We consider two problems where Borel summation based methods can be used to obtain information about solutions to differential equations. In the first problem, we analyze the initial value problem for the Boussinesq equation for fluid motion and temperature field as well as the magnetic Benard equation which models electromagnetic effects on fluid flow under some simplifying assumptions. This method has previously been applied to the NavierStokes equation, which is a limiting case for each of these equations. We show that this approach can be used to prove local existence for the Boussinesq and magnetic Benard equation, in two or three dimensions. We prove that an equivalent system of integral equations in each case has a unique solution, which is exponentially bounded for p on the positive real line, p being the Laplace dual variable of 1/t. This implies local existence of a classical solution to the Boussinesq and magnetic Benard equations in a complex tregion that includes a real positive time axis segment. Further, it is shown that within this real time interval, for analytic initial data and forcing f, the solution remains analytic and has the same analyticity strip width. Further, under these conditions, the solution is Borel summable, implying that the formal series in time is Gevrey1 asymptotic for small t. We also determine conditions on the computed solution to the integral equation in each case over a finite interval in p that results in a better estimate for existence time for the corresponding solution to the partial differential equation.
The second problem is to give rigorous bounds on the Stokes constant values for a nonlinear ordinary differential equation (ODE) that arises in the context of selection of limiting finger width in viscous fingering the so called SaffmanTaylor problem. Specifically, it was proved that the selected finger width asymptotically corresponds to values of a parameter C in a nonlinear ODE such that the Stokes constant on the real positive line vanishes. The full asymptotic expansion for the solution includes not only inverse powers of the independent variable, but also exponentially small corrections. We prove rigorous estimates on the Stokes constant and find intervals in C for which the Stokes constant vanishes, in agreement with earlier numerical computations.
