This paper improves some of the results of the previous paper [J. I. Díaz et al., Discrete Contin. Dyn. Syst. 38, No. 2, 509–546 (2018; Zbl 1374.35178)]. The aim of the authors is to study the problem \[ \begin{aligned} Au=-\Delta u+\vec U(x),\;\nabla u+ V(x)u= f(x)\quad &\text{in }\Omega,\\ u=0\quad &\text{on }\partial\Omega,\end{aligned} \] where \(\Omega\) is an open bounded set in \(\mathbb{R}^n\), the potential \(V(x)\) is very singular, i.e., \(V(x)\geq ,d^{-r}\), \(r\geq 2\), \(d= \text{dist}(x,\partial\Omega)\) and \(\vec U(x)\) is a convective flow defined in the weak setting. The authors propose two equivalent definitions of a “very weak solution \(u\) in the sense of Brezis” and a “very weak distributional solution \(u\)” for \(f\in L^1(\Omega,\delta)\). The existence and uniqueness of a very weak solution \(u\) of the equation \(Au=f\) (new result) is proved even if no boundary conditions are prescribed.
On the other hand, it is shown that \(u\) necessarily satisfies in a suitable way the condition \(u=0\) on \(\partial\Omega\). Some new results concerning the \(m\)-accretivity of \(A\) are verified. For example
a) if \(\vec U\in L^\infty(\Omega)\) is a given convective flow and \(V\geq 0\) is locally integrable, then \(A\) is \(m\)-accretive for any \(0\leq \alpha<1\) in \(L^1(\Omega,\delta^\alpha)\),
b) if \(\alpha=1\), \(V\geq c\delta^{-2}\), then \(A\) is still \(m\)-accretive in \(L^1(\Omega,\delta)\).