SAME SPEED›to move at the same speed as someone or something that is moving forward so that you stay level with them: She was walking so fast I couldn't keep up with her.
MAKE PROGRESS›to increase or make progress at the same speed as something or someone else so that you stay at the same level as them: Prices have been rising very fast and wages haven't kept up.
UNDERSTAND›to be able to understand or deal with something that is happening or changing very fast: I feel it's important to keep up with current events.