# The use of dissipation in parabolic equations

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 $$W^{s,q}$$.

• I recommend adding the equation to the question. – Tommi Brander Dec 31 '18 at 7:41