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...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...

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...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 ...

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...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...

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...

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...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 |...

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...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 ...

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...

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...

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...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 ...

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...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...

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 ...

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...

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 ...

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...

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...

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...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...

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 ...

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... 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...

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... 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...

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...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...

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...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...

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... 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...

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.... 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...

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...

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...

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...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 ...

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...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...

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...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 ...

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...

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...

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...

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...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 = ...

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...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 ...

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 ...

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...

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 ...

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...

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 ...

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...

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...

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...

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...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...

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...

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...

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...

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...

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 ...

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...

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...

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...

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...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...

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...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...

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...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 ...

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...

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...

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...

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...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...

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... 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...

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...

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, ...

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...

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...

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...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 ...

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... 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...

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...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...

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...

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...

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... 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...

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...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...

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...