On The Boltzmann Equation: Entropy And Convergence To Equilibrium

The paper studies the irreversible behavior of the solution of the spatially homo-geneous Boltzmann equation, including the evolution of its entropy and the problem of convergence to equilibrium. Using the entropy method and a kind of weak moment estimate, we prove the strong convergence to equilibrium for all possible collision ker-nels as long as it has a hard-potential type of upper bound and a log type of lower bound.We specially study the case that the initial value has an infinite entropy, namely H(fo)=+∞. Under the hard sphere and hard potential model, it is found that:for some initial values having infinite entropy, the entropy of corresponding conservative solu-tions will become finite since a certain time, which is called entropy production; but for the other initial values also having infinite entropy, the entropy of corresponding conservative solutions will keep infinite all the time, namely the entropy production doesn't happen. Especially for those initial values that are radially symmetric and locally bounded, we obtain a set of rules, which can completely judge whether the entropy production will happen on the corresponding conservative solutions. If it hap-pens, the rules can also provide an estimate of its happening time. The conclusion is different under the Maxwell model, where we proved:as long as the initial value has infinite entropy, the entropy of corresponding conservative solutions will keep infinite all the time, namely the entropy production never happen. The main tools we use are the Duhamel formula of the solution and an upper bound estimate of Q+(f,f)(v,t), which change the estimation of H(f)(t) to the estimation of an integral of∫0 similar to H(f0).Wennberg mentioned in one paper a sufficient condition for a distribution to have infinite entropy dissipation:the distribution vanishes on some open subset of the ve-locity space R3. We find that this intuitively obvious conclusion is not trivial to prove. Using the geometric property of velocity transformation of particle collision, we give a strict proof of the above sufficient condition.