SQG 方程是描述地球物理流体力学的一个重要模型，这一方程无论是在理论研究,还是在气象学和海洋学领域都起着至关重要的作用。我们首先证明有界光滑区域上的SQG 方程的光滑解在粘性情况下的局部适定性和整体存在性，继而再借助Lp 有界性和线性对流扩散的相关定理，证明其在无粘情况下的光滑解的局部存在性，最后根据粘性情况下的相关推理同样得到唯一性，从而完成了无粘情况下的局部适定性证明。

The SQG equation is an important model to describe geophysical fluid dynamics, which plays a vital role in theoretical research, meteorology and oceanography. In bounded domains with smooth boundary, the smooth solution for the equation has local well-posedness. First, we prove the local well-posedness and global existence in the viscid case, and then with some relevant theorem of the Lp  bounds and Linear advection–diffusion, we prove the local existence of the smooth solution in the inviscid case. Finally we obtain the uniqueness according to the similar reasoning in the viscid case, so as to complete the proof of the local well-posedness in the inviscid case.