  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 $L^{p}$-$L^{q}$ decay estimates of the fractional heat
kernel, we establish the local existence and regularity of solutions to the 2D
dissipative surface quasi-geostrophic equation in the variable Lebesgue space.
