Stability and Exponential Convergence for the Boltzmann Equation
Doctoral thesis, 1993
This thesis is devoted to the long-time behaviour of the nonlinear Boltzmann equation for a class of molecular interactions, including inverse power laws for hard potentials. It has been known for some time now, that under natural conditions on the initial data, the space homogeneous Boltzmann equation possesses a unique, global solution, and that this solution tends strongly to the equilibrium distribution as time tends to infinity. These results concern L1-solutions, as well as solutions in spaces of the type intersection between L1 and Lq. More recently, it was established that in L1 and for strictly hard potentials, the trend to equilibrium is exponential in time. We prove here, that the same result holds in intersection between L1 and Lq, and extend the L1-result to include pseudo-Maxwellian molecules.
The general theory for the space non-homogeneous Boltzmann equation is much less developed. We consider here the concept of 'nearly space homogeneous solutions'. Existing theory for such solutions state that, in L* and for strictly hard potentials, if initial data are sufficiently close to being space homogeneous and possess some regularity with respect to the space variable, then the Cauchy problem for the Boltzmann equation is well posed, and the solution tends, exponentially in time, to the equilibrium.
This thesis contains a proof that the same type of result holds in Lq. Moreover, the previous L*-theory is extended so as to include the case of pseudo-Maxwellian molecules.
In the space homogeneous case as well as in the non-homogeneous case, the proof is carried out by splitting the Boltzmann equation into a high-velocity part and a low-velocity part. The linearized Boltzmann equation is essential in the study of the low-velocity part, and as part of the thesis we develop the spectral theory for the linearized Boltzmann operator and the evolution of the corresponding semigroup.
To prove existence of solutions to the high-velocity part, a contraction argument is used; for this purpose we prove a number of new estimates for the collision operator.
As part of the proof of these results for pseudo-Maxwellian molecules, we prove that, for such molecules, if the initial data for the space homogeneous equation possess L*-moments of order s, then the solution possesses globally bounded moments of order s', for all s' < s; the analogous result for strictly hard potentials is well-known.