The Ill-posedness and Well-posedness for the Inviscid Boussinesq Equations

Abstract

By constructing a counterexample based on a modified DiPerna–Majda type shear flow, we show that the solution map for the inviscid Boussinesq equation fails to be continuous from C^1,α to C(0, T;C^1,α) for any 0 < α < 1. In contrast, we establish local well-posedness and continuity of the solution map in the little Hölder space c^1,α, where c^1,α is the completion of C_c^∞ for the norm C^1,α. In some sense, this dichotomy highlights that the discontinuity in C^1,α is sharp, arising precisely from the non-separability of the classical Hölder space.

Keywords

Harmonic analysis / inviscid Boussinesq equations / discontinuity dependence / Hölder space / 35Q35

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Peking University