Statement of the Problem
We wish to prove $L^p$-norm decay of a function $\theta \in C([0,\infty) ; L^1 \cap L^p(\mathbb{R}^2)) \cap C^1((0,\infty) ; L^1 \cap L^p(\mathbb{R}^2)) $, where $p \in [2,\infty).$
Skipping all the work, which I will explain more in the next section, we have reached the equation:
$$\frac{1}{p} \partial_t ||\theta(t)||_{L^p(\mathbb{R}^2)} + \int_{\mathbb{R}^2} |\theta(t)|^{p-2} \theta(t) (-\Delta)^{\alpha/2}\theta(t) \text{d}x = 0, $$
where $(-\Delta)^{\alpha/2}\theta = \mathcal{F}^{-1}[|\xi|^{2\alpha} \hat{\theta}]$ is the Fractional Laplacian, with $\alpha \in (0,2)$.
I now wish to obtain some inequality of the form:
$$ \frac{1}{p} \partial_t ||\theta(t)||_{L^p(\mathbb{R}^2)} + f(||\theta(t)||_{L^p(\mathbb{R}^2)}) \leq 0. $$
This was easy to do in the case $\alpha = 2$! We could just use basic properties of derivatives and integration by parts to show that:
$$ |\theta|^{p-2} \theta (-\Delta)^{\alpha/2}\theta = \frac{-4}{p^2}(p-1) |\nabla(|\theta|^{p/2})|^2. \ \ \ \ \ \ \text{(1)}$$
This gave us $ \frac{1}{p} \partial_t ||\theta(t)||_{L^p(\mathbb{R}^2)} + \frac{-4}{p}(p-1) ||\nabla(|\theta(t)|^{p/2})||_{L^2(\mathbb{R}^2)}^2 \leq 0. $
We then used the useful inequality found here (Lemma 1, page 11) which states:
$$ \partial_t||\theta(t)||_{L^p(\mathbb{R}^2)}^{\frac{p^2}{p-1}} \leq C||\theta(t)||_{L^1(\mathbb{R}^2)}^{\frac{p}{p-1}} ||\nabla(|\theta(t)|^{p/2})||_{L^2(\mathbb{R}^2)}^2. $$
By a previous result, the $L^1$-norm of $\theta(t)$ is bounded by a constant. Thus we get the inequality needed, and need only solve the simple ODE:
$$ \partial_t ||\theta(t)||_{L^p(\mathbb{R}^2)}^p + K||\theta(t)||_{L^p(\mathbb{R}^2)}^{\frac{p^2}{p-1}} \leq 0. $$
I now wish to obtain a similar result in the Fractional Laplacian case. However, it seems we do not have such nice properties of the Fractional Laplacian to be able to rewrite the integral term like we did in (1). Is there something similar to (1) that we can do here? Otherwise, is there some inequality on the integral term $\int_{\mathbb{R}^2} |\theta(t)|^{p-2} \theta(t) (-\Delta)^{\alpha/2}\theta(t) \text{d}x$ which we can use to get a similar result?
Background to This Problem
The function $\theta$ is constructed as a solution to the Quasi-Geostrophic Equation
$$ \begin{cases} \partial_t \theta + (-\Delta)^{\alpha/2} \theta + (u\cdot \theta) \theta = 0, \ \text{in} \ (0,\infty) \times \mathbb{R}^2, \\
\theta|_{t=0} = \theta_0 \in L^1( \mathbb{R}^2) \cap L^p( \mathbb{R}^2), \ p \in (2,\infty).
\end{cases} $$
where $\alpha \in (1, 2]$, and $u = (R_2\theta(t),-R_1\theta(t))$, where $R_i$ is the $i^{\text{th}}$ Riesz transform.
It is shown using Banach's Fixed Point Theorem that there exists a unique global-in-time solution
$$ \theta(t) = P_\alpha(t) \ast \theta_0 – \int^{t}_{0} P_\alpha(t-s) \ast (u \cdot \nabla)\theta(s) \text{d}s, $$
where $P_\alpha$ is the kernel generated by the Fractional Laplacian version of the Heat Equation.
The case $\alpha = 2$ is easy. We just adapt all arguments from the paper linked above to the QG Equation. In the more general case, we have so far proven that there exists a unique global solution, and that the $L^1$ and $L^p$-norms of $\theta(t)$ are bounded by the norms of $\theta_0$. Our next step is to show decay of the $L^p$-norm over time, which is described in the 'statement of the problem' above.
Best Answer
I have found an a suitable inequality for this problem, in this paper, Lemma 2.4:
Let $0 \leq \alpha \leq 2$, $x \in \mathbb{R}^2$, and $ \theta, \Lambda^\alpha \theta \in L^p $, with $p = 2^n$. Then:
$ \int_{\mathbb{R}^2} |\theta|^{p-2} \theta \Lambda^\alpha \theta \, \text{d}x \geq \frac{1}{p} \int_{\mathbb{R}^2} | \Lambda^{\alpha / 2} \theta^{p/2} |^2 \, \text{d}x $.
Where $\Lambda^{\alpha} = (-\Delta)^{\alpha / 2}$