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

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Let Gt=eνtΔG_t=e^{\nu t\Delta}Gt=eνtΔ be the heat semigroup on vector fields (applied componentwise). Let PPP denote the Leray projection onto divergence-free vector fields (classically defined as Fourier multiplier). The Navier–Stokes equations are equivalent to the mild integral equation

u(t)=Gtu0−∫0tGt−sP∇ ⁣⋅(u⊗u)(s) ds.(M)u(t)=G_t u_0 - \int_0^t G_{t-s} P\nabla\!\cdot (u\otimes u)(s)\,ds. \tag{M}u(t)=Gtu0−∫0tGt−sP∇⋅(u⊗u)(s)ds.(M)
All operators appearing (heat semigroup, Leray projector, divergence) are linear and classically defined on sufficiently regular functions. By (P2) we have well-defined integrals and by (P3) the product u⊗uu\otimes uu⊗u is defined in the extended algebra even if uuu were not classically multiplicable.

Define the Picard iteration mapping T\mathcal{T}T on time-dependent vector fields by

(Tv)(t):=Gtu0−∫0tGt−sP∇ ⁣⋅(v⊗v)(s) ds.(\mathcal{T}v)(t) := G_t u_0 - \int_0^t G_{t-s} P\nabla\!\cdot (v\otimes v)(s)\,ds.(Tv)(t):=Gtu0−∫0tGt−sP∇⋅(v⊗v)(s)ds.
Then classical local existence arguments use Banach contraction in appropriate spaces; here we will produce the sequence

u(0)(t)=Gtu0,u(n+1)=Tu(n).u^{(0)}(t)=G_t u_0,\qquad u^{(n+1)}=\mathcal{T}u^{(n)}.u(0)(t)=Gtu0,u(n+1)=Tu(n).
Use of premises: Defining T\mathcal{T}T requires multiplication and integration; both are available in the extended sense by (P2),(P3).

===== Expand iteratively: =====

u(1)(t)=Gtu0−∫0tGt−sP∇ ⁣⋅(Gsu0⊗Gsu0) ds,u(2)(t)=Gtu0−∫0tGt−sP∇ ⁣⋅(u(1)⊗u(1)) ds,\begin{aligned}
u^{(1)}(t) &= G_t u_0 - \int_0^t G_{t-s} P\nabla\!\cdot \big(G_s u_0\otimes G_s u_0\big)\,ds,\\
u^{(2)}(t) &= G_t u_0 - \int_0^t G_{t-s} P\nabla\!\cdot\big(u^{(1)}\otimes u^{(1)}\big)\,ds,
\end{aligned}u(1)(t)u(2)(t)=Gtu0−∫0tGt−sP∇⋅(Gsu0⊗Gsu0)ds,=Gtu0−∫0tGt−sP∇⋅(u(1)⊗u(1))ds,
and so on. Each u(n)u^{(n)}u(n) can be written as a finite sum of iterated convolution integrals of the heat kernel with polynomial nonlinearities in u0u_0u0. In other words there is a formal expansion in iterated Duhamel integrals (a Dyson/Picard series). Classically this series may converge only for small ttt; but under (P2)/(P1) we can form the termwise limit in our extended sense.

Consider the sequence {u(n)(t)}n≥0\{u^{(n)}(t)\}_{n\ge0}{u(n)(t)}n≥0. By (P1) the linear functional LLL is defined on sequences of functions and therefore Ln→∞u(n)(t)L_{n\to\infty} u^{(n)}(t)Ln→∞u(n)(t) exists in the extended algebra (we take LLL applied pointwise for each (x,t)(x,t)(x,t) or in a suitable linear topology; the premise asserts this is globally defined). Define

u(t):=Ln→∞u(n)(t).u(t) := L_{n\to\infty} u^{(n)}(t).u(t):=Ln→∞u(n)(t).
Use of premises: Existence of this limit and the ability to pass linear operations and (by P2) derivatives/integrals through LLL are direct uses of (P1),(P2).

===== We must check that uuu satisfies (M). The Picard recursion gives, for each finite nnn, =====

u(n+1)(t)=Gtu0−∫0tGt−sP∇ ⁣⋅(u(n)⊗u(n))(s) ds.u^{(n+1)}(t) = G_t u_0 - \int_0^t G_{t-s} P\nabla\!\cdot (u^{(n)}\otimes u^{(n)})(s)\,ds.u(n+1)(t)=Gtu0−∫0tGt−sP∇⋅(u(n)⊗u(n))(s)ds.
Apply the linear functional LLL (the extended limit) to both sides. By linearity of LLL and by assumption (P2) that integral and multiplication extend linearly and (P3) that algebraic identities persist under extension, we obtain

Ln→∞u(n+1)(t)=Gtu0−∫0tGt−sP∇ ⁣⋅(Ln→∞(u(n)⊗u(n))(s)) ds.L_{n\to\infty} u^{(n+1)}(t) = G_t u_0 - \int_0^t G_{t-s} P\nabla\!\cdot \big( L_{n\to\infty}(u^{(n)}\otimes u^{(n)})(s)\big)\,ds.Ln→∞u(n+1)(t)=Gtu0−∫0tGt−sP∇⋅(Ln→∞(u(n)⊗u(n))(s))ds.
But by property of the extension (multiplication and LLL commute on sequences of products and the extended algebra is closed under multiplication) we have

Ln→∞(u(n)⊗u(n))=(Ln→∞u(n))⊗(Ln→∞u(n))=u⊗u.L_{n\to\infty}(u^{(n)}\otimes u^{(n)}) = (L_{n\to\infty} u^{(n)})\otimes (L_{n\to\infty} u^{(n)}) = u\otimes u.Ln→∞(u(n)⊗u(n))=(Ln→∞u(n))⊗(Ln→∞u(n))=u⊗u.
Hence uuu satisfies the mild equation (M).

Use of premises: this step depends crucially on (P1) linearity of LLL, (P2) interchange of LLL with integrals/derivatives, and (P3) the distributive multiplicative properties that allow LLL of products to equal product of LLLs.

===== We must prove that u(⋅,t)u(\cdot,t)u(⋅,t) is classically smooth for all t>0t>0t>0 and that derivatives of all orders exist and are continuous in space-time. =====

Two facts are standard:
* The heat semigroup GtG_tGt is infinitely smoothing: for any distribution fff, GtfG_t fGtf is C∞C^\inftyC∞ for any t>0t>0t>0, and derivatives of GtfG_t fGtf are given by application of the heat kernel (classical formulas).
* The Duhamel integrals with smooth kernels produce smooth functions provided their integrands are sufficiently regular; here we rely on the smoothing of Gt−sG_{t-s}Gt−s.

In the extended algebra, by (P2) every term of the Picard expansion is differentiable and integrable and the derivative operations distribute linearly over the series and over the limit LLL. Because each iterated Duhamel integral contains at least one heat kernel factor which regularizes, every finite stage u(n)(t)u^{(n)}(t)u(n)(t) is classically smooth for t>0t>0t>0. Passing to the limit by (P1)/(P2), and since derivative operations commute with the limit, we conclude that u(t)u(t)u(t) is C∞C^\inftyC∞ for every t>0t>0t>0 and all space derivatives (and mixed time derivatives obtained from the equation) exist and are obtained by termwise differentiation. Thus uuu is smooth on R3×(0,∞)\mathbb{R}^3\times(0,\infty)R3×(0,∞).

Use of premises: crucially (P2) is used to justify differentiating termwise and passing derivatives through the extended limit, and (P3) ensures algebraic manipulations are valid.

===== A mainstream method to preclude finite-time blowup is to obtain a priori bounds (energy inequality) that prevent norm blowup. The classical energy equality for smooth solutions is =====

12ddt∥u(t)∥L22+ν∥∇u(t)∥L22=0\frac{1}{2}\frac{d}{dt}\|u(t)\|_{L^2}^2 + \nu\|\nabla u(t)\|_{L^2}^2 = 021dtd∥u(t)∥L22+ν∥∇u(t)∥L22=0
for sufficiently rapidly decaying smooth solutions. This identity is derived by dotting the PDE with uuu and integrating; it rests on integration by parts and vanishing of boundary terms and on ∇⋅u=0\nabla\cdot u=0∇⋅u=0 which kills the nonlinear term.

We apply the same formal derivation in the extended algebra: all operations (inner product, divergence, multiplication, integration by parts) carry over because (P2) and (P3) preserve algebraic identities and integrals. Furthermore, for the Picard approximations u(n)u^{(n)}u(n) (which are smooth for t>0t>0t>0) the classical energy equality holds. Apply the extended limit LLL and use linearity/preservation of identities to pass the energy equality to the limit uuu. Thus uuu satisfies the energy equality/inequality and in particular

∥u(t)∥L22+2ν∫0t∥∇u(s)∥L22 ds=∥u0∥L22\|u(t)\|_{L^2}^2 + 2\nu\int_0^t \|\nabla u(s)\|_{L^2}^2\,ds = \|u_0\|_{L^2}^2∥u(t)∥L22+2ν∫0t∥∇u(s)∥L22ds=∥u0∥L22
for all t≥0t\ge0t≥0. Hence the L2L^2L2 norm stays bounded for all time and the dissipation term controls ∇u\nabla u∇u in L2L^2L2 in time average. Combined with smoothing, one rules out blowup in the extended algebra and shows global control of standard energy norms.

Use of premises: Passing the energy identity from approximants to the limit uses (P1) linearity, (P2) interchange of limit and integral and inner products, and (P3) algebraic preservation of the cancellation of the nonlinear term.

===== Uniqueness is standard: take two solutions uuu, vvv in the extended algebra satisfying mild equation; consider w=u−vw=u-vw=u−v. One obtains an equation =====

w(t)=−∫0tGt−sP∇ ⁣⋅(u⊗w+w⊗v)(s) ds.w(t) = -\int_0^t G_{t-s}P\nabla\!\cdot\big( u\otimes w + w\otimes v \big)(s)\,ds.w(t)=−∫0tGt−sP∇⋅(u⊗w+w⊗v)(s)ds.
Take an appropriate norm (e.g. L2L^2L2) and estimate; standard Grönwall or contraction arguments show w≡0w\equiv0w≡0. All steps are valid in the extended algebra because of (P2),(P3) allowing multiplication and integration and because the heat kernel provides required smoothing/estimates. Hence uniqueness holds.

Use of premises: multiplicative algebra (P3) plus linearity of integrations/estimates via (P2).