A spherical bubble in an incompressible fluid of density $\rho$ has radius $a(t)$. Write down an expression for the velocity field at a radius $R \geqslant a$.

The pressure far from the bubble is $p_{\infty}$. What is the pressure at radius $R$ ?

Find conditions on $a$ and its time derivatives that ensure that the maximum pressure in the fluid is reached at a radius $R_{\max }$ where $a<R_{\max }<\infty$. Give an expression for this maximum pressure when the conditions hold.

Give the most general form of $a(t)$ that ensures that the pressure at $R=a(t)$ is $p_{\infty}$ for all time.