Existence of unimodal solutions of the generalized Constantin–Lax–Majda equation with viscosity

Source abstract

Abstract In this paper, we study the stationary problem for the generalized Constantin–Lax–Majda equation with viscosity and an external force as a one-dimensional model of two-dimensional fluid flow. The equation is second-order, but it involves nonlocal terms defined by the Hilbert transform. A numerical study by Kim, Miyaji, and Okamoto (Jpn. J. Ind. Appl. Math. 35:1065–1083, 2018) has revealed that the equation has unimodal stationary solutions at large Reynolds numbers. In this model, a solution and its second antiderivative are regarded as vorticity and the sign change of the stream function, respectively. The solution is said to be unimodal if the stream function has exactly one maximum and one minimum in the interval. We present a computer-assisted proof of the existence of unimodal solutions. Applying a fixed-point theorem in infinite-dimensional function spaces and utilizing interval arithmetic, we have successfully proved the existence for the Reynolds number $$R = 10^4$$ R = 10 4 .