Factorization of non-symmetric operators and exponential \(H\)-theorem.

*(English. French summary)*Zbl 06889665Consider the fully nonlinear Boltzmann equation
\[
\partial_t f+v\cdot\nabla_xf= Q(f,f),\quad t\geq 0,\quad v\in\mathbb{R}^3,\quad x\in \pi^3
\]
(3D torus) with hard spheres collision kernel, periodic boundary conditions, and nonnegative initial data “close enough” to the Maxwell equilibrium \(\mu\) or to a spatially homogeneous profile.

Under some boundedness and continuity assumptions the authors prove \(\| f)t-\mu\|\leq Ce^{-\lambda t}\), \(t> 0\), and \(\int f_t\log(f_t/\mu)\,dx\,dv\leq Ce^{-\lambda t}\), \(t>0\) (\(H\)-theorem), where \(\|\cdot\|\) denotes the \(L^1_v L^\infty_x(1+|v|^2)\)-norm, the integral is taken over \(\mathbb{T}^3\times\mathbb{R}^3\), \(C>0\) is a constructive constant, and \(\lambda>0\) a sharp rate given by the spectral gap of the linearized flow in \(L^2(\mu^{-1/2})\).

The basic part of the proof is given in an abstract operator theoretic setting: Let \(E\) and \({\mathcal E}\) be two Banach spaces, \(E\subsetneq E\) dense, \(L\) and \({\mathcal L}\) generators of \(C_0\)-semigroups \(S\) and \({\mathcal S}\) on \(E\) and \({\mathcal E}\), respectively, the spectra of \(L\) and \({\mathcal L}\) contained in but not equal to \(\mathbb{C}\). Suppose \(L{\mathcal L}|_E\) and \({\mathcal L}= {\mathcal A}+{\mathcal B}\), where the spectrum of \({\mathcal B}\) is well-localized and the iterated convolution \(({\mathcal A}{\mathcal S}_{{\mathcal B}})^{*^n}\), \({\mathcal S}_{{\mathcal B}}\) denoting the semigroup generated by \({\mathcal B}\), maps \({\mathcal E}\) to \(E\) with proper time-decay control for some \(n\in\mathbb{N}^*\).

Using a factorization argument on the resolvents and the semigroups, it is shown that that \({\mathcal L}\) inherits most of the spectral properties of \(L\) and that explicit estimates of the rate of decay of the semigroup \({\mathcal S}\) can be computed from the corresponding estimates for \(S\). That is applied to the kinetic Fokker-Planck operator and the linearized Boltzmann operator with hard sphere interaction. Considerable effort goes into showing that the latter actually fits into the abstract framework.

In both cases, spectral gap estimates for the respective semigroups are obtained. The singularities of the linearized Boltzmann flow are also studied using the factorization method. Given the results obtained for the linearized Boltzmann equation, the case of the fully nonlinear equation is handled in a perturbation setting.

Under some boundedness and continuity assumptions the authors prove \(\| f)t-\mu\|\leq Ce^{-\lambda t}\), \(t> 0\), and \(\int f_t\log(f_t/\mu)\,dx\,dv\leq Ce^{-\lambda t}\), \(t>0\) (\(H\)-theorem), where \(\|\cdot\|\) denotes the \(L^1_v L^\infty_x(1+|v|^2)\)-norm, the integral is taken over \(\mathbb{T}^3\times\mathbb{R}^3\), \(C>0\) is a constructive constant, and \(\lambda>0\) a sharp rate given by the spectral gap of the linearized flow in \(L^2(\mu^{-1/2})\).

The basic part of the proof is given in an abstract operator theoretic setting: Let \(E\) and \({\mathcal E}\) be two Banach spaces, \(E\subsetneq E\) dense, \(L\) and \({\mathcal L}\) generators of \(C_0\)-semigroups \(S\) and \({\mathcal S}\) on \(E\) and \({\mathcal E}\), respectively, the spectra of \(L\) and \({\mathcal L}\) contained in but not equal to \(\mathbb{C}\). Suppose \(L{\mathcal L}|_E\) and \({\mathcal L}= {\mathcal A}+{\mathcal B}\), where the spectrum of \({\mathcal B}\) is well-localized and the iterated convolution \(({\mathcal A}{\mathcal S}_{{\mathcal B}})^{*^n}\), \({\mathcal S}_{{\mathcal B}}\) denoting the semigroup generated by \({\mathcal B}\), maps \({\mathcal E}\) to \(E\) with proper time-decay control for some \(n\in\mathbb{N}^*\).

Using a factorization argument on the resolvents and the semigroups, it is shown that that \({\mathcal L}\) inherits most of the spectral properties of \(L\) and that explicit estimates of the rate of decay of the semigroup \({\mathcal S}\) can be computed from the corresponding estimates for \(S\). That is applied to the kinetic Fokker-Planck operator and the linearized Boltzmann operator with hard sphere interaction. Considerable effort goes into showing that the latter actually fits into the abstract framework.

In both cases, spectral gap estimates for the respective semigroups are obtained. The singularities of the linearized Boltzmann flow are also studied using the factorization method. Given the results obtained for the linearized Boltzmann equation, the case of the fully nonlinear equation is handled in a perturbation setting.

Reviewer: Heinrich Hering (Rockenberg)

##### MSC:

47J35 | Nonlinear evolution equations |

37L05 | General theory of infinite-dimensional dissipative dynamical systems, nonlinear semigroups, evolution equations |

76P05 | Rarefied gas flows, Boltzmann equation in fluid mechanics |

35Q84 | Fokker-Planck equations |

47D06 | One-parameter semigroups and linear evolution equations |

35P15 | Estimates of eigenvalues in context of PDEs |