Buyers have private information on auctioning divisible goods. Linearity could be a useful property on measuring their marginal utility on those goods or on their bidding strategies under such a share auction environment. This paper establishes an auction model with independent private-values paradigm (IPVP) where bidders have linear demand. A mechanism design approach is applied to explore the optimal share auction in this model. I discuss the most popular auction formats in practice, including Vickrey auction (VA), uniform-price auction (UPA) and discriminatory price auction (DPA). The ex-post equilibriums on explicit solutions are achieved. I found VA does not generally constitute an optimal mechanism as expected even in a symmetric scenario. Furthermore, I rank the different auction formats in terms of revenue and social efficiency. The more private information bidders keep, the lower revenue VA generates to seller, and it could be even inferior to UPA or DPA. My study aggregates dispersed private information with linearity and is robust to distributional assumption.