$L^2$ norm for solutions of evolution equations driven by different elliptic operators

Question

Let $u$ be a solution of the heat equation $$u_t - \Delta u = 0, \qquad t >0, \ x \in \mathbb T^d$$ and $v$ be a solution of the bi-harmonic heat equation $$v_t +\Delta^2 v = 0, \qquad t >0, \ x \in \mathbb T^d$$ with the same initial data $f$. Is it true that, for every fixed time $T >0$, $$\|v(T,\cdot)\|_{L^2} \le \|u(T,\cdot)\|_{L^2}$$ holds? That is, in some sense, $v$ is "more dissipated"?

Answer 1

Not necessarily. I mean, it depends upon the torus you consider. Notice that in the case of the standard one ${\mathbb T}^d={\mathbb R}^d/{\mathbb Z}^d$, the answer is positive. But if you torus is ${\mathbb R}^d/a{\mathbb Z}^d$, then it is positive if $a\le2\pi$ and negative otherwise. The reason is that both semi-groups $H_t$ and $B_t$ are co-diagonal with an orthogonal eigenbasis, but they have eigenvalues $e^{-t\mu}$ and $e^{-t\mu^2}$, where $\mu_0=0<\mu_1,\ldots$ are the eigenvalues of $-\Delta$. To have the required inequality is equivalent to having $\mu^2\ge\mu$ for every eigenvalue, that is $\mu_1\ge1$. Whence the condition on the size of the torus.

More generally, if ${\mathbb T}^d={\mathbb R}^d/\Lambda$ where $\Lambda$ is a lattice, the inequality amounts to the fact that every non-zero element of the dual lattice $\Lambda^*$ has norm $\ge\frac1{2\pi}$. Here $\Lambda^*$ is the set of points $\alpha\in{\mathbb R}^d$ such that $\langle \alpha, p\rangle\in{\mathbb Z}$ for every $p\in\Lambda$.

Edit. Here are the details. Let me consider a general torus $T={\mathbb R}^d/\Lambda$ where $\Lambda$ is a lattice. Both the Laplacian and its square are diagonal in the Fourier basis of exponentials $$\phi_\alpha(x):=c_\alpha\exp(2i\pi\alpha\cdot x),\qquad \alpha\in\Lambda^*,$$ where the constant $c_\alpha$ normalizes: $\|\phi_\alpha\|_{L^2}=1$.

The corresponding eigenvalues are $4\pi^2|\alpha|^2$, respectively $(4\pi^2|\alpha|^2)^2$. Decomposing an arbitrary data $$u(0,\cdot)=\sum_{\alpha\in\Lambda^*}a_\alpha\phi_\alpha,$$ we have $$u(T,\cdot)=\sum_{\alpha\in\Lambda^*}e^{4\pi^2|\alpha|^2T}a_\alpha\phi_\alpha,\qquad v(T,\cdot)=\sum_{\alpha\in\Lambda^*}e^{(4\pi^2|\alpha|^2)^2T}a_\alpha\phi_\alpha.$$ Using the fact that the Fourier basis is $L^2$-orthogonal, we find $$\|u(T)\|_{L^2}^2=\sum_{\alpha\in\Lambda^*}e^{8\pi^2|\alpha|^2T}|a_\alpha|^2,\qquad \|v(T)\|_{L^2}^2=\sum_{\alpha\in\Lambda^*}e^{32(\pi^2|\alpha|^2)^2T}|a_\alpha|^2.$$

The required inequality is equivalent to $$e^{32(\pi^2|\alpha|^2)^2T}\le e^{8\pi^2|\alpha|^2T}$$ for $T>0$ and every $\alpha$, that is to $2\pi|\alpha|\ge1$ for every non-zero element of $\Lambda^*$.