The Boltzmann equation, Besov spaces, and optimal time decay rates inRxn

Abstract

Abstract. We prove that k-th order derivatives of perturbative classical solutions to the hard and soft potential Boltzmann equation (without the angular cut-off assumption) in the whole space, R n x with n ≥ 3, converge in large time to the global Maxwellian with the optimal decay rate of O t

Large time decay estimates for the Muskat equation

“…This inequality (35) can be seen in [31,Lemma 4.2]. We will however give a short proof of (35) for completeness.…”

A sharp time-weighted inequality for the compressible Navier–Stokes–Poisson system in the critical L framework

“…Taking advantage of (1.9) and the definition of functional D p (t), we see that Proof. Let us first recall the following Gagliardo-Nirenberg type inequality in [19] (or see [1,24,25]):…”

Large-time behavior of the two-species relativistic Landau–Maxwell system inRx3

“…There is also a method originated in [11], which is to use a negative Sobolev space Ḣ −s (s ≥ 0) and interpolations among negative derivative and positive derivatives, such as [27,28]. In the same spirit as this method, recently [25] introduced the Besov space and applied a time weighted energy estimate combined with a time-regularity comparison via dyadic decomposition to derive the optimal decay rates of the perturbative solutions to Boltzmann equation without angular cut-off assumption as long as initially the Ḃ − ,∞ 2 L 2 ξ -norm of the perturbation is bounded. Especially, the optimal time-decay rates the m-th (m ≥ 0) order derivatives were studied in [25,27,28,[30][31][32].…”

“…In the same spirit as this method, recently [25] introduced the Besov space and applied a time weighted energy estimate combined with a time-regularity comparison via dyadic decomposition to derive the optimal decay rates of the perturbative solutions to Boltzmann equation without angular cut-off assumption as long as initially the Ḃ − ,∞ 2 L 2 ξ -norm of the perturbation is bounded. Especially, the optimal time-decay rates the m-th (m ≥ 0) order derivatives were studied in [25,27,28,[30][31][32]. However, we notice that in all these results, the influence of the magnetic field B were not considered.…”