First, a quick summary of what to know about viscous (or dissipative) Burgers' equation $$ u_t-u_{xx}=(u^2)_x. \tag{1}\label{1}$$ Recall that $\dot H^{-1/2}(\mathbb R)$ is a scaling-critical Sobolev space.
Local well-posedness in $\dot H^{-1/2}(\mathbb R)$ is known, as well as in $H^s(\mathbb R)$, $s>-1/2$ (the same arguments should work for both the homogeneous and non-homogeneous cases).
Concerning global well-posedness, I don't know a reference, but it's clear that smooth solutions are global thanks to the $L^\infty$ norm of the solution being non-increasing, which together with the fact that all solutions become smooth at any positive time, should be enough to imply global existence for all data in $H^{-1/2}(\mathbb R)$ (just a rough idea). Note that formally, all $L^p$ norms of the solution are non-increasing.
Now, consider Burgers' equation with forcing $$ u_t-u_{xx}=(u^2)_x+f_x, \tag{2}\label{2} $$ where $f$ is a given space-time function/distribution. It can be proved with standard fixed point techniques that if $f$ belongs to a scaling critical space like $L^{3/2}(\mathbb R\times [0,\infty))$ or $L^2([0,\infty);\dot H^{-1/2}(\mathbb R))$, there exists a unique solution $u\in C([0,T);\dot H^{-1/2}(\mathbb R))\cap L^3_{loc}([0,T);L^3(\mathbb R))$ with $u|_{t=0}\equiv 0$, where $T$ is small enough depending on $f$. In this case, we also have global existence for small data (that is, if $f$ is small in one of the above norms, the above holds for $T=\infty$; actually, for small data we also have $u\in L^3(\mathbb R\times[0,\infty))$). I suspect that global existence holds for large data as well, but I don't see how to prove it. The main problem is that the solution $u$ is not smooth for positive times in general, so the argument as in the standard case does not apply so easily. It seems clear, though, that if in addition $f$ is a bit more regular, $f\in L^2(\mathbb R^2)$, then one can have the previous statement with $T=\infty$, because in this case the solution also lies in $C([0,T));L^2(\mathbb R))$, and the usual energy estimates still lead to the conclusion that $\|u(t)\|_{L^2}$ is bounded for all times by the $L^2$ norm of $f$.
So, my question is: does anyone know how to prove that global solutions of $\eqref{2}$ always exist for large data $f$ in $L^{3/2}(\mathbb R\times [0,\infty))$ or $L^2([0,\infty);\dot H^{-1/2}(\mathbb R))$? Also, if you have a reference for global existence at critical regularity for the equation without forcing $\eqref{1}$, that would be nice to know. Many thanks.
