Annals of Applied Probability

Upper bounds for the function solution of the homogeneous $2D$ Boltzmann equation with hard potential

Vlad Bally

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Abstract

We deal with $f_{t}(dv)$, the solution of the homogeneous $2D$ Boltzmann equation without cutoff. The initial condition $f_{0}(dv)$ may be any probability distribution (except a Dirac mass). However, for sufficiently hard potentials, the semigroup has a regularization property (see Probab. Theory Related Fields 151 (2011) 659–704): $f_{t}(dv)=f_{t}(v)\,dv$ for every $t>0$. The aim of this paper is to give upper bounds for $f_{t}(v)$, the most significant one being of type $f_{t}(v)\leq Ct^{-\eta}e^{-\vert v\vert^{\lambda}}$ for some $\eta,\lambda>0$.

Article information

Source
Ann. Appl. Probab., Volume 29, Number 3 (2019), 1929-1961.

Dates
Received: May 2018
Revised: August 2018
First available in Project Euclid: 19 February 2019

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1550566847

Digital Object Identifier
doi:10.1214/18-AAP1451

Mathematical Reviews number (MathSciNet)
MR3914561

Zentralblatt MATH identifier
07057471

Subjects
Primary: 60H07: Stochastic calculus of variations and the Malliavin calculus 60J75: Jump processes 82C40: Kinetic theory of gases

Keywords
Boltzmann equation without cutoff Hard potentials interpolation criterion integration by parts

Citation

Bally, Vlad. Upper bounds for the function solution of the homogeneous $2D$ Boltzmann equation with hard potential. Ann. Appl. Probab. 29 (2019), no. 3, 1929--1961. doi:10.1214/18-AAP1451. https://projecteuclid.org/euclid.aoap/1550566847


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References

  • [1] Alexandre, R., Desvillettes, L., Villani, C. and Wennberg, B. (2000). Entropy dissipation and long-range interactions. Arch. Ration. Mech. Anal. 152 327–355.
    Zentralblatt MATH: 0968.76076
    Digital Object Identifier: doi:10.1007/s002050000083
  • [2] Bally, V. and Caramellino, L. (2017). Convergence and regularity of probability laws by using an interpolation method. Ann. Probab. 45 1110–1159.
    Zentralblatt MATH: 1377.60066
    Digital Object Identifier: doi:10.1214/15-AOP1082
    Project Euclid: euclid.aop/1490947315
  • [3] Bally, V., Caramellino, L. and Cont, R. (2016). Stochastic Integration by Parts and Functional Itô Calculus. Advanced Courses in Mathematics. CRM Barcelona. Birkhäuser/Springer, Cham.
  • [4] Bally, V. and Clément, E. (2011). Integration by parts formula and applications to equations with jumps. Probab. Theory Related Fields 151 613–657.
    Zentralblatt MATH: 1243.60045
    Digital Object Identifier: doi:10.1007/s00440-010-0310-y
  • [5] Bally, V. and Fournier, N. (2011). Regularization properties of the 2D homogeneous Boltzmann equation without cutoff. Probab. Theory Related Fields 151 659–704.
    Zentralblatt MATH: 1237.82038
    Digital Object Identifier: doi:10.1007/s00440-010-0311-x
  • [6] Bennett, C. and Sharpley, R. (1988). Interpolation of Operators. Pure and Applied Mathematics 129. Academic Press, Boston, MA.
    Zentralblatt MATH: 0647.46057
  • [7] Bichteler, K., Gravereaux, J.-B. and Jacod, J. (1987). Malliavin Calculus for Processes with Jumps. Stochastics Monographs 2. Gordon and Breach Science Publishers, New York.
    Zentralblatt MATH: 0706.60057
  • [8] Bismut, J.-M. (1983). Calcul des variations stochastique et processus de sauts. Z. Wahrsch. Verw. Gebiete 63 147–235.
    Zentralblatt MATH: 0494.60082
    Digital Object Identifier: doi:10.1007/BF00538963
  • [9] Bouleau, N. and Denis, L. (2015). Dirichlet Forms Methods for Poisson Point Measures and Lévy Processes. Probability Theory and Stochastic Modelling 76. Springer, Cham.
  • [10] De Marco, S. (2011). Smoothness and asymptotic estimates of densities for SDEs with locally smooth coefficients and applications to square root-type diffusions. Ann. Appl. Probab. 21 1282–1321.
    Zentralblatt MATH: 1246.60082
    Digital Object Identifier: doi:10.1214/10-AAP717
    Project Euclid: euclid.aoap/1312818837
  • [11] Debussche, A. and Fournier, N. (2013). Existence of densities for stable-like driven SDE’s with Hölder continuous coefficients. J. Funct. Anal. 264 1757–1778.
    Zentralblatt MATH: 1272.60032
    Digital Object Identifier: doi:10.1016/j.jfa.2013.01.009
  • [12] Debussche, A. and Romito, M. (2014). Existence of densities for the 3D Navier–Stokes equations driven by Gaussian noise. Probab. Theory Related Fields 158 575–596.
    Zentralblatt MATH: 06288061
    Digital Object Identifier: doi:10.1007/s00440-013-0490-3
  • [13] Fournier, N. (2000). Existence and regularity study for two-dimensional Kac equation without cutoff by a probabilistic approach. Ann. Appl. Probab. 10 434–462.
    Zentralblatt MATH: 1056.60052
    Digital Object Identifier: doi:10.1214/aoap/1019487350
    Project Euclid: euclid.aoap/1019487350
  • [14] Fournier, N. (2015). Finiteness of entropy for the homogeneous Boltzmann equation with measure initial condition. Ann. Appl. Probab. 25 860–897.
    Zentralblatt MATH: 1322.82013
    Digital Object Identifier: doi:10.1214/14-AAP1012
    Project Euclid: euclid.aoap/1424355132
  • [15] Fournier, N. and Mouhot, C. (2009). On the well-posedness of the spatially homogeneous Boltzmann equation with a moderate angular singularity. Comm. Math. Phys. 289 803–824.
    Zentralblatt MATH: 1175.76129
    Digital Object Identifier: doi:10.1007/s00220-009-0807-3
  • [16] Fournier, N. and Printems, J. (2010). Absolute continuity for some one-dimensional processes. Bernoulli 16 343–360.
    Zentralblatt MATH: 1248.60062
    Digital Object Identifier: doi:10.3150/09-BEJ215
    Project Euclid: euclid.bj/1274821074
  • [17] Gamba, I. M., Panferov, V. and Villani, C. (2009). Upper Maxwellian bounds for the spatially homogeneous Boltzmann equation. Arch. Ration. Mech. Anal. 194 253–282.
    Zentralblatt MATH: 1273.76373
    Digital Object Identifier: doi:10.1007/s00205-009-0250-9
  • [18] Graham, C. and Méléard, S. (1999). Existence and regularity of a solution of a Kac equation without cutoff using the stochastic calculus of variations. Comm. Math. Phys. 205 551–569.
  • [19] Guerin, H., Meleard, S. and Nualart, E. (2006). Exponential estimates for spatially homogeneous Landau equations via the Malliavin calculus. J. Funct. Anal. 2 649–677.
    Zentralblatt MATH: 1108.60047
    Digital Object Identifier: doi:10.1016/j.jfa.2006.01.017
  • [20] Ishikawa, Y. (2013). Stochastic Calculus of Variations for Jump Processes. De Gruyter Studies in Mathematics 54. de Gruyter, Berlin.
    Zentralblatt MATH: 1301.60003
  • [21] Kolokoltsov, V. N. (2006). On the regularity of solutions to the spatially homogeneous Boltzmann equation with polynomially growing collision kernel. Adv. Stud. Contemp. Math. (Kyungshang) 12 9–38.
    Zentralblatt MATH: 1132.35338
  • [22] Lapeyre, B., Pardoux, É. and Sentis, R. (2003). Introduction to Monte-Carlo Methods for Transport and Diffusion Equations. Oxford Texts in Applied and Engineering Mathematics 6. Oxford Univ. Press, Oxford.
    Zentralblatt MATH: 1136.65133
  • [23] Léandre, R. (1985). Régularité de processus de sauts dégénérés. Ann. Inst. Henri Poincaré Probab. Stat. 21 125–146.
  • [24] Tanaka, H. (1978/79). Probabilistic treatment of the Boltzmann equation of Maxwellian molecules. Z. Wahrsch. Verw. Gebiete 46 67–105.
    Zentralblatt MATH: 0389.60079
    Digital Object Identifier: doi:10.1007/BF00535689

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