In this paper, we consider the cauchy problem for the 2D Groma–Balogh model (Acta Mater 47:3647–3654, 1999). From the works Cannone et al. (Arch Ration Mech Anal 196:71–96, 2010) and El Hajj (Ann Inst Henri Poincare Anal Nonlineaire 27:21–35, 2010), one can see global well-posedness for this model is an open question. However, we can prove longtime well-posedness. In particular, we show that this model admits a unique solution with the lifespan \(T^\star \) satisfying \(T^\star \log ^2(1+T^\star )\gtrsim \epsilon ^{-2}\) if the initial data is of size \(\epsilon \). To achieve this, we first establish some new decay estimates concerning the operator \(e^{-{\mathcal {R}}_{12}^2t}\). Then, we prove the longtime well-posedness by utilizing the weak dissipation to deal with the nonlinear terms.
['Renhui Wan', 'Jiecheng Chen']
Longtime Well-posedness for the 2D Groma–Balogh Model