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Decay of solutions to the Cauchy problem for the linearized Boltzmann equation with an unbounded external force potential. (English) Zbl 0817.35116

The time evolution of rarefied gas in the external-force field coming out of a potential \(\varphi= \varphi(x)\) is described by the nonlinear Boltzmann equation \[ f_ t+ \lambda(\varphi)f= Q(f,f) \tag{1} \] where \(f= f(t,x, \xi)\) is the unknown density of the gas particles, and \(\Lambda (\varphi)\) and \(Q\) are linear and nonlinear operator respectively. If (1) is linearized around the equilibrium state \(M= \exp(- \varphi(x)- | \xi|^ 2/ 2)\), substituting \(f= M+ M^{1/2} u\) and dropping the nonlinear term, the following linearized Boltzmann equation is obtained: (2) \(u_ t= B(\varphi) u\) where \(B(\varphi)= A(\varphi)+ L_ 1 (\varphi)\), \(A(\varphi)=- \Lambda(\varphi)+ e^{-\varphi} (-\nu)\), \(L_ 1 (\varphi)= e^{-\varphi} K\), \(\nu\) is a multiplication operator and \(K\) is a selfadjoint compact operator.
The paper studies the order of the decay of solutions to the Cauchy problem related to (2) by means of the order of the decay of solution of the Cauchy problem \(u_ t= A(\varphi) u\), \(u(0)= u_ 0\) which is simpler than (2). Semigroup techniques are used and also the analysis of the spectrum of the operator under consideration is made.
Reviewer: S.Totaro (Firenze)

MSC:

35Q72 Other PDE from mechanics (MSC2000)
76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics
47D06 One-parameter semigroups and linear evolution equations
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References:

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[3] DOI: 10.1007/BF01197579 · Zbl 0434.76065 · doi:10.1007/BF01197579
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