Moment classification of infinite energy solutions to the homogeneous Boltzmann equation

Morimoto, Yoshinori; Wang, Shuaikun; Yang, Tong

doi:10.1142/s0219530515500232

Cited by 6 publications

(14 citation statements)

References 14 publications

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“…It should be noted (see [19], ex.) that K 2 = F (P 2 (R 3 )), which is a key of our construction of the measure valued solution for the finite energy initial datum.…”

Section: Existence Under Cut-off Assumptionmentioning

confidence: 99%

“…It was proved in [20] that the smoothing effect of measure valued solutions always occurs except for a single Dirac mass initial datum. After that, the results about the existence and smoothing effect of solutions have been improved in [18,19] by introducing more precise characteristic function spaces M α and M α , which satisfy relations…”

Section: Existence Under Cut-off Assumptionmentioning

confidence: 99%

“…Then, since it follows from K 2 = F (P 2 (R 3 )) (see [19], ex.) that |v| 2 dF t is bounded, for any ǫ > 0, we can select R ǫ,t > 0 large enough such that…”

Section: Proofmentioning

confidence: 99%

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Measure Valued Solutions to the Spatially Homogeneous Boltzmann Equation Without Angular Cutoff

2016

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A uniform approach is introduced to study the existence of measure valued solutions to the homogeneous Boltzmann equation for both hard potential with finite energy, and soft potential with finite or infinite energy, by using Toscani metric. Under the non-angular cutoff assumption on the crosssection, the solutions obtained are shown to be in the Schwartz space in the velocity variable as long as the initial data is not a single Dirac mass without any extra moment condition for hard potential, and with the boundedness on moments of any order for soft potential.

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“…It should be noted (see [19], ex.) that K 2 = F (P 2 (R 3 )), which is a key of our construction of the measure valued solution for the finite energy initial datum.…”

Section: Existence Under Cut-off Assumptionmentioning

confidence: 99%

Section: Existence Under Cut-off Assumptionmentioning

confidence: 99%

See 1 more Smart Citation

Measure Valued Solutions to the Spatially Homogeneous Boltzmann Equation Without Angular Cutoff

2016

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“…Recently, Morimoto, Wang and Yang in [14] introduce a new classification on the characteristic functions and prove the smoothing effect under this measure initial datum, see also [15] and [2]. For the Shubin space, we have…”

Section: Introductionmentioning

confidence: 96%

The Cauchy problem for the radially symmetric homogeneous Boltzmann equation with Shubin class initial datum and Gelfand–Shilov smoothing effect

2017

Journal of Differential Equations

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In this paper, we study the Cauchy problem for radially symmetric homogeneous non-cutoff Boltzmann equation with Maxwellian molecules, the initial datum belongs to Shubin space of the negative index which can be characterized by spectral decomposition of the harmonic oscillators. The Shubin space of the negative index contains the measure functions. Based on this spectral decomposition, we construct the weak solution with Shubin class initial datum, we also prove that the Cauchy problem enjoys Gelfand-Shilov smoothing effect, meaning that the smoothing properties are the same as the Cauchy problem defined by the evolution equation associated to a fractional harmonic oscillator.

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“…Then ( M α , dis α,β,ǫ ) is a complete metric space. In [12], the well-posedness of the Cauchy problem (1.1)-(1.5) is established in M α . In this paper, we are devoted to characterizing the Fourier images of spaces P 2k−2+α (R d ).…”

Section: Introductionmentioning

confidence: 99%

Probability Measures with Finite Moments and the Homogeneous Boltzmann Equation

Cho

Morimoto

Wang³

et al. 2016

SIAM J. Math. Anal.

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We characterize probability measure with finite moment of any order in terms of the symmetric difference operators of their Fourier transforms. By using our new characterization, we prove the continuitystands for the density of unique measure-valued solution (F t ) t≥0 of the Cauchy problem for the homogeneous non-cutoff Boltzmann equation, with Maxwellian molecules, corresponding to a probability measure initial datum F 0 satisfying |v| 2k−2+α dF 0 (v) < ∞, 0 ≤ α < 2, k = 2, 3, 4, · · · , provided that F 0 is not a single Dirac mass.MSC: Primary 35Q20, 76P05; secondary 35H20, 82B40, 82C40

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Moment classification of infinite energy solutions to the homogeneous Boltzmann equation

Cited by 6 publications

References 14 publications

Measure Valued Solutions to the Spatially Homogeneous Boltzmann Equation Without Angular Cutoff

Measure Valued Solutions to the Spatially Homogeneous Boltzmann Equation Without Angular Cutoff

The Cauchy problem for the radially symmetric homogeneous Boltzmann equation with Shubin class initial datum and Gelfand–Shilov smoothing effect

Probability Measures with Finite Moments and the Homogeneous Boltzmann Equation

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