## Global existence and full regularity of the Boltzmann equation without angular cutoff

Venue: | Comm. Math. Phys |

Citations: | 14 - 10 self |

### Citations

240 |
A review of mathematical topics in collisional kinetic theory
- Villani
- 2002
(Show Context)
Citation Context ...ption, avoiding the non-integrable angular singularity of the cross-sections, see for example [22, 23, 34, 39, 40, 46, 55, 48, 49, 70] to cite only a few, further references being given in the review =-=[73]-=-. However, except for the hard sphere model, for most of the other molecule interaction potentials such as the inverse power laws recalled above, the cross section B(v − v∗, σ) is a non-integral funct... |

218 |
On the Cauchy problem for Boltzmann equations : global existence and stability
- DiPerna, Lions
- 1989
(Show Context)
Citation Context ...derivation in 1872. A large number of mathematical works have been performed under the Grad’s cutoff assumption, avoiding the non-integrable angular singularity of the cross-sections, see for example =-=[22, 23, 34, 39, 40, 46, 55, 48, 49, 70]-=- to cite only a few, further references being given in the review [73]. However, except for the hard sphere model, for most of the other molecule interaction potentials such as the inverse power laws ... |

120 |
The uncertainty principle
- Fefferman
- 1983
(Show Context)
Citation Context ...upling of the transport operator with the collision operator, which is similar to the Landau equation studied in [25]. To overcome this difficulty, a generalized uncertainty principle à la Fefferman =-=[38]-=- (see also [56, 57, 58]) was introduced in [8, 9] for the study of smoothing effects of the linearized and spatially inhomogeneous Boltzmann equation with non-cutoff cross sections. In order to comple... |

117 | On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations
- Villani
- 1998
(Show Context)
Citation Context ... be very useful for the study on the mathematical theory regarding the regularizing effect for the spatially homogeneous problems for which the theory can now be considered as quite satisfactory, cf. =-=[6, 7, 16, 24, 29, 30, 32, 47, 59, 60, 71]-=-, and the references therein, see also for a much more detailed discussion [5]. Compared to the spatially homogeneous problems, the original spatially inhomogeneous Boltzmann equation is of course phy... |

105 |
The theory of the nonlinear spatially uniform Boltzmann equation for Maxwell molecules
- Bobylev
- 1988
(Show Context)
Citation Context ...0) − Re ĝ2(ξ−) )̂ µ(2ξ)dσdξ. = C ∫ R3 ∫ S2 b ( ξ |ξ| · σ ) ∫ R 3 v g2(v) ( 1 − cos(ξ− · v) ) dv̂µ(2ξ)dσdξ. 14 R. ALEXANDRE, Y. MORIMOTO, S. UKAI, C.-J. XU, AND T. YANG We now use Bobylev’s technique =-=[18]-=- to have ∫ S2 b ( ξ |ξ| · σ ) ψ(ξ− · v)dσ = ∫ S2 b ( v |v| · σ ) ψ(ξ · v−)dσ, so that B2 ≥ C ∫ R 3 v g2(v) (∫ R3 ∫ S2 b ( v |v| · σ )( 1 − cos(ξ · v−) )̂ µ(2ξ)dσdξ ) dv = C ∫ R 3 v g2(v) (∫ S2 b ( v |... |

91 |
Compactness in Boltzmann’s equation via Fourier integral operators and applications
- Lions
- 1994
(Show Context)
Citation Context ...derivation in 1872. A large number of mathematical works have been performed under the Grad’s cutoff assumption, avoiding the non-integrable angular singularity of the cross-sections, see for example =-=[22, 23, 34, 39, 40, 46, 55, 48, 49, 70]-=- to cite only a few, further references being given in the review [73]. However, except for the hard sphere model, for most of the other molecule interaction potentials such as the inverse power laws ... |

81 |
On the existence of global solutions of a mixed problem for the nonlinear Boltzman equation
- Ukai
- 1974
(Show Context)
Citation Context ...available results, the mathematical theory for non-cutoff cross-sections is so far not satisfactory. This is in sharp contrast to the cutoff case, for which the theories have been well developed, see =-=[19, 20, 21, 34, 36, 46, 52, 53, 67, 68, 70]-=- and the references therein. For the study of the regularizing effect, one of the main difficulties comes from the coupling of the transport operator with the collision operator, which is similar to t... |

79 | On the spatially homogeneous Landau equation for hard potentials
- Desvillettes, Villani
(Show Context)
Citation Context ...cerning only the existence. An important step was initiated by the works of Desvillettes and his collaborators in 1990s, showing partial regularization results for some simplified kinetic models, cf. =-=[26, 27, 28, 29, 33, 31, 72]-=-. After the well known result of DiPerna and Lions [34] for the cutoff case, Lions was able to show the gain of regularity of solutions in the Landau case [50], which is a model arising as a grazing l... |

71 |
The Landau equation in a periodic box
- Guo
(Show Context)
Citation Context ...oduction of a new non-isotropic norm which captures the main feature of the singularity in the cross-section. This new norm is in fact the counterpart of the coercive norm which was introduced by Guo =-=[45]-=- as an essential step for Landau equation. It is not known if there is any equivalence of this norm to some Sobolev norm, in contrast to the case of the Landau equation. However, since it is designed ... |

53 |
About the regularization properties of the non cut-off Kac equation
- Desvillettes
- 1995
(Show Context)
Citation Context ...cerning only the existence. An important step was initiated by the works of Desvillettes and his collaborators in 1990s, showing partial regularization results for some simplified kinetic models, cf. =-=[26, 27, 28, 29, 33, 31, 72]-=-. After the well known result of DiPerna and Lions [34] for the cutoff case, Lions was able to show the gain of regularity of solutions in the Landau case [50], which is a model arising as a grazing l... |

49 |
The Boltzmann equation in the whole space
- Guo
(Show Context)
Citation Context ...derivation in 1872. A large number of mathematical works have been performed under the Grad’s cutoff assumption, avoiding the non-integrable angular singularity of the cross-sections, see for example =-=[22, 23, 34, 39, 40, 46, 55, 48, 49, 70]-=- to cite only a few, further references being given in the review [73]. However, except for the hard sphere model, for most of the other molecule interaction potentials such as the inverse power laws ... |

47 | Smoothess of the solution of the spatially homogeneous Boltzmann equation without cutoff
- Desvillettes, Wennberg
(Show Context)
Citation Context ... be very useful for the study on the mathematical theory regarding the regularizing effect for the spatially homogeneous problems for which the theory can now be considered as quite satisfactory, cf. =-=[6, 7, 16, 24, 29, 30, 32, 47, 59, 60, 71]-=-, and the references therein, see also for a much more detailed discussion [5]. Compared to the spatially homogeneous problems, the original spatially inhomogeneous Boltzmann equation is of course phy... |

46 |
Asymptotic theory of the Boltzmann equation II Rarefied Gas Dynamics
- Grad
- 1962
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45 | On the Boltzmann equation for long-range interactions
- Alexandre, Villani
(Show Context)
Citation Context ... be very useful for the study on the mathematical theory regarding the regularizing effect for the spatially homogeneous problems for which the theory can now be considered as quite satisfactory, cf. =-=[6, 7, 16, 24, 29, 30, 32, 47, 59, 60, 71]-=-, and the references therein, see also for a much more detailed discussion [5]. Compared to the spatially homogeneous problems, the original spatially inhomogeneous Boltzmann equation is of course phy... |

44 | Entropy dissipation and long-range interactions.
- Alexandre, Desvillettes, et al.
- 2000
(Show Context)
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42 | Global classical solutions of the Boltzmann equation without angular cut-off
- Gressman, Strain
(Show Context)
Citation Context ...y variable. Note added in September, 2010: Several new results have been announced along the same line of development since the submission of the current paper. For the reader’s references we mention =-=[42, 43, 44, 13, 14, 15]-=-. The main difference of the results is the range of admissible values of γ: γ > −1 − 2s in the first 3 papers and γ > max(−3,−3/2− 2s) in the latter 4 paper. 2. Non-isotropic norms In this section, w... |

34 |
Solutions of the Boltzmann equation, in patterns and waves-qualitative analysis of differential equations
- Ukai
- 1986
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32 |
Intermolecular forces of infinite range and the Boltzmann equation
- Arkeryd
- 1981
(Show Context)
Citation Context ... at least to the work by Pao in 1970s [64] which is about the spectrum of the linearized operator. In 1980s, the existence of weak solutions to the spatially homogeneous case was proved by Arkeryd in =-=[17]-=- for the mild singular case, that is, when 0 < s < 12 , and by using an abstract Cauchy-Kovalevskaya theorem, Ukai in [69] proved the local existence of solutions to the inhomogeneous equation, in the... |

32 |
Local solutions in Gevrey classes to the nonlinear Boltzmann equation without cutoff,”
- Ukai
- 1984
(Show Context)
Citation Context ... of weak solutions to the spatially homogeneous case was proved by Arkeryd in [17] for the mild singular case, that is, when 0 < s < 12 , and by using an abstract Cauchy-Kovalevskaya theorem, Ukai in =-=[69]-=- proved the local existence of solutions to the inhomogeneous equation, in the space of functions which are analytic in x and Gevrey in v. For a long time, the mathematical study of singular cross-sec... |

30 | Spectral gap and coercivity estimates for linearized Boltzmann collision operators without angular cutoff
- Mouhot, Strain
(Show Context)
Citation Context ...inearized collision operator through the H-theorem, but also well fits for the upper bound estimation on the nonlinear collision operator. Here, we would like to mention the work by Mouhot and Strain =-=[62, 63]-=- about the gain of moment in a linearized context due to the singularity in the cross-section. Such a gain of moment which is well described by the new non-isotropic norm is in fact crucial for the pr... |

30 | The Vlasov-Maxwell-Boltzmann system in the whole space
- Strain
(Show Context)
Citation Context ...he non-isotorpic norms (2.2.1). We also note that his method has been generalized to various directions. Among them, a few are the external force case [35, 37], the Vlassov-Maxwell-Boltzmann equation =-=[65]-=-, the soft potential case [66, 74] and the Landau equation [45, 74]. Independently of his method which is based on the macro-micro decomposition near a global Maxwellian, the energy method based on th... |

28 |
Terraneo, Propagation of Gevrey regularity for solutions of Boltzmann equation for Maxwellian molecules
- Desvillettes, Furioli, et al.
- 2008
(Show Context)
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28 |
Exponential decay for soft potentials near
- Strain, Guo
- 2009
(Show Context)
Citation Context .... We also note that his method has been generalized to various directions. Among them, a few are the external force case [35, 37], the Vlassov-Maxwell-Boltzmann equation [65], the soft potential case =-=[66, 74]-=- and the Landau equation [45, 74]. Independently of his method which is based on the macro-micro decomposition near a global Maxwellian, the energy method based on the macro-micro decomposition around... |

27 | Explicit coercivity estimates for the linearized Boltzmann and Landau operators
- Mouhot
(Show Context)
Citation Context ...inearized collision operator through the H-theorem, but also well fits for the upper bound estimation on the nonlinear collision operator. Here, we would like to mention the work by Mouhot and Strain =-=[62, 63]-=- about the gain of moment in a linearized context due to the singularity in the cross-section. Such a gain of moment which is well described by the new non-isotropic norm is in fact crucial for the pr... |

26 |
Regularity estimates via the entropy dissipation for the spatially homogeneous Boltzmann equation without cut-off
- Villani
- 1999
(Show Context)
Citation Context ...cerning only the existence. An important step was initiated by the works of Desvillettes and his collaborators in 1990s, showing partial regularization results for some simplified kinetic models, cf. =-=[26, 27, 28, 29, 33, 31, 72]-=-. After the well known result of DiPerna and Lions [34] for the cutoff case, Lions was able to show the gain of regularity of solutions in the Landau case [50], which is a model arising as a grazing l... |

24 |
Hypoellipticity for a class of kinetic equations
- Morimoto, Xu
(Show Context)
Citation Context ...ple, it should be similar to the case when one replaces the collision operator in the Boltzmann equation by a fractional Laplacian in velocity variable, that is, a fractional Kolmogorov-type equation =-=[61]-=-. Certainly, the results of Lions [51] and Desvillettes have influenced the research in this direction. It is therefore not surprising that a systematic approach, using the entropy dissipation and/or ... |

23 | A proof of the smoothing properties of the positive part of Boltzmann’s kernel
- Bouchut, Desvillettes
- 1998
(Show Context)
Citation Context ...available results, the mathematical theory for non-cutoff cross-sections is so far not satisfactory. This is in sharp contrast to the cutoff case, for which the theories have been well developed, see =-=[19, 20, 21, 34, 36, 46, 52, 53, 67, 68, 70]-=- and the references therein. For the study of the regularizing effect, one of the main difficulties comes from the coupling of the transport operator with the collision operator, which is similar to t... |

19 |
A review of Boltzmann equation with singular kernels
- Alexandre
(Show Context)
Citation Context ...ete, let us now review some previous works related to the Boltzmann equation in the context of such singular (or non-cutoff) cross-sections. For other references and comments, readers are referred to =-=[5, 73]-=-. THE BOLTZMANN EQUATION WITHOUT ANGULAR CUTOFF 3 The mathematical study for the Boltzmann equation, without assuming Grad’s cutoff assumption, can be traced back at least to the work by Pao in 1970s ... |

18 |
Kinetic Equations and Asymptotic
- Bouchut, Golse, et al.
- 2000
(Show Context)
Citation Context ...available results, the mathematical theory for non-cutoff cross-sections is so far not satisfactory. This is in sharp contrast to the cutoff case, for which the theories have been well developed, see =-=[19, 20, 21, 34, 36, 46, 52, 53, 67, 68, 70]-=- and the references therein. For the study of the regularizing effect, one of the main difficulties comes from the coupling of the transport operator with the collision operator, which is similar to t... |

18 |
About the use of the Fourier transform for the Boltzmann equation
- Desvillettes
- 2003
(Show Context)
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18 |
Regularity and compactness for Boltzmann collision operators without angular cut-off
- Lions
- 1998
(Show Context)
Citation Context ...when one replaces the collision operator in the Boltzmann equation by a fractional Laplacian in velocity variable, that is, a fractional Kolmogorov-type equation [61]. Certainly, the results of Lions =-=[51]-=- and Desvillettes have influenced the research in this direction. It is therefore not surprising that a systematic approach, using the entropy dissipation and/or the smoothing property of the gain par... |

16 |
Nonlinear stability of rarefaction waves for the Boltzmann equation, preprint
- Liu, Yang, et al.
(Show Context)
Citation Context ...sition near a global Maxwellian, the energy method based on the macro-micro decomposition around a local Maxwellian has also been developed with application to the classical fluid dynamical equations =-=[52, 53, 54]-=-. Further references are found in these paper. 46 R. ALEXANDRE, Y. MORIMOTO, S. UKAI, C.-J. XU, AND T. YANG Introduce the macro-micro decomposition near the absolute Maxwellian µ: g = Pg + (I − P)g = ... |

15 |
Asymptotic equivalence of the Navier-Stokes and nonlinear Boltzmann equations
- Grad
- 1965
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15 |
A direct method for the regularity of the gain term in the Boltzmann equation
- Lu
- 1998
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14 | On the singularities of the global small solutions of the full Boltzmann equation, Monatsh
- Boudin, Desvillettes
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14 |
The Boltzmann equation and its applications, Applied Mathematical Sciences n ffi 67
- Cercignani
- 1988
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14 |
Regularization properties of the 2-dimensional non radially symmetric non cutoff spatially homogeneous Boltzmann equation for Maxwellian molecules, Trans. Theory Stat
- Desvillettes
- 1997
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14 |
The Boltzmann equation. I. Uniqueness and local existence
- Kaniel, Shinbrot
- 1978
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13 |
solutions globales de l’équation de Boltzmann dans l’espace tout entier et dans le demi-espace
- Ukai, Les
- 1976
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12 |
Smoothing effects for classical solutions of the full Landau equation
- Chen, Desvillettes, et al.
(Show Context)
Citation Context ...or the study of the regularizing effect, one of the main difficulties comes from the coupling of the transport operator with the collision operator, which is similar to the Landau equation studied in =-=[25]-=-. To overcome this difficulty, a generalized uncertainty principle à la Fefferman [38] (see also [56, 57, 58]) was introduced in [8, 9] for the study of smoothing effects of the linearized and spatia... |

12 | A combination of energy method and spectral analysis for study of equations of gas motion, Front
- Duan, Ukai, et al.
(Show Context)
Citation Context ...r and related terms are carried out in terms of the non-isotorpic norms (2.2.1). We also note that his method has been generalized to various directions. Among them, a few are the external force case =-=[35, 37]-=-, the Vlassov-Maxwell-Boltzmann equation [65], the soft potential case [66, 74] and the Landau equation [45, 74]. Independently of his method which is based on the macro-micro decomposition near a glo... |

12 |
On Boltzmann and Landau equations, Philos
- Lions
- 1994
(Show Context)
Citation Context ...etic models, cf. [26, 27, 28, 29, 33, 31, 72]. After the well known result of DiPerna and Lions [34] for the cutoff case, Lions was able to show the gain of regularity of solutions in the Landau case =-=[50]-=-, which is a model arising as a grazing limit of the Boltzmann equation. It was then expected that this kind of singular cross sections should lead to smoothing effect on solutions, that is, the solut... |

12 |
The positivity of Schrödinger operators and the hypoellipticity of second order degenerate elliptic operators
- Morimoto, Morioka
- 1997
(Show Context)
Citation Context ...ransport operator with the collision operator, which is similar to the Landau equation studied in [25]. To overcome this difficulty, a generalized uncertainty principle à la Fefferman [38] (see also =-=[56, 57, 58]-=-) was introduced in [8, 9] for the study of smoothing effects of the linearized and spatially inhomogeneous Boltzmann equation with non-cutoff cross sections. In order to complete the full regularizat... |

11 | On a model Boltzmann equation without angular cutoff
- Desvillettes, Golse
(Show Context)
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11 |
Gevrey smoothing effect of solutions for spatially homogeneous nonlinear Boltzmann equation without angular cutoff,”
- Morimoto, Ukai
- 2010
(Show Context)
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10 | anisotropic estimates for the Boltzmann collision operator and its entropy production, Adv
- Gressman, Strain, et al.
(Show Context)
Citation Context ...y variable. Note added in September, 2010: Several new results have been announced along the same line of development since the submission of the current paper. For the reader’s references we mention =-=[42, 43, 44, 13, 14, 15]-=-. The main difference of the results is the range of admissible values of γ: γ > −1 − 2s in the first 3 papers and γ > max(−3,−3/2− 2s) in the latter 4 paper. 2. Non-isotropic norms In this section, w... |

9 |
C.-J.Xu, T.Yang, Uncertainty principle and kinetic equations
- Alexandre, Ukai
- 2008
(Show Context)
Citation Context ...sion operator, which is similar to the Landau equation studied in [25]. To overcome this difficulty, a generalized uncertainty principle à la Fefferman [38] (see also [56, 57, 58]) was introduced in =-=[8, 9]-=- for the study of smoothing effects of the linearized and spatially inhomogeneous Boltzmann equation with non-cutoff cross sections. In order to complete the full regularization process, recently, in ... |

9 | Stability of the Boltzmann equation with potential forces on
- Duan
(Show Context)
Citation Context ...r and related terms are carried out in terms of the non-isotorpic norms (2.2.1). We also note that his method has been generalized to various directions. Among them, a few are the external force case =-=[35, 37]-=-, the Vlassov-Maxwell-Boltzmann equation [65], the soft potential case [66, 74] and the Landau equation [45, 74]. Independently of his method which is based on the macro-micro decomposition near a glo... |

8 | Safadi M., Littlewood Paley decomposition and regularity issues in Boltzmann equation homogeneous equations. II. Non cutoff case and non Maxwellian molecules. Discrete Contin - Alexandre, El |

8 | The Gevrey hypoellipticity for a class of kinetic equations
- Chen, Li, et al.
(Show Context)
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8 |
Regularization for the non Cutoff 2D Radially Symmetric Boltzmann Equation with a Velocity Dependant
- Desvillettes
- 1996
(Show Context)
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8 | Strain,Global Strong Solutions of the Boltzmann Equation without Angular Cut-off, submitted
- Gressman, Robert
(Show Context)
Citation Context ...croscopic component. 6 R. ALEXANDRE, Y. MORIMOTO, S. UKAI, C.-J. XU, AND T. YANG Note: After finishing this paper, we were informed by R. Strain of his recent paper in collaboration with P. Gressmann =-=[41]-=-, showing also the existence of global solutions to the Cauchy problem by using different approach. Notice that their solution is in different function space which does not lead to full regularity bec... |

7 |
Some solutions of the Boltzmann equation without angular cutoff
- Alexandre
(Show Context)
Citation Context ...ng and mathematically more challenging. For existence of weak solutions, we mention two results regarding the Cauchy problem. One is about the local existence between two moving Maxwellians proved in =-=[3]-=- by constructing the upper and lower solutions, another is the global existence of renormalized solutions with defect measures shown in [16] where the solutions become weak solutions if the defect mea... |

7 |
The Mathematical theory of Dilute gases, Applied mathematical sciences, 106
- Cercignani, Illner, et al.
- 1994
(Show Context)
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7 |
C.-J.Xu and T.Yang, Regularity of solutions to the spatially homogeneous Boltzmann equation without Angular cutoff. Discrete Contin
- Morimoto
(Show Context)
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6 |
Xu and T.Yang, Boltzmann equation without angular cutoff in the whole space: III, Qualitative properties of solutions
- Alexandre, Morimoto, et al.
- 51
(Show Context)
Citation Context ...y variable. Note added in September, 2010: Several new results have been announced along the same line of development since the submission of the current paper. For the reader’s references we mention =-=[42, 43, 44, 13, 14, 15]-=-. The main difference of the results is the range of admissible values of γ: γ > −1 − 2s in the first 3 papers and γ > max(−3,−3/2− 2s) in the latter 4 paper. 2. Non-isotropic norms In this section, w... |

6 |
Propagation of singularities in the solutions to the Boltzmann equation near equilibrium
- Duan, Li, et al.
(Show Context)
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5 |
C.-J.Xu and T.Yang, Regularizing effect and local existence for noncutoff Boltzmann equation
- Alexandre, Ukai
(Show Context)
Citation Context ...lized solutions with defect measures shown in [16] where the solutions become weak solutions if the defect measures vanish. On the other hand, the local existence of classical solutions was proved in =-=[12]-=- in some weighted Sobolev spaces. However, in view of the above available results, the mathematical theory for non-cutoff cross-sections is so far not satisfactory. This is in sharp contrast to the cu... |

5 |
Y.Morimoto, S.Ukai and T.Yang, Regularity of solutions for spatially homogeneous Boltzmann equation without Angular cutoff. Kinetic and Related Models
- Huo
- 2008
(Show Context)
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5 |
Estimates for degenerate Schrödinger operators and hypoellipticity for infinitely degenerate elliptic operators
- Morimoto
- 1992
(Show Context)
Citation Context ...ransport operator with the collision operator, which is similar to the Landau equation studied in [25]. To overcome this difficulty, a generalized uncertainty principle à la Fefferman [38] (see also =-=[56, 57, 58]-=-) was introduced in [8, 9] for the study of smoothing effects of the linearized and spatially inhomogeneous Boltzmann equation with non-cutoff cross sections. In order to complete the full regularizat... |

5 |
Boltzmann collision operator with inverse power intermolecular potential
- Pao
- 1974
(Show Context)
Citation Context .... THE BOLTZMANN EQUATION WITHOUT ANGULAR CUTOFF 3 The mathematical study for the Boltzmann equation, without assuming Grad’s cutoff assumption, can be traced back at least to the work by Pao in 1970s =-=[64]-=- which is about the spectrum of the linearized operator. In 1980s, the existence of weak solutions to the spatially homogeneous case was proved by Arkeryd in [17] for the mild singular case, that is, ... |

4 |
The uncertainty principle and hypoelliptic operators
- Morimoto
- 1987
(Show Context)
Citation Context ...ransport operator with the collision operator, which is similar to the Landau equation studied in [25]. To overcome this difficulty, a generalized uncertainty principle à la Fefferman [38] (see also =-=[56, 57, 58]-=-) was introduced in [8, 9] for the study of smoothing effects of the linearized and spatially inhomogeneous Boltzmann equation with non-cutoff cross sections. In order to complete the full regularizat... |

2 |
Around 3D Boltzmann operator without cutoff. A New formulation
- Alexandre
(Show Context)
Citation Context ...ides this, we still need some functional estimates on the Boltzmann collision operators. The first one, given below, is about the boundedness of the collision operator in weighted Sobolev spaces, see =-=[1, 2, 4, 5, 12, 47]-=- . Theorem 2.1. Assume that the cross-section satisfies (1.2) with 0 < s < 1 . Then for any m ∈ R and any α ∈ R, there exists C > 0 such that (2.1.2) ‖Q( f , g)‖Hmα (R3v ) ≤ C‖ f ‖L1α++2s(R3v )‖g‖Hm+2... |

2 |
C.-J.Xu and T.Yang, Uncertainty principle and regularity for Boltzmann type equations
- Alexandre, Ukai
- 2007
(Show Context)
Citation Context ...sion operator, which is similar to the Landau equation studied in [25]. To overcome this difficulty, a generalized uncertainty principle à la Fefferman [38] (see also [56, 57, 58]) was introduced in =-=[8, 9]-=- for the study of smoothing effects of the linearized and spatially inhomogeneous Boltzmann equation with non-cutoff cross sections. In order to complete the full regularization process, recently, in ... |

2 |
Xu and T.Yang, Qualitative properties of solutions to the Boltzmann equation without angular cutoff, preprint
- Alexandre, Morimoto, et al.
(Show Context)
Citation Context ... and the non-negativity of the solutions. This new method for proving non-negativity can be applied to the case with angular cutoff. For more detail discussion on the non-negativity problem, refer to =-=[15]-=-. In Section 6, the full regularity is proved along the approach of [12]. Finally, the global existence of the solution will be given in the last section. For this, the macro-micro decomposition intro... |

2 |
Micro-macro decompositions and positivity of shock profiles
- Liu, Yu
(Show Context)
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1 |
Remarks on 3D Boltzmann linear operator without cutoff
- Alexandre
- 1999
(Show Context)
Citation Context ...ides this, we still need some functional estimates on the Boltzmann collision operators. The first one, given below, is about the boundedness of the collision operator in weighted Sobolev spaces, see =-=[1, 2, 4, 5, 12, 47]-=- . Theorem 2.1. Assume that the cross-section satisfies (1.2) with 0 < s < 1 . Then for any m ∈ R and any α ∈ R, there exists C > 0 such that (2.1.2) ‖Q( f , g)‖Hmα (R3v ) ≤ C‖ f ‖L1α++2s(R3v )‖g‖Hm+2... |

1 |
Integral kernel estimates for a linear singular operator linked with Boltzmann equation
- Alexandre
- 1975
(Show Context)
Citation Context ...ides this, we still need some functional estimates on the Boltzmann collision operators. The first one, given below, is about the boundedness of the collision operator in weighted Sobolev spaces, see =-=[1, 2, 4, 5, 12, 47]-=- . Theorem 2.1. Assume that the cross-section satisfies (1.2) with 0 < s < 1 . Then for any m ∈ R and any α ∈ R, there exists C > 0 such that (2.1.2) ‖Q( f , g)‖Hmα (R3v ) ≤ C‖ f ‖L1α++2s(R3v )‖g‖Hm+2... |

1 |
C.-J.Xu and T.Yang, Local existence for non-cutoff Boltzmann equation
- Alexandre, Ukai
(Show Context)
Citation Context ... UKAI, C.-J. XU, AND T. YANG and upper bounds of the collision operators in Sobolev space, together with the estimation on the commutators with these pseudo-differential operators. More precisely, in =-=[10, 11, 12]-=-, for classical solutions, we established the hypo-ellipticity of the Boltzmann operator, using the generalized version of the uncertainty principle. The present work is a continuation of our collabor... |

1 |
T.Yang and S.-H.Yu, Energy method for Boltzmann equation
- Liu
(Show Context)
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1 |
Convergence rate for the Boltzmann and Landau equations with soft potentials
- Yu
(Show Context)
Citation Context .... We also note that his method has been generalized to various directions. Among them, a few are the external force case [35, 37], the Vlassov-Maxwell-Boltzmann equation [65], the soft potential case =-=[66, 74]-=- and the Landau equation [45, 74]. Independently of his method which is based on the macro-micro decomposition near a global Maxwellian, the energy method based on the macro-micro decomposition around... |