I'm considering an equation in Sobolev spaces and stuck at a dissipation term. After constructing my desired Sobolev norm $W^{s,q}$, on the left hand side of the equation, I have $$\Vert \nabla (\Lambda^s u)^{q/2} \Vert_{L^{2}(\mathbb R^2)}^{2}$$ from the dissipation, while I want to use this term to absorb the term $$\Vert \nabla (\Lambda^s u)\Vert_{L^{2}(\mathbb R^2)}^{2}$$ on the RHS, where $\Lambda^s$ is the fractional derivative with $s>1$. It works when $q=2$, i.e., in $H^s$, but I don't know how to extend this $q$ to any $q>2$, i.e., to consider in $W^{s,q}$.
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$\begingroup$ I recommend adding the equation to the question. $\endgroup$ – Tommi Brander Dec 31 '18 at 7:41