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Consider the following classical PDE in $R^n$: $$ \partial_tu(t,x)+\Delta u(t,x)+b(t,x)\cdot\nabla u(t,x)=f(t,x),\quad u(0,x)=0. $$ Is there any references on solving the above equation by using the Littlewood-Paley theory? More precisely, I wonder whether the following result is known or not: $$ f\in L^p(R_+\times R^n),\quad b\in L^\infty(R_+;B^\alpha_{q,\infty}(R^n)) $$ with $p>1$ and some conditions on $\alpha,q$ (especially for $\alpha<0$), then there exists a unique solution $u$ to the above equation.

Many thanks for the help!

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You may find Chapter 3, in particular section 3.4, of the book Fourier Analysis and Nonlinear Partial Differential Equations by H. Bahouri, J.Y. Chemin, and R. Danchin helpful.

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