Well-Posedness of the 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 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 $1\le t$, the periodic classes $C_{\mathrm{per}}$ and $C^{2}_{\mathrm{per}}$, and the restriction $v|_{Q}$ are the ones fixed there; the Laplacian $\Delta v$ of a map $v$ of class $C^{2}$ on $\mathbb{R}^{n}$ is that of The Laplacian of a Twice Continuously Differentiable Function (i), the set $\mathbb{R}^{n}$ being open in itself by claim 1 of Euclidean Space Is Open in Itself. A representative of a class $U\in L^{t}(\mathbb{T}^{n})$ is a member $u$ of $\mathcal{L}^{t}(\mathbb{T}^{n})$ with $[u]=U$. Weak partial derivatives of classes are those of The Weak Partial Derivative on the Torus, and $H^{1}(\mathbb{T}^{n})$, with its inner product $\langle\,\cdot\,,\cdot\,\rangle_{H^{1}}$ and norm $\lVert\,\cdot\,\rVert_{H^{1}}$, is the Sobolev space fixed there; $\langle\,\cdot\,,\cdot\,\rangle_{L^{2}}$, $\lVert\,\cdot\,\rVert_{L^{2}}$ and $d_{L^{2}}$ are the inner product, norm and distance of $L^{2}(\mathbb{T}^{n})$. For a map $v$ defined on $Q$ or on $\mathbb{R}^{n}$, the pointwise cube $v^{3}$ is the map with the same domain whose value at $y$ is the third power of $v(y)$; pointwise sums, differences and real multiples of maps with a common domain are formed valuewise. The metric $d_{\mathbb{R}}$ on $\mathbb{R}$ is the one fixed in The Real Numbers: Standing Notation and Background (i).

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; and $\mathrm{Sym}(H^{1}(\mathbb{T}^{n}))$ denotes the set of bounded symmetric bilinear forms on $H^{1}(\mathbb{T}^{n})$ fixed in Hilbert Triples: Standing Notation and Background (vi).

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 $L(X)=-\kappa X$. Then the following hold.

1. (The data)(i) $B$ 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

$$F(X,r,p,Y)=\lambda_{0}\,r+\tfrac{1}{2}\bigl(\lVert p\rVert_{L^{2}}\bigr)^{2}+\bigl\langle AX+B(X)+L(X),\,p\bigr\rangle_{L^{2}}-g(X)$$

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

2. (Well-posedness)(ii) 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 $X\in L^{2}(\mathbb{T}^{n})$.

3. (The drift on twice continuously differentiable periodic functions)(iii) Let $v\in C^{2}_{\mathrm{per}}$ and set $X=[v|_{Q}]$. Then $X\in D(A)$, the map $-\Delta v+b\,v^{3}-(\kappa-1)\,v$ lies in $C_{\mathrm{per}}$ and its restriction to $Q$ lies in $\mathcal{L}^{2}(\mathbb{T}^{n})$, and

$$AX+B(X)+L(X)=\bigl[\bigl(-\Delta v+b\,v^{3}-(\kappa-1)\,v\bigr)\big|_{Q}\bigr].$$