The Boltzmann equation for uniform shear flow
arXiv:2008.02551 · doi:10.1007/s00205-021-01717-5
Abstract
The uniform shear flow for the rarefied gas is governed by the time-dependent spatially homogeneous Boltzmann equation with a linear shear force. The main feature of such flow is that the temperature may increase in time due to the shearing motion that induces viscous heat and the system becomes far from equilibrium. For Maxwell molecules, we establish the unique existence, regularity, shear-rate-dependent structure and non-negativity of self-similar profiles for any small shear rate. The non-negativity is justified through the large time asymptotic stability even in spatially inhomogeneous perturbation framework, and the exponential rates of convergence are also obtained with the size proportional to the second order shear rate. The analysis supports the numerical result that the self-similar profile admits an algebraic high-velocity tail that is the key difficulty to overcome in the proof.
arXiv comment51 pages
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Cited by in corpus (4)
- Self-similar profiles for homoenergetic solutions of the Boltzmann equation for non-cutoff Maxwell molecules
- On smooth solutions to the thermostated Boltzmann equation with deformation
- Stability of Rarefaction Waves for the Non-cutoff Vlasov-Poisson-Boltzmann System with Physical Boundary
- On a class of self-similar solutions of the Boltzmann equation