Wiktionary
vb. 1 To manage to remain beside or just behind someone or something that is moving away from one. 2 (context figuratively English) To manage to remain up to date with trends, fashions, etc. 3 To manage to follow an argument, a discussion, etc.
Usage examples of "keep up with".
One day, you may be able to fly cross-country by will alone, but for now, if you fell out of the car, I'd have to stop and pick you up, 'cause you couldn't run fast enough to keep up with me.
I shortened my stride so she could keep up with me and from time to time I looked down at her.
The wipers thumped at their top speed but were barely able to keep up with the whirling snow.
Moreover, he could keep up with the young men, which was more than Grandpa Merriwether could do, he told them gleefully.
She didn't know how Sean kept his balance, much less how he kept dancing so lightly, and yet she who, a mere five weeks before, had barely been able to walk without doubling over with lung spasms could now-almost-keep up with him.
None of the street children had been able to keep up with the skimmer as it set about on its lawful errand, but they followed as far as they could: right up to the rancid water.