I am given the following function: fix any $p$ and $\beta \in (0,1)$ $$ u \in L^{p-\beta}(0,T;W^{1,p-\beta}(\Omega)) \quad \text{and} \quad \frac{du}{dt} \in L^{\frac{p-\beta}{p-1}}(0,T; W^{-1,\frac{p-\beta}{p-1}}(\Omega)) $$

Is there a nice approximation $u_k$ to $u$ such that $$ u_k \in L^{p}(0,T;W^{1,p}(\Omega))$$ $$ \frac{du_k}{dt} \in L^{\frac{p}{p-1}}(0,T; W^{-1,\frac{p}{p-1}}(\Omega))$$ $$u_k \rightarrow u \quad \text{in} \ L^{p-\beta}(0,T;W^{1,p-\beta}(\Omega))$$ $$\frac{du_k}{dt} \rightarrow \frac{du}{dt} \quad \text{in} \ L^{\frac{p-\beta}{p-1}}(0,T; W^{-1,\frac{p-\beta}{p-1}}(\Omega))$$

I think $u_k = u \ast \eta_{k}$ where $\eta_k$ is the standard mollifier should give this, but how do I go about proving this?


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