Existence, uniqueness, analyticity, and Borel summability for Boussinesq equations

Through Borel summation methods, we analyze the Boussinesq equations for coupled fluid velocity and temperature fields:equation(1)ut−νΔu=−P[u⋅∇u−ae2Θ]+fut−νΔu=−P[u⋅∇u−ae2Θ]+fΘt−μΔΘ=−u⋅∇Θ.Θt−μΔΘ=−u⋅∇Θ. We prove that an equivalent system of integral equations in the Borel variable p∈R+p∈R+ dual to 1/t1/t has a unique solution in a class of exponentially bounded functions, implying the existence of a classical solution to (1) in a complex t-region that includes a real positive time axis segment. For analytic initial data and forcing, it is shown that the solution is Borel summable, implying that formal series in powers of t is Gevrey-1 asymptotic, and within the time interval of existence, the solution remains analytic with the same analyticity strip width as the initial data and forcing. We also determine conditions on the integral equation solution that improve the estimate for existence time.