The acceleration of charged particles with a short laser pulse is a revolutionary acceleration technology, but many restrictions limit their advantages. Wave breaking and dephasing are most important of these restrictions which weaken acceleration efficiency in the laser Wakefield acceleration scheme. The purpose of this paper is to demonstrate that flying focus laser Wakefield acceleration (FF-LWFA) can confirm this effectiveness. Flying focus creates in the focal region of a hyper-chromatic lens when the chirped laser pulse passing through them. Therefore, by controlling the lens chromaticity and initial chirp characteristics, the focal point velocity can be controlled and adjusted near the speed of light. For a realistic working parameter, two-dimensional simulation results show the flying focus pulse can retain its shape and energy within a few centimeters (one order of magnitude greater than Rayleigh's length). We introduce a velocity interval around velocity of light in which dephasing and wave breaking are restricted and high acceleration gradients (about a hundred of GeV/m) are achieved in the flying focus region. Moreover, in the optimized condition for negative chirp, wakefield amplitude is increased 50 %. Under this ideal situation, 0.5 % of background electrons accelerate to 2GeV in the 5 cm, where the acceleration gradient reaches to, 40GeV/m, and about 0.2 % of them accelerate up to 10GeV.
