**Proof plan.** The partial [Fourier transform](/page/Fourier%20Transform) in $x$ decouples the equation into modes $\hat{\theta}(t,k,y) = e^{-ikv(y)t}\hat{\theta}_{\mathrm{in}}(k,y)$ (Step 1). The $\dot{H}^{-1}(\mathbb{T} \times (0,1))$ norm decomposes via Parseval in $x$ into per-mode dual norms $\|\hat{\theta}(t,k,\cdot)\|_{(H^1_k)^*}^2$ (Step 2). For each $k \neq 0$, the dual pairing against a [test function](/page/Test%20Function) $g \in H^1_k$ produces an oscillatory [integral](/page/Integral) with phase $v(y)$ and parameter $kt$. The one-dimensional Sobolev embedding $H^1((0,1)) \hookrightarrow C^0([0,1])$ ensures the amplitude is continuous, and a [partition of unity](/page/Partition%20of%20Unity) on the compact interval $[0,1]$ subordinate to the non-degeneracy condition localises the integral to regions where a [derivative](/page/Derivative) of $v$ is bounded below (Step 3). The [Van der Corput lemma](/theorems/637) then gives $|kt|^{-1/m}$ decay per piece (Claim 1, Step 4). Taking the minimum with the trivial bound and summing over $k$ using $\langle kt \rangle \geq \langle t \rangle$ for $|k| \geq 1$ yields the global estimate (Claim 2, Step 5). **Step 1 (Partial Fourier transform and mode decoupling).** Taking the Fourier transform of $\partial_t \theta + v(y)\partial_x \theta = 0$ in $x \in \mathbb{T}$ gives, for each $k \in \mathbb{Z}$: \begin{align*} \partial_t \hat{\theta}(t, k, y) + ikv(y)\,\hat{\theta}(t, k, y) = 0, \end{align*} an ODE in $t$ for each fixed $(k,y)$ with explicit solution \begin{align*} \hat{\theta}(t, k, y) = e^{-ikv(y)t}\,\hat{\theta}_{\mathrm{in}}(k, y). \end{align*} The $k = 0$ mode satisfies $\hat{\theta}(t, 0, y) = \hat{\theta}_{\mathrm{in}}(0, y) = 0$ by the zero $x$-average assumption and is excluded throughout. **Step 2 (Parseval decomposition of the norms).** The Parseval decomposition $\|\psi\|_{\dot{H}^1}^2 = \sum_{k \neq 0}\|\hat{\psi}(k,\cdot)\|_{H^1_k}^2$ and $\|\psi\|_{\dot{H}^{-1}}^2 = \sum_{k \neq 0}\|\hat{\psi}(k,\cdot)\|_{(H^1_k)^*}^2$ follows from orthogonality of the $x$-Fourier modes in $L^2(\mathbb{T})$, exactly as stated in the definitions. The $\dot{H}^{-1}$ decomposition uses the Cauchy–Schwarz decoupling argument: $\|\psi\|_{\dot{H}^{-1}} = \sup_{\|\phi\|_{\dot{H}^1} \leq 1}|\langle \psi, \phi\rangle_{L^2}|$, Parseval in $x$ gives $\langle \psi, \phi\rangle_{L^2} = \sum_{k \neq 0}\langle \hat{\psi}(k,\cdot), \hat{\phi}(k,\cdot)\rangle_{L^2_y}$, and optimising $\hat{\phi}(k,\cdot) = \lambda_k g_k$ with $\|g_k\|_{H^1_k} = 1$ subject to $\sum \lambda_k^2 \leq 1$ decouples the supremum mode by mode via Cauchy–Schwarz on $\ell^2$. **Step 3 (Partition of unity on the compact interval $[0,1]$).** The non-degeneracy condition $\sum_{j=1}^m |v^{(j)}(y)| > 0$ on $(0,1)$ and the [continuity](/page/Continuity) of $v \in C^m([0,1])$ give the following. For each $y_0 \in [0,1]$, there exists $j(y_0) \in \{1, \ldots, m\}$ such that $|v^{(j(y_0))}(y_0)| > 0$. By continuity, there exists an open interval $U_{y_0} \ni y_0$ (open in $\mathbb{R}$) and a constant $c_{y_0} > 0$ such that $|v^{(j(y_0))}(y)| \geq c_{y_0}$ for all $y \in U_{y_0} \cap [0,1]$. The collection $\{U_{y_0}\}_{y_0 \in [0,1]}$ is an open cover of the compact [set](/page/Set) $[0,1]$. By the Heine–Borel theorem, there exists a finite subcover $\{I_\ell\}_{\ell=1}^L$ with associated indices $j_\ell \in \{1, \ldots, m\}$ and constants $c_\ell > 0$ satisfying $|v^{(j_\ell)}(y)| \geq c_\ell$ for all $y \in I_\ell \cap [0,1]$. Let $\{\chi_\ell\}_{\ell=1}^L$ be a smooth partition of unity on $[0,1]$ subordinate to $\{I_\ell\}$: each $\chi_\ell \in C^\infty_c(I_\ell)$, $0 \leq \chi_\ell \leq 1$, and $\sum_{\ell=1}^L \chi_\ell(y) = 1$ for all $y \in [0,1]$. **Step 4 (Per-mode oscillatory estimate).** [claim:Per Mode Decay Estimate] For each $k \in \mathbb{Z} \setminus \{0\}$ and each $t > 0$: \begin{align*} \|\hat{\theta}(t, k, \cdot)\|_{(H^1_k)^*} \leq C\,\min\!\left(1,\; |kt|^{-1/m}\right)\|\hat{\theta}_{\mathrm{in}}(k, \cdot)\|_{H^1_k}, \end{align*} where $C > 0$ depends only on $v$ and $m$. [/claim] [proof] Let $g: (0,1) \to \mathbb{C}$ with $\|g\|_{H^1_k} \leq 1$. The dual pairing is \begin{align*} \int_0^1 \hat{\theta}(t,k,y)\,\overline{g(y)}\,d\mathcal{L}^1(y) = \int_0^1 e^{-ikv(y)t}\,\hat{\theta}_{\mathrm{in}}(k,y)\,\overline{g(y)}\,d\mathcal{L}^1(y). \end{align*} **Trivial bound.** Since $|e^{-ikv(y)t}| = 1$ and $|k| \geq 1$ implies $\|g\|_{L^2} \leq |k|^{-1}\|g\|_{H^1_k} \leq 1$: \begin{align*} \left|\int_0^1 e^{-ikv(y)t}\,\hat{\theta}_{\mathrm{in}}(k,y)\,\overline{g(y)}\,d\mathcal{L}^1(y)\right| \leq \|\hat{\theta}_{\mathrm{in}}(k,\cdot)\|_{L^2}\,\|g\|_{L^2} \leq \|\hat{\theta}_{\mathrm{in}}(k,\cdot)\|_{H^1_k}. \end{align*} **Oscillatory bound.** Since $|k| \geq 1$, we have $\|g\|_{H^1((0,1))}^2 = \|g\|_{L^2}^2 + \|g'\|_{L^2}^2 \leq k^2\|g\|_{L^2}^2 + \|g'\|_{L^2}^2 = \|g\|_{H^1_k}^2 \leq 1$. By the one-dimensional Sobolev embedding $H^1((0,1)) \hookrightarrow C^0([0,1])$, the [function](/page/Function) $g$ is continuous and bounded on $[0,1]$. Similarly, $\hat{\theta}_{\mathrm{in}}(k,\cdot) \in H^1_k$ implies $\hat{\theta}_{\mathrm{in}}(k,\cdot) \in H^1((0,1)) \hookrightarrow C^0([0,1])$. Define the amplitude $a(y) := \hat{\theta}_{\mathrm{in}}(k,y)\,\overline{g(y)}$, which belongs to $H^1((0,1)) \hookrightarrow C^0([0,1])$. Using the partition of unity from Step 3: \begin{align*} \int_0^1 e^{-ikv(y)t}\,a(y)\,d\mathcal{L}^1(y) = \sum_{\ell=1}^L \int_0^1 e^{-ikv(y)t}\,\chi_\ell(y)\,a(y)\,d\mathcal{L}^1(y). \end{align*} On each piece, the amplitude $\chi_\ell \cdot a$ is compactly supported inside $I_\ell \cap [0,1]$ and belongs to $C^0$. To apply the [Van der Corput lemma](/theorems/637), which requires a $C^1$ amplitude, we approximate: for each $\delta > 0$, let $a_\delta \in C^1_c(I_\ell)$ with $\|\chi_\ell a - a_\delta\|_{L^1} < \delta$ and $\|a_\delta'\|_{L^1} \leq C\|(\chi_\ell a)'\|_{L^1}$ (by density of $C^1_c$ in $W^{1,1}$, which holds since $\chi_\ell a \in H^1 \subset W^{1,1}$ and has compact support). On $I_\ell$, the phase $v$ satisfies $|v^{(j_\ell)}(y)| \geq c_\ell > 0$ with $j_\ell \leq m$. The [Van der Corput lemma](/theorems/637) with exponent $j_\ell$ gives \begin{align*} \left|\int_{I_\ell} e^{-ikv(y)t}\,a_\delta(y)\,d\mathcal{L}^1(y)\right| \leq c_{j_\ell}\,|kt|^{-1/j_\ell}\,\|a_\delta'\|_{L^1(I_\ell)}. \end{align*} Passing $\delta \to 0$ (the left-hand side converges since $a_\delta \to \chi_\ell a$ in $L^1$, and the right-hand side is bounded by $c_{j_\ell}|kt|^{-1/j_\ell}C\|(\chi_\ell a)'\|_{L^1}$): \begin{align*} \left|\int_{I_\ell} e^{-ikv(y)t}\,\chi_\ell(y)\,a(y)\,d\mathcal{L}^1(y)\right| \leq c_{j_\ell}\,|kt|^{-1/j_\ell}\,C\,\|(\chi_\ell a)'\|_{L^1(I_\ell)}. \end{align*} Since $j_\ell \leq m$ and $|kt| \geq 1$, we have $|kt|^{-1/j_\ell} \leq |kt|^{-1/m}$. By the product rule: \begin{align*} \|(\chi_\ell a)'\|_{L^1} \leq \|\chi_\ell'\|_{L^\infty}\|a\|_{L^1} + \|\chi_\ell\|_{L^\infty}\|a'\|_{L^1}. \end{align*} For the $L^1$ norms: since $(0,1)$ has finite measure, $\|a\|_{L^1} \leq \|a\|_{L^2} \leq \|\hat{\theta}_{\mathrm{in}}(k,\cdot)\|_{L^\infty}\|g\|_{L^2} + \|\hat{\theta}_{\mathrm{in}}(k,\cdot)\|_{L^2}\|g\|_{L^\infty}$, and the $L^\infty$ factors are controlled by the Sobolev embedding $H^1((0,1)) \hookrightarrow L^\infty((0,1))$. Similarly for $\|a'\|_{L^1}$ using the product rule $a' = \hat{\theta}_{\mathrm{in}}'\,\overline{g} + \hat{\theta}_{\mathrm{in}}\,\overline{g'}$. Combining: \begin{align*} \|(\chi_\ell a)'\|_{L^1} \leq C_\ell\,\|\hat{\theta}_{\mathrm{in}}(k,\cdot)\|_{H^1((0,1))}\,\|g\|_{H^1((0,1))}, \end{align*} where $C_\ell$ depends on $\chi_\ell$ and the Sobolev embedding constant. Since $\|\hat{\theta}_{\mathrm{in}}(k,\cdot)\|_{H^1((0,1))} \leq \|\hat{\theta}_{\mathrm{in}}(k,\cdot)\|_{H^1_k}$ and $\|g\|_{H^1((0,1))} \leq \|g\|_{H^1_k} \leq 1$ (both because $|k| \geq 1$), summing over $\ell = 1, \ldots, L$: \begin{align*} \left|\int_0^1 e^{-ikv(y)t}\,a(y)\,d\mathcal{L}^1(y)\right| \leq C\,|kt|^{-1/m}\,\|\hat{\theta}_{\mathrm{in}}(k,\cdot)\|_{H^1_k}, \end{align*} where $C = \sum_{\ell=1}^L c_{j_\ell}\,C\,C_\ell$ depends only on $v$ and $m$ (through the partition of unity, the non-degeneracy constants $c_\ell$, and the Sobolev embedding). Taking the supremum over $\|g\|_{H^1_k} \leq 1$ and combining with the trivial bound gives the claimed per-mode estimate. [/proof] **Step 5 (Global estimate).** [claim:Global Mixing Estimate] \begin{align*} \|\theta(t)\|_{\dot{H}^{-1}(\mathbb{T} \times (0,1))} \leq C\,\langle t \rangle^{-1/m}\,\|\theta_{\mathrm{in}}\|_{\dot{H}^1(\mathbb{T} \times (0,1))}. \end{align*} [/claim] [proof] By Claim 1, for each $k \in \mathbb{Z} \setminus \{0\}$: \begin{align*} \|\hat{\theta}(t,k,\cdot)\|_{(H^1_k)^*} &\leq C\min(1, |kt|^{-1/m})\,\|\hat{\theta}_{\mathrm{in}}(k,\cdot)\|_{H^1_k} \leq C'\langle kt \rangle^{-1/m}\,\|\hat{\theta}_{\mathrm{in}}(k,\cdot)\|_{H^1_k}, \end{align*} where $C'$ depends on $C$ and $m$ (the interpolation $\min(1, s^{-1/m}) \leq 2^{1/m}\langle s \rangle^{-1/m}$ for $s \geq 0$ holds because $\langle s \rangle^{1/m} \leq 2^{1/m}\max(1, s^{1/m})$). Since $|k| \geq 1$ for all $k \in \mathbb{Z} \setminus \{0\}$: \begin{align*} \langle kt \rangle^2 = 1 + k^2t^2 \geq 1 + t^2 = \langle t \rangle^2, \end{align*} so $\langle kt \rangle^{-1/m} \leq \langle t \rangle^{-1/m}$. Squaring and summing over $k$ using the Parseval decomposition (Step 2): \begin{align*} \|\theta(t)\|_{\dot{H}^{-1}(\mathbb{T} \times (0,1))}^2 &= \sum_{k \neq 0}\|\hat{\theta}(t,k,\cdot)\|_{(H^1_k)^*}^2 \\ &\leq (C')^2\,\langle t \rangle^{-2/m}\sum_{k \neq 0}\|\hat{\theta}_{\mathrm{in}}(k,\cdot)\|_{H^1_k}^2 \\ &= (C')^2\,\langle t \rangle^{-2/m}\,\|\theta_{\mathrm{in}}\|_{\dot{H}^1(\mathbb{T} \times (0,1))}^2. \end{align*} Taking square roots: $\|\theta(t)\|_{\dot{H}^{-1}} \leq C'\langle t \rangle^{-1/m}\|\theta_{\mathrm{in}}\|_{\dot{H}^1}$. [/proof] [remark:Couette Flow On The Unbounded Strip] For the Couette profile $v(y) = y$ on the **unbounded** domain $\mathbb{T} \times \mathbb{R}$, a sharper and more explicit estimate is obtained by a direct Fourier computation that avoids Van der Corput entirely. The full Fourier transform in both $x$ and $y$ converts the solution into a frequency shift: $\hat{\theta}(t, k, \eta) = \hat{\theta}_{\mathrm{in}}(k, \eta + kt)$ where $(k,\eta) \in \mathbb{Z} \times \mathbb{R}$. In this setting the Sobolev norms use the weight $(k^2 + \eta^2)^s$ and the pointwise bound \begin{align*} \frac{1}{(1 + (\eta/k)^2)(1 + (t - \eta/k)^2)} \leq \frac{4}{1 + t^2} \end{align*} (proved by a three-case analysis: $|\eta/k| \geq |t|/2$, $|t - \eta/k| \geq |t|/2$, or both — the third case being empty by the triangle inequality) yields \begin{align*} \|\theta(t)\|_{\dot{H}^{-1}(\mathbb{T} \times \mathbb{R})} \leq 2\,\langle t \rangle^{-1}\,\|\theta_{\mathrm{in}}\|_{\dot{H}^1(\mathbb{T} \times \mathbb{R})}. \end{align*} The explicit constant $C = 2$ and the optimal decay rate $m = 1$ reflect the fact that $v'(y) = 1 \neq 0$ everywhere. More generally, for any $s > 0$, the same method gives $\|\theta(t)\|_{\dot{H}^{-s}} \lesssim \langle t \rangle^{-s}\|\theta_{\mathrm{in}}\|_{\dot{H}^s}$. [/remark]