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The motion of rarefied gases for uniform shear flow at the kinetic level isgoverned by the spatially homogeneous Boltzmann equation with a deformationforce. In the paper we study the corresponding Cauchy problem with initial dataof finite mass and energy for the collision kernel in case of hard potentials$0<\gamma\leq 1$ under the cutoff assumption. We prove the global existence andlarge time behavior of solutions provided that the force strength $\alpha>0$ issmall enough. In particular, when the initial perturbation is of order$\alpha^m$ for $m>2$, we make a rigorous justification of the uniform-in-timeasymptotic expansion of solutions up to order $\alpha^2$ under a homoenergeticself-similar scaling that can capture the increase of temperature$\theta(t)\sim (1+\gamma \varrho_0\alpha^2 t)^{2/\gamma}$ when time tends toinfinity, where $\varrho_0>0$ is a strictly positive constant depending only onthe deformation force and the linearized collision operator. Specifically, weestablish $$ \theta^{3/2}(t)F(t,\theta^{1/2}(t)v)= \mu+\alpha \sqrt{\mu}G_1(t,v)+\alpha^2 \sqrt{\mu}G_2(t,v)+O(1)\alpha^m(1+\gamma \varrho_0\alpha^2t)^{-2} $$ as $t\to\infty$, where $\mu$ is a global Maxwellian and $G_1,G_2$are microscopic bounded functions that can be explicitly determined and decayin time as $G_1\sim (1+\gamma \varrho_0\alpha^2 t)^{-1}$ and $G_2\sim (1+\gamma\varrho_0\alpha^2 t)^{-2}$.

Uniform shear flow via the Boltzmann equation with hard potentials