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

We work in the setting of The Flat Torus: Standing Notation, used here with a natural number $n$ satisfying $1\le n$ and $n\le3$; the initial segments $[n]$, Euclidean space $\mathbb{R}^{n}$ with its norm $\lVert\,\cdot\,\rVert$, the integer lattice $\mathbb{Z}^{n}$, the cell $Q$, the classes $\mathcal{L}^{t}(\mathbb{T}^{n})$ and spaces $L^{t}(\mathbb{T}^{n})$ with the class map $[\,\cdot\,]$ for a real number $t$ with $1\le t$, and the restriction $v|_{Q}$ are the ones fixed there. Let $\mathbb{N}$ be the set of natural numbers, let $\pi$ be the real number of The Number Pi (i) (the wrapping map of The Flat Torus: Standing Notation (ii) is not used here), let $2=1+1$ and $4=2+2$, and for a real number $s$ write $s^{2}=ss$. A representative of a class $U\in L^{t}(\mathbb{T}^{n})$ is a member $u$ of $\mathcal{L}^{t}(\mathbb{T}^{n})$ with $[u]=U$. Let $H^{1}(\mathbb{T}^{n})$, with its inner product $\langle\,\cdot\,,\cdot\,\rangle_{H^{1}}$ and norm $\lVert\,\cdot\,\rVert_{H^{1}}$, be the Sobolev space fixed there, and let $\langle\,\cdot\,,\cdot\,\rangle_{L^{2}}$, $\lVert\,\cdot\,\rVert_{L^{2}}$ and $d_{L^{2}}$ be the inner product, norm and distance of $L^{2}(\mathbb{T}^{n})$. Let $E_{k}$ for $k\in\mathbb{Z}^{n}$ be the classes of The Trigonometric System on the Torus is Orthonormal (i), which lie in $H^{1}(\mathbb{T}^{n})$ by The Trigonometric System is Continuously Differentiable and Orthogonal in the Sobolev Space of the Torus (ii). Finite sums $\sum_{j=1}^{N}$ are those of Finite Sum Notation in a Field.

We work also in the setting of Hilbert Triples: Standing Notation and Background, used here with the Hilbert triple $(H,V,A)$ taken to be the one of The Square-Integrable and Sobolev Spaces of the Torus Form a Hilbert Triple (i), that is, with $H=L^{2}(\mathbb{T}^{n})$, with $V=H^{1}(\mathbb{T}^{n})$ carrying $\langle\,\cdot\,,\cdot\,\rangle_{H^{1}}$, and with $A$ the form operator determined by these data; its standing hypothesis Hilbert Triples: Standing Notation and Background (ii) holds for this triple by that clause. Accordingly $\langle\,\cdot\,,\cdot\,\rangle_{H}$ is $\langle\,\cdot\,,\cdot\,\rangle_{L^{2}}$ and $|\cdot|_{H}$ is $\lVert\,\cdot\,\rVert_{L^{2}}$, $|\cdot|_{V}$ is $\lVert\,\cdot\,\rVert_{H^{1}}$, and $D(A)\subseteq H^{1}(\mathbb{T}^{n})$ is the domain of the form operator, as fixed in Hilbert Triples: Standing Notation and Background (iii). The set $L^{2}(\mathbb{T}^{n})$ is open in itself by Hilbert Triples: Standing Notation and Background (v), so that the trace of $D(A)$ on it is $D(A)$ itself; $\mathrm{Sym}(H^{1}(\mathbb{T}^{n}))$ denotes the set of bounded symmetric bilinear forms on $H^{1}(\mathbb{T}^{n})$ fixed in Hilbert Triples: Standing Notation and Background (vi), and $0_{V}$ the zero vector of $H^{1}(\mathbb{T}^{n})$. Square-summability of a sequence in $V$, the sum $\sigma(f)$ and the trace $\mathrm{Tr}_{f}$ are those of Square-Summable Sequences in the Form Space of a Hilbert Triple and the Trace of a Form along Them.

Let $b,\kappa,\lambda_{0},C_{g}\in\mathbb{R}$ satisfy $0\le b$, $0<\lambda_{0}$ and $0\le C_{g}$, let $\omega_{g}$ be a modulus of continuity, and let $g:H^{1}(\mathbb{T}^{n})\to\mathbb{R}$ satisfy

$$|g(X)|\le C_{g}\quad\text{for every }X\in H^{1}(\mathbb{T}^{n}),\qquad |g(X)-g(Y)|\le\omega_{g}\bigl(\lVert X-Y\rVert_{H^{1}}\bigr)\quad\text{for all }X,Y\in H^{1}(\mathbb{T}^{n}).$$

Let $B:H^{1}(\mathbb{T}^{n})\to L^{2}(\mathbb{T}^{n})$ be the cube map fixed there for this $b$, whose value at $X$ is $B(X)=b\,[x^{3}]$ for $x$ any representative of $X$, and let $L:L^{2}(\mathbb{T}^{n})\to L^{2}(\mathbb{T}^{n})$ be the map whose value at $X$ is $L(X)=-\kappa X$.

Let $\theta,\nu\in\mathbb{R}$ satisfy $0\le\theta\le1$ and $0\le\nu$. Let $N\in\mathbb{N}$ and let $c:[N]\to\mathbb{R}$ and $w:[N]\to\mathbb{Z}^{n}$ be maps, with values written $c_{j}$ and $w_{j}$. Let $f=(f_{j})_{j\in\mathbb{N}}$ be the sequence in $H^{1}(\mathbb{T}^{n})$ with

$$f_{j}=c_{j}\,E_{w_{j}}\ \ \text{for }j\in[N],\qquad f_{j}=0_{V}\ \ \text{for }j\in\mathbb{N}\text{ with }N<j,$$

which is well defined because $H^{1}(\mathbb{T}^{n})$ is a linear subspace of $L^{2}(\mathbb{T}^{n})$ by The Sobolev Space of Once Weakly Differentiable Square-Integrable Classes on the Torus (i) and $[N]$ is the set of $j\in\mathbb{N}$ with $j\le N$, so that exactly one of $j\le N$ and $N<j$ holds for each $j$, by claim 3 of Properties of the Order on the Natural Numbers. Then the following hold.

1. (The noise)(i) The sequence $f$ is square-summable in $V$, with

$$\sigma(f)=\sum_{j=1}^{N}c_{j}^{2}\bigl(1+4\pi^{2}\lVert w_{j}\rVert^{2}\bigr),$$

and for every $Y\in\mathrm{Sym}(H^{1}(\mathbb{T}^{n}))$

$$\mathrm{Tr}_{f}Y=\sum_{j=1}^{N}c_{j}^{2}\,Y\bigl(E_{w_{j}},E_{w_{j}}\bigr).$$

2. (The data)(ii) $L^{2}(\mathbb{T}^{n})$ is not finite-dimensional as a vector space over $\mathbb{R}$, by The Lebesgue Space of Square-Integrable Functions on the Torus is Not Finite-Dimensional; $B$ is a monotone nonlinearity for $(H,V,A)$ by The Cube Map is a Monotone Nonlinearity on the Sobolev Hilbert Triple of the Torus (v); and $L$ is Lipschitz with constant $|\kappa|$ from $(L^{2}(\mathbb{T}^{n}),d_{L^{2}})$ to itself, the number $|\kappa|$ being nonnegative. Consequently the function $F$ on $D(A)\times\mathbb{R}\times L^{2}(\mathbb{T}^{n})\times\mathrm{Sym}(H^{1}(\mathbb{T}^{n}))$ whose value at $(X,r,p,Y)$ is

$$F(X,r,p,Y)=\lambda_{0}\,r-\tfrac{\nu}{2}\,\mathrm{Tr}_{f}Y+\tfrac{\theta}{2}\bigl(\lVert p\rVert_{L^{2}}\bigr)^{2}+\bigl\langle AX+B(X)+L(X),\,p\bigr\rangle_{L^{2}}-g(X)$$

is defined, and the hypotheses of Well-Posedness for a Viscous Hamilton-Jacobi Equation with a Monotone Nonlinearity on a Hilbert Triple are satisfied by the present data, with $\ell=|\kappa|$ there.

3. (Well-posedness)(iii) There is 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

$$|u(X)|\le\frac{C_{g}}{\lambda_{0}}\qquad\text{for every }X\in L^{2}(\mathbb{T}^{n}),$$

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); the quotient above is the quotient of $C_{g}$ by the nonzero $\lambda_{0}$, and is nonnegative. Moreover, if $u'$ is a viscosity solution of $F$ on $L^{2}(\mathbb{T}^{n})$ that is continuous on $L^{2}(\mathbb{T}^{n})$ and for which some $C''\in\mathbb{R}$ satisfies $|u'(X)|\le C''$ for every $X\in L^{2}(\mathbb{T}^{n})$, then $u'(X)=u(X)$ for every $X\in L^{2}(\mathbb{T}^{n})$.