Regularity of weak solutions to a class of fourth order parabolic variational inequality problems arising from swap option pricing

This article concerns the regularity of weak solutions for a variational inequality problem constructed by a fourth-order parabolic operator which has received much attention recently. We first consider the internal regular estimate of weak solutions using the difference type test function. Then, the near edge regularity and global regularity of weak solutions are analyzed by using the finite cover principle. Since the quadratic gradient of the weak solution does not satisfy the conditions for a test function, we have constructed a test function using a spatial difference operator to complete the proof of regularity. The results show that the weak solution has a second order regularity and an $L^{\mathrm{\infty }}(0,T;H^2(\mathrm{\Omega }))$ estimation independent of the lower order norm of the weak one.