Consider the equation $$u' + Au = f$$ $$u|_{\partial \Omega} = 0$$ $$u(0) = u_0$$ where $A:L^p(0,T;W^{1,p}_0) \to L^q(0,T;W^{-1,q})$ is some monotone nonlinear operator (with additional assumptions). Using the Galerkin method, one can show $u \in L^p(0,T;W_0^{1,p})$ with $u' \in L^q(0,T;W^{-1,q})$ .

Now I want to show that $u' \in L^\infty(0,T;W^{1,p})$ under additional regularity of the data $f$ and $u_0$.

If $A$ were linear, I could simply take the basis functions of the Galerkin expansion to be the eigenfunctions of $A$ and test the Galerkin approximation with $Au_n'$ which is a valid test function. This will lead me to a uniform bound on $Au_n$.

Alternatively one could differentiate the PDE to find the PDE that $v=u'$ satisfies and then one shows rigourously that $v$ has to be $u'$.

How does this work in a nonlinear setting? Can someone recommend me a text? Thank you.

Consider the equation $$u' + Au = f$$ $$u|_{\partial \Omega} = 0$$ $$u(0) = u_0$$ where $A:L^p(0,T;W^{1,p}_0) \to L^q(0,T;W^{-1,q})$ is some monotone nonlinear operator (with additional assumptions). Using the Galerkin method, one can show $u \in L^p(0,T;W_0^{1,p})$ with $u' \in L^q(0,T;W^{-1,q})$ .

Now I want an bound on $\lVert Au \rVert_{L^\infty(0,T;L^2)}$ depending on the data.

If $A$ were the Laplacian, I could simply take the basis functions of the Galerkin expansion to be the eigenfunctions of $A$ and test the Galerkin approximation with $Au_n'$ which is a valid test function. This will lead me to a uniform bound on $Au_n$.

How does this work in a nonlinear setting? Can someone recommend me a text? Thank you.