13

I have a problem understanding the meaning of a distributional solution. Let me tell you the context the problem appeared: I read through some papers by DiPerna and Lions concerning the Cauchy Problem for Boltzmann Equations ("R.J. DiPerna and P.L. Lions. On the Cauchy problem for Boltzmann equations: global existence and weak stability. Ann. Math., 130"), available from JSTOR: http://www.jstor.org/stable/1971423 for those with subscription.

Shortly summarized they show that classical solutions of the Boltzmann Equation converge weakly in $L^1$ to a renormalized solution and from this they deduce global existence of a solution to the Cauchy Problem.

What makes this so attractive? Well they also show that $f$ is a distributional solution if and only if $f$ is a renormalized solution, they also show that this is also equivalent that $f$ is mild solution.

Question: What is purpose for doing that? What can I deduce from the knowledge that if $f$ is renormalized solution than it has to be also a distributional solution?

REMARK: In the context of this question I also put another one on mathoverflow (where I do not understand the first part of a proof: https://mathoverflow.net/questions/132871/cauchy-problem-for-boltzmann-equations); the question has since been removed.

Community
  • 1
Montaigne
  • 885

1 Answers1

0

In their paper, they show that under the assumption that $Q^{\pm}(f,f)\in L^1_{\mathrm{loc}}((0,\infty)\times \mathbb{R}^N \times \mathbb{R}^N)$, a function $f\in L^1_{\mathrm{loc}}((0,\infty)\times \mathbb{R}^N \times \mathbb{R}^N)$ is a renormalized solution if and only if it is a distributional solution (Lemma II.1). However, the assumption that $Q^{\pm}(f,f)\in L^1_{\mathrm{loc}}((0,\infty)\times \mathbb{R}^N \times \mathbb{R}^N)$ is really (really!) hard to verify in general. This is because, roughly speaking, $Q^{\pm}(f,f)\sim f^2$ and the conservation of mass, momentum, energy, and the entropy inequality do not by themselves guarantee that $f^2 \in L^1_{\mathrm{loc}}((0,\infty)\times \mathbb{R}^N \times \mathbb{R}^N)$.

Note that if $Q^{\pm}(f,f)\in L^1_{\mathrm{loc}}((0,\infty)\times \mathbb{R}^N \times \mathbb{R}^N)$ is not true, we don't even know whether $Q^{\pm}(f,f)$ makes sense as a distribution. So what they proved in Lemma II.1 is that if the notion of distributional solutions ever makes sense (which seems difficult, and it is precisely why the notion of renormalized solutions is introduced), it is equivalent to the notion of renormalized solutions.

Kai
  • 131