Maxwellian Decay Research Articles

Local existence of polynomial decay solutions to the Boltzmann equation for soft potentials

The existence of classical solutions to the Cauchy problem for the Boltzmann equation without angular cutoff has been extensively studied in the framework when the solution has Maxwellian decay in the velocity variable, cf. [6, 8] and the references therein. In this paper, we prove local existence of solutions with polynomial decay in the velocity variable for the Boltzmann equation with soft potential. In the proof, the singular change of variables between post- and pre-collision velocities plays an important role, as well as the regular one introduced in the celebrated cancelation lemma by Alexandre–Desvillettes–Villani–Wennberg [Entropy dissipation and long-range interactions, Arch. Ration. Mech. Anal. 152 (2000) 327–355].

Gevrey regularity for solutions of the non-cutoff Boltzmann equation: The spatially inhomogeneous case

In this paper we consider the non-cutoff Boltzmann equation in the spatially inhomogeneous case. We prove the propagation of Gevrey regularity for the so-called smooth Maxwellian decay solutions to the Cauchy problem of spatially inhomogeneous Boltzmann equation, and obtain Gevrey regularity of order 1/(2s) in the velocity variable v and order 1 in the space variable x. The strategy relies on our recent results for the spatially homogeneous case [T.-F. Zhang and Z. Yin, Gevrey regularity of spatially homogeneous Boltzmann equation without cutoff, J. Differential Equations 253 (4) (2012), 1172–1190. http://dx.doi.org/10.1016/j.jde.2012.04.023]. Rather, we need much more intricate analysis additionally in order to handle with the coupling of the double variables. Combining with the previous result mentioned above, it gives a characterization of the Gevrey regularity of the particular kind of solutions to the non-cutoff Boltzmann.

Gevrey regularity of spatially homogeneous Boltzmann equation without cutoff

In this paper, we study the Gevrey regularity of spatially homogeneous Boltzmann equation without angular cutoff. We prove the propagation of Gevrey regularity for C∞ solutions with the Maxwellian decay to the Cauchy problem of spatially homogeneous Boltzmann equation. The idea we use here is based on the framework of Morimoto–Ukaiʼs recent paper (see [Y. Morimoto, S. Ukai, Gevrey smoothing effect of solutions for spatially homogeneous nonlinear Boltzmann equation without angular cutoff, J. Pseudo-Differ. Oper. Appl. 1 (2010) 139–159]), but we extend the range of the index γ satisfying γ+2s∈(−1,1), s∈(0,1/2) and in this case we consider the kinetic factor in the form of Φ(v)=|v|γ instead of 〈v〉γ as Morimoto and Ukai did before.

Maxwellian Decay for Well-balanced Approximations of a Super-characteristic Chemotaxis Model

We focus on the numerical simulation of a one-dimensional model of chemotaxis dynamics (proposed by Greenberg and Alt [Trans. Amer. Math. Soc., 300 (1987), pp. 235–258] in a bounded domain by means of a previously introduced well-balanced (WB) and asymptotic-preserving (AP) scheme [L. Gosse, J. Math. Anal. Appl., (2011)]. We are especially interested in studying the decay onto numerical steady-states for two reasons: (1) conventional upwind schemes have been shown to stabilize onto spurious non-Maxwellian states (with a very big mass flux; see, e.g., [F. Guarguaglini et al., Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), pp. 39–76]) and (2) the initial data lead to a dynamic which is mostly super-characteristic in the sense of [S. Jin and M. Katsoulakis, SIAM J. Appl. Math., 61 (2000), pp. 273–292]; thus the stability results of Gosse do not apply. A reflecting boundary condition which is compatible with the well-balanced character is presented; a mass-preservation property is proved and some results on super-characteristic relaxation are recalled. Numerical experiments with coarse computational grids are presented in detail: they illustrate the bifurcation diagrams of Guarguaglini et al., which relate the total initial mass of cells with the time-asymptotic values of the chemoattractant substance on each side of the domain. It is shown that the WB scheme stabilizes correctly onto zero-mass flow rate (hence Maxwellian) steady-states which agree with the aforementioned bifurcation diagrams. The evolution in time of residues is commented for every considered test case.

Gevrey smoothing effect of solutions for spatially homogeneous nonlinear Boltzmann equation without angular cutoff

There have been extensive studies on the regularizing effect of solutions to the Boltzmann equation without angular cutoff assumption, for both spatially homogeneous and inhomogeneous cases, by noticing the fact that non cutoff Boltzmann collision operator behaves like the fractional power of the Laplace operator. As a further study on the problem in the spatially homogeneous situation, in this paper, we consider the Gevrey regularity of C ∞ solutions with the Maxwellian decay to the Cauchy problem of spatially homogeneous Boltzmann equation for modified hard potentials, by using analytic techniques developed in Alexandre et al. (J Funct Anal 255:2013–2066, 2008; Arch Ration Mech Anal, doi: 10.1007/s00205-010-0290-1 , 2010), Huo et al. (Kinet Relat Models 1:453–489, 2008) and Morimoto et al. (Discrete Contin Dyn Syst Ser A 24:187–212, 2009).

Secondary Stress Relaxation Mechanism in Ion-Containing Polymers I. A New Analytical Method

With a view to isolating the secondary component from the complexed stress relaxation, a method originally proposed by Tobolsky and used for the study of the chemical stress relaxation in covalently bonded networks was applied to ion-containing polymers to which time-temperature superposition had been previously found inapplicable. A few examples were chosen from the two different groups of ionomers to demonstrate the applicability of this method. Although the secondary component in. one of styrene–sodium methacrylate copolymers (PS–NaMA) with 7.7 mol% of ions cannot be separated by this method at low temperatures, the secondary mechanism can be isolated at high temperatures. In other ionomers with more ions, the secondary mechanism is isolated by this method for the entire temperature ranges. The secondary relaxation is due to the yielding of the ionic structure, and these ionomers studied here approximately follow Maxwellian decay of stress. The activation energy obtained for the secondary mechanism of each ionomer, which ranges from 26 to 31 kcal mol−1, may correspond to the energy required to break the ionic interactions in the ionic clusters.

The stress relaxation of the 1,2‐syndiotactic polybutadiene thermoplastic elastomer

Abstract The stress relaxation of a low‐crystallinity type thermoplastic elastomer (TPR), 1,2‐syndiotactic polybutadiene, was studied. By using the continuous and intermittent stress relaxation method, the Co60 r‐ray irradiated samples were tested from 30°C to 106°C under air, nitrogen or oxygen atmosphere. The relative stress relaxation curve of this low crystallinity kind of elastomer can be expressed by two Maxwellian decay terms. f(t)/f(0)=o 1 exp (‐k 1 t) + o 2 exp (‐k 2 t) Since there are only a few C=C double bonds in the polymer main chain, the oxidation of C=C in the side chain does not influence the strength of the stress relaxation. Thus the chemical relaxation can be ignored in this kind of thermoplastic elastomer. These two Maxwellian terms are both of physical stress relaxation, namely, flexible chain slippage and crystalline domain disintegration (breaking), respectively:

The Expression of a More General Chemical Relaxation Curve than Maxwellian Decay Curve

The Expression of a More General Chemical Relaxation Curve than Maxwellian Decay Curve

Chemical Stress Relaxation of Cis-1, 4 Polybutadiene

Polybutadiene is considered to be most promising elastomers in synthetic rubber industry as the result of remarkable improvement made on this material in recent years so that it contains now nearly all cis-1, 4 in its structure. The chemorheological study of this material is in this way becoming a very interesting subject.For the measurement of stress relaxation two kinds of samples were used, the samples that were subjected to thoroughgoing extraction in aceton and benzene at room temperature after complete vulcanization, and therefore free from all chemical residue that might act as cross-linking agents, and the samples that were not subjected to extraction, and therefore retain chemical substances that might act as cross-linking agents. As was found by A.V. Tobolsky and others with natural rubber when f(t)/f(0) was plotted against log t, all cis-1, 4 polybutadiene showed the Maxwellian decay with approximate exactness in unextracted samples as is shown in Figs. 2 and 3, and with rough exactness in extracted samples as is shown in Figs. 4 and 5. It is clear from these figures that both cis 4 and Ameripol CB decay more rapidly in unextracted samples than in extracted samples at the temperature used in the present experiment. This phenomenon will be accounted for by the fact that the chemical impurities that remain in unextracted samples will accerelate the scission reaction. The samples shown in Figs. 2, 3, 4, and 5 are replotted in the form of log f(t)/f(0) vs. t, in Figs. 6, 7, 8, and 9, respectively.From the slopes of the straight lines in Figs. 6, 7, 8, and 9, the values of k' can be obtained, the functions of T as the Maxwellian decay curve is expressed by equation (1).f(t)/f(0)=e-k't (1)As such functional dependence of k' on temperature is known to be expressed by the Arhenius equation (2), the values of Eact were obtained for the four samples, the unextracted cis 4, and Ameripol CB, the extracted cis 4 and Ameripol CB.k'=Aexp(-Eact/RT) (2)In this way the activation energy Eact for the four samples can be calculated, using the plotting in the form of log k' vs. 1/T as shown in Fig. 10.The activation energy Eact of the four samples is roughly equal to the value (30±2)kcal/mol as was obtained by A.V. Tobolsky and others with natural rubber being approximately independent of the amounts of cis-1, 4 structure, and of the existence of chemical impurities.Various samples of cis 4 and Ameripol CB of different initial cross-link density, i.e., different values of n(0) were prepared. The n(0) values for the samples were determined by measuring the initial stress f(0) at 25% on the equation of state for rubber as shown by equation (3).f(0)=n(0)RT[(Lo/Lu)2-(Lu/Lo)] (3)Here, Lo/Lu equals to 25% elongation.The shapes of the stress relaxation curves of the various samples having different values of n(0) are shown in Fig. 11.It is already known that if cleavage occurs randomly at the links of the network chains, the decay of the stress relaxation curves rapidly decrease in n(0).According to our experiment shown in Fig. 11, however, the decay of the stress relaxation curves appears to be independent of the initial density of n(0). The fact that f(t)/f(0) is independent of n(0) indicates that cleavage takes place at or adjacent to the network junctures for all cis-1, 4 polybutadiene.

The viscoelastic properties of crosslinked poly(ethylene tetrasulfide) polymers. II

Stress relaxation studies were made on poly(ethylene tetrasulfide) crosslinked during the polymerization reaction by the use of trifunctional organic chloride. At very low crosslink densities a certain amount of stress-relaxation due to chain slippage occurred. Over the middle range of crosslinking density stress-relaxation was due to tetrasulfide bond interchange exclusively, and these samples followed Maxwellian decay of stress. The specific rate constant for bond interchange ( k) is given by k = 5.86 × 10 12 (exp −25900/ RT). The value of the activation energy obtained is discussed in the light of the various energy values reported for the scission of sulfur-sulfur bonds in sulfur chains and rings.