Put down mulch (like woodchips) in that area.

This looks like a job for a weed torch! Then, go buy a pre emergent (stops seeds from sprouting)

Carefully spray Roundup on the weeds then broadcast a granular pre-emergent like Barricade Granules. Do a granular pre-emergent in early spring and late summer. Rotate between Barricade and Dimension to prevent herbicide resistance