Ask Mr. Smarty Plants

Frankly, that would be a hostile environment for just about any groundcover we can think of. Foot traffic, trash cans and flooding are a lot to ask a plant to tolerate. If the dirt is level with the curb, our suggestion might not work, but if the level of the dirt is lower than the sidewalk and curb, we would suggest some nice round river rock. Not gravel, as this would be just as likely to wash away in the flooding as dirt or plants. You don't want to pave it over because that would deny oxygen and nutrients from the soil around the roots of your trees. We often suggest mulch as groundcover for difficult situations like that, but mulch would be down the street and into the ocean in the first hard rain.

Often the best plant for holding soil and preventing erosion is a grass of some sort. Carex pensylvanica (Pennsylvania sedge) is not a true grass but is grasslike. Grasses are good because they have long fibrous roots that can anchor the plant and help hold the dirt in place. Unfortunately, those long fibrous roots are going to take a while to develop and if you have that many floods every year, it seems unlikely that any plant would be able to get itself properly established in time to do you some good. The rounded rock would have spaces around it to permit water and air to get to the tree roots; of course, that would also permit a certain amount of weeds to creep up, but vigilance and a little pulling out should take care of that.

Here are links to some articles with suggestions for a rock groundcover:

Decorative Ground Cover Rocks

How to Use Rocks as a Ground Cover to Control Your Weeds

Using River Rock for Ground Cover

We also feel that carex or most other plants would be overwhelmed by flooding. You would likely find yourself doing the planting over and over again. In the meantime, is there any chance that your municipality or area might find a way to prevent that periodic flooding?