Well-posedness of the 2D surface quasi-geostrophic equation in variable Lebesgue spaces

In this paper, we are mainly concerned with the well-posedness of the dissipative surface quasi-geostrophic equation in the framework of variable Lebesgue spaces. Based on some analytical results developed in the variable Lebesgue spaces and the Lᵖ-Lq decay estimates of the fractional heat kernel, we establish, for the 2D dissipative surface quasi-geostrophic equation, the global well-posedness in the space Lp(·)2/α-1(R²,L∞(0,∞)) and the local well-posedness in the space Lq(·)(0,T; Lᵖ(R²)).