Self-similar Asymptotics for a Modified Maxwell–Boltzmann Equation in Systems Subject to Deformations

On smooth solutions to the thermostated Boltzmann equation with deformation

“…These prototypical solutions not only indicate the existence of invariant manifolds of molecular dynamics but also give a new insight into the relation between atomic forces and nonequilibrium behavior of the gas. Recently, James-Nota-Velázquez [25][26][27] and Bobylev-Nota-Velázquez [10] provided the systematic mathematical study of the subject. Motivated by those works, the authors of this paper [14] also considered the smoothness and asymptotic stability of self-similar solutions to the Boltzmann equation for the uniform shear flow in case of the Maxwell molecule.…”

On Smooth Solutions to the Thermostated Boltzmann Equation with Deformation

“…To the best of our knowledge, [10] seems the only known result on the large time asymptotics to the self-similar profile in weak topology, see also [3] for a further study to provide explicit estimates of the smallness of the matrix A. Following [10,25], in the case of Maxwell molecule, the authors of this paper [14] constructed smooth self-similar profiles for the shear flow problem on the Boltzmann equation and proved the dynamical stability of the stationary solution via a perturbation approach.…”

“…These prototypical solutions not only indicate the existence of invariant manifolds of molecular dynamics but also give a new insight into the relation between atomic forces and nonequilibrium behavior of the gas. Recently, James et al [25][26][27] and Bobylev et al [10] provided the systematic mathematical study of the subject. Motivated by those works, the authors of this paper also considered the smoothness and asymptotic stability of self-similar solutions to the Boltzmann equation for the uniform shear flow in case of the Maxwell molecule [14].…”

“…In the meantime, it is discussed in [26,27] that there is a balance between the hyperbolic term and collision term for the Boltzmann equation describing homogenergetic flow and the corresponding long time asymptotic behavior depends on which term is dominated in large time. By combining the Fourier transform method and moments argument, a more recent progress has been achieved by Bobylev et al [10], where the authors proved the self-similar asymptotics of solutions in large time for the Boltzmann equation with a general deformation force under a smallness condition on the matrix A, and they also showed that the selfsimilar profile can have the finite polynomial moments of higher order as long as the norm of A is getting smaller. To the best of our knowledge, [10] seems the only known result on the large time asymptotics to the self-similar profile in weak topology, see also [3] for a further study to provide explicit estimates of the smallness of the matrix A.…”

“…By combining the Fourier transform method and moments argument, a more recent progress has been achieved by Bobylev et al [10], where the authors proved the self-similar asymptotics of solutions in large time for the Boltzmann equation with a general deformation force under a smallness condition on the matrix A, and they also showed that the selfsimilar profile can have the finite polynomial moments of higher order as long as the norm of A is getting smaller. To the best of our knowledge, [10] seems the only known result on the large time asymptotics to the self-similar profile in weak topology, see also [3] for a further study to provide explicit estimates of the smallness of the matrix A. Following [10,25], in the case of Maxwell molecule, the authors of this paper [14] constructed smooth self-similar profiles for the shear flow problem on the Boltzmann equation and proved the dynamical stability of the stationary solution via a perturbation approach.…”