Proof of Well-Posedness of the Viscous Allen-Cahn Hamilton-Jacobi Equation on the Torus

Each result cited is universally quantified over the data in its own statement, and is applied to the data named here. The field axioms of $\mathbb{R}$ are used freely for associativity, commutativity and distributivity, and for $s\cdot1=s$. By Finite Sum Notation in a Field a finite sum is determined by the map on the initial segment it is formed over, so two such sums agree as soon as the two maps agree; this is used without further comment.

Claim 1. (A series whose terms vanish beyond a fixed index.) Let $M\in\mathbb{N}$ and let $(a_{k})_{k\in\mathbb{N}}$ be a sequence of real numbers with $a_{k}=0$ for every $k\in\mathbb{N}$ with $M<k$. Then the series $\sum_{k=1}^{\infty}a_{k}$ converges, with sum $\sum_{k=1}^{M}a_{k}$.

Proof of Claim 1. For $m\in\mathbb{N}$ let $s_{m}=\sum_{k=1}^{m}a_{k}$ be the partial sums of the series. We first show that $s_{M+j}=s_{M}$ for every $j\in\mathbb{N}$, by Principle of Induction for the Natural Numbers applied to the set $T$ of natural numbers $j$ for which $s_{M+j}=s_{M}$.

For $j=1$ we have $M+1=S(M)$ by claim 1 of Arithmetic of Addition on the Natural Numbers, where $S$ is the successor map of Natural Numbers; by claim 1 of Properties of Finite Sums, $s_{S(M)}=s_{M}+a_{S(M)}$; and $M<S(M)$ by claim 5 of Properties of the Order on the Natural Numbers, so $a_{S(M)}=0$ and $s_{M+1}=s_{M}$. Thus $1\in T$.

Let $j\in T$. Then $M+S(j)=S(M+j)$ by Natural Numbers, and $s_{S(M+j)}=s_{M+j}+a_{S(M+j)}$ by claim 1 of Properties of Finite Sums. Moreover $M<M+j$ by claim 6 of Properties of the Order on the Natural Numbers and $M+j<S(M+j)$ by claim 5 of that lemma, so $M<S(M+j)$ by the transitivity of the strict order in claim 1 of that lemma; hence $a_{S(M+j)}=0$ and $s_{M+S(j)}=s_{M+j}=s_{M}$. Thus $S(j)\in T$, and $T=\mathbb{N}$.

Now let $\varepsilon$ be a positive real number and let $m\in\mathbb{N}$ satisfy $M\le m$. By the trichotomy in claim 3 of Properties of the Order on the Natural Numbers exactly one of $m<M$, $m=M$ and $M<m$ holds, and the first is impossible: $m<M$ would give $m\le M$ by claim 1 of that lemma, hence $m=M$ by the antisymmetry in claim 2 of that lemma together with $M\le m$, and then $m<m$, contradicting the irreflexivity in claim 2. So either $m=M$, or $M<m$ and then $m=M+j$ for some $j\in\mathbb{N}$ by claim 7 of that lemma; in both cases $s_{m}=s_{M}$ by the previous paragraph. Hence $|s_{m}-s_{M}|=|0|=0<\varepsilon$, using claim 1 of Properties of the Absolute Value in an Ordered Field. Taking $M$ itself as the index required by Limit of a Sequence of Real Numbers, the sequence $(s_{m})_{m\in\mathbb{N}}$ converges to $s_{M}$; so by Series of Real Numbers (ii) the series converges with sum $s_{M}=\sum_{k=1}^{M}a_{k}$. This proves Claim 1.

Claim 2. (The norms of the terms of the noise.) For $j\in[N]$ one has $|f_{j}|_{V}^{2}=c_{j}^{2}\bigl(1+4\pi^{2}\lVert w_{j}\rVert^{2}\bigr)$, and for $j\in\mathbb{N}$ with $N<j$ one has $|f_{j}|_{V}^{2}=0$.

Proof of Claim 2. The norm of $H^{1}(\mathbb{T}^{n})$ satisfies $\lVert X\rVert_{H^{1}}^{2}=\langle X,X\rangle_{H^{1}}$ by that clause, and is absolutely homogeneous by Elementary Identities in a Real Inner Product Space (iv), so for $j\in[N]$

$$|f_{j}|_{V}^{2}=\lVert c_{j}E_{w_{j}}\rVert_{H^{1}}^{2}=\bigl(|c_{j}|\,\lVert E_{w_{j}}\rVert_{H^{1}}\bigr)^{2}=|c_{j}|^{2}\,\langle E_{w_{j}},E_{w_{j}}\rangle_{H^{1}}=c_{j}^{2}\bigl(1+4\pi^{2}\lVert w_{j}\rVert^{2}\bigr),$$

using claim 1 of Nonnegativity of Squares in an Ordered Field for $|c_{j}|^{2}=c_{j}^{2}$ and The Trigonometric System is Continuously Differentiable and Orthogonal in the Sobolev Space of the Torus (iii), taken with $k=m=w_{j}$, for the last factor. For $j$ with $N<j$ we have $f_{j}=0_{V}$; since $0_{V}=0\cdot0_{V}$ and the inner product is linear in its first argument by Real Inner Product Spaces (i), $\langle 0_{V},0_{V}\rangle_{H^{1}}=0\cdot\langle 0_{V},0_{V}\rangle_{H^{1}}=0$ by claim 1 of Zero Products and Elementary Identities in a Field, so $|f_{j}|_{V}^{2}=0$. This proves Claim 2.

Proof of claim 1 of the statement. Apply Claim 1 with $M=N$ to the sequence $a_{k}=|f_{k}|_{V}^{2}$, whose terms vanish for $N<k$ by Claim 2. It gives that $\sum_{k=1}^{\infty}|f_{k}|_{V}^{2}$ converges with sum $\sum_{k=1}^{N}|f_{k}|_{V}^{2}$; so $f$ is square-summable in $V$ by Square-Summable Sequences in the Form Space of a Hilbert Triple and the Trace of a Form along Them (i), and $\sigma(f)$, being that sum, equals $\sum_{j=1}^{N}c_{j}^{2}(1+4\pi^{2}\lVert w_{j}\rVert^{2})$ by Claim 2.

Now let $Y\in\mathrm{Sym}(H^{1}(\mathbb{T}^{n}))$ and put $a_{k}=Y(f_{k},f_{k})$. By Bounded Symmetric Bilinear Forms on a Real Inner Product Space: Norm, Order, Identity Form and Restriction (i), which gives additivity and homogeneity in the first argument together with symmetry, a scalar may be taken out of either argument; so $a_{j}=Y(c_{j}E_{w_{j}},c_{j}E_{w_{j}})=c_{j}^{2}\,Y(E_{w_{j}},E_{w_{j}})$ for $j\in[N]$, and $a_{j}=Y(0_{V},0_{V})=0\cdot 0\cdot Y(0_{V},0_{V})=0$ for $N<j$, using $0_{V}=0\cdot0_{V}$ and claim 1 of Zero Products and Elementary Identities in a Field. Claim 1, applied with $M=N$ to this sequence, gives that $\sum_{k=1}^{\infty}Y(f_{k},f_{k})$ converges with sum $\sum_{j=1}^{N}c_{j}^{2}Y(E_{w_{j}},E_{w_{j}})$. By Square-Summable Sequences in the Form Space of a Hilbert Triple and the Trace of a Form along Them (ii) the real number $\mathrm{Tr}_{f}Y$ is the sum of that series, so it equals $\sum_{j=1}^{N}c_{j}^{2}Y(E_{w_{j}},E_{w_{j}})$.

Proof of claim 2 of the statement. That $L^{2}(\mathbb{T}^{n})$ is not finite-dimensional as a vector space over $\mathbb{R}$ is The Lebesgue Space of Square-Integrable Functions on the Torus is Not Finite-Dimensional, applied with the present $n$.

The data $b$, $\kappa$, $\lambda_{0}$, $C_{g}$, $\omega_{g}$, $g$, $B$ and $L$ here satisfy exactly the hypotheses imposed on them in Well-Posedness of the Allen-Cahn Hamilton-Jacobi Equation on the Torus, and the Hilbert triple, the settings and the readings of $\langle\,\cdot\,,\cdot\,\rangle_{L^{2}}$, $\lVert\,\cdot\,\rVert_{H^{1}}$, $D(A)$ and $\mathrm{Sym}(H^{1}(\mathbb{T}^{n}))$ are the ones adopted there. That $B$ is a monotone nonlinearity for $(H,V,A)$ is The Cube Map is a Monotone Nonlinearity on the Sobolev Hilbert Triple of the Torus (v), applied to the present $b$ and the present Hilbert triple; and Well-Posedness of the Allen-Cahn Hamilton-Jacobi Equation on the Torus (i) applies and gives that $L$ is Lipschitz with the nonnegative constant $|\kappa|$ from $(L^{2}(\mathbb{T}^{n}),d_{L^{2}})$ to itself.

We check the remaining hypotheses of Well-Posedness for a Viscous Hamilton-Jacobi Equation with a Monotone Nonlinearity on a Hilbert Triple, read for the Hilbert triple $(H,V,A)$ fixed above with $\ell=|\kappa|$. The standing hypothesis of Hilbert Triples: Standing Notation and Background holds for this triple, as recorded in the statement. The reals $\lambda_{0}$ and $C_{g}$ satisfy $0<\lambda_{0}$ and $0\le C_{g}$, $\omega_{g}$ is a modulus of continuity, and $g:V\to\mathbb{R}$ is bounded by $C_{g}$ and has modulus $\omega_{g}$ with respect to $|\cdot|_{V}=\lVert\,\cdot\,\rVert_{H^{1}}$, all by hypothesis. The reals $\theta$ and $\nu$ satisfy $0\le\theta\le1$ and $0\le\nu$ by hypothesis, and $f$ is square-summable in $V$ by claim 1 of the statement, so the trace $\mathrm{Tr}_{f}$ is defined by Square-Summable Sequences in the Form Space of a Hilbert Triple and the Trace of a Form along Them (ii). Every hypothesis of that theorem is therefore met, and in particular the displayed function $F$ is the function denoted $F$ there, hence is defined on $D(A)\times\mathbb{R}\times L^{2}(\mathbb{T}^{n})\times\mathrm{Sym}(H^{1}(\mathbb{T}^{n}))$.

Proof of claim 3 of the statement. By claim 2 of the statement the hypotheses of Well-Posedness for a Viscous Hamilton-Jacobi Equation with a Monotone Nonlinearity on a Hilbert Triple hold for the present data. Its Well-Posedness for a Viscous Hamilton-Jacobi Equation with a Monotone Nonlinearity on a Hilbert Triple (i) provides a function $u:L^{2}(\mathbb{T}^{n})\to\mathbb{R}$ that is a viscosity solution of $F$ on $L^{2}(\mathbb{T}^{n})$, satisfies $|u(X)|\le C$ for every $X$, where $C$ is the quotient of $C_{g}$ by $\lambda_{0}$ and is nonnegative, and is uniformly continuous on $L^{2}(\mathbb{T}^{n})$ with respect to $d_{L^{2}}$ and the metric of Real Hilbert Spaces: Standing Notation and Background (i). Its Well-Posedness for a Viscous Hamilton-Jacobi Equation with a Monotone Nonlinearity on a Hilbert Triple (iii), applied to this $u$, gives that any viscosity solution $u'$ of $F$ on $L^{2}(\mathbb{T}^{n})$ that is continuous on $L^{2}(\mathbb{T}^{n})$ and bounded by some $C''\in\mathbb{R}$ satisfies $u'(X)=u(X)$ for every $X\in L^{2}(\mathbb{T}^{n})$. These are the two assertions of the claim.