋅(u⊗v) satisfies ∥B(u,v)∥Hs−1≤C∥u∥Hs∥v∥Hs\|B(u,v)\|_{H^{s-1}} \le C\|u\|_{H^s}\|v\|_{H^s}∥B(u,v)∥Hs−1≤C∥u∥Hs∥v∥Hs (product and derivative estimates).

Tell me which one you want first and I will produce the step-by-step derivation.