## The Annals of Applied Probability

### Finiteness of entropy for the homogeneous Boltzmann equation with measure initial condition

Nicolas Fournier

#### Abstract

We consider the $3D$ spatially homogeneous Boltzmann equation for (true) hard and moderately soft potentials. We assume that the initial condition is a probability measure with finite energy and is not a Dirac mass. For hard potentials, we prove that any reasonable weak solution immediately belongs to some Besov space. For moderately soft potentials, we assume additionally that the initial condition has a moment of sufficiently high order ($8$ is enough) and prove the existence of a solution that immediately belongs to some Besov space. The considered solutions thus instantaneously become functions with a finite entropy. We also prove that in any case, any weak solution is immediately supported by $\mathbb{R}^{3}$.

#### Article information

Source
Ann. Appl. Probab., Volume 25, Number 2 (2015), 860-897.

Dates
First available in Project Euclid: 19 February 2015

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1424355132

Digital Object Identifier
doi:10.1214/14-AAP1012

Mathematical Reviews number (MathSciNet)
MR3313757

Zentralblatt MATH identifier
1322.82013

#### Citation

Fournier, Nicolas. Finiteness of entropy for the homogeneous Boltzmann equation with measure initial condition. Ann. Appl. Probab. 25 (2015), no. 2, 860--897. doi:10.1214/14-AAP1012. https://projecteuclid.org/euclid.aoap/1424355132

#### References

• [1] Aldous, D. (1978). Stopping times and tightness. Ann. Probability 6 335–340.
• [2] Alexandre, R. (2009). A review of Boltzmann equation with singular kernels. Kinet. Relat. Models 2 551–646.
• [3] Alexandre, R., Desvillettes, L., Villani, C. and Wennberg, B. (2000). Entropy dissipation and long-range interactions. Arch. Ration. Mech. Anal. 152 327–355.
• [4] Alexandre, R. and Elsafadi, M. (2009). Littlewood–Paley theory and regularity issues in Boltzmann homogeneous equations. II. Non cutoff case and non Maxwellian molecules. Discrete Contin. Dyn. Syst. 24 1–11.
• [5] Bally, V. and Fournier, N. (2011). Regularization properties of the 2D homogeneous Boltzmann equation without cutoff. Probab. Theory Related Fields 151 659–704.
• [6] Carleman, T. (1933). Sur la théorie de l’équation intégrodifférentielle de Boltzmann. Acta Math. 60 91–146.
• [7] Cercignani, C. (1988). The Boltzmann Equation and Its Applications. Applied Mathematical Sciences 67. Springer, New York.
• [8] Chen, Y. and He, L. (2011). Smoothing estimates for Boltzmann equation with full-range interactions: Spatially homogeneous case. Arch. Ration. Mech. Anal. 201 501–548.
• [9] 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.
• [10] Desvillettes, L. (1993). Some applications of the method of moments for the homogeneous Boltzmann and Kac equations. Arch. Rational Mech. Anal. 123 387–404.
• [11] Desvillettes, L. (1995). About the regularizing properties of the non-cut-off Kac equation. Comm. Math. Phys. 168 417–440.
• [12] Desvillettes, L. (1997). Regularization properties of the $2$-dimensional non-radially symmetric non-cutoff spatially homogeneous Boltzmann equation for Maxwellian molecules. Transport Theory Statist. Phys. 26 341–357.
• [13] Desvillettes, L. and Mouhot, C. (2009). Stability and uniqueness for the spatially homogeneous Boltzmann equation with long-range interactions. Arch. Ration. Mech. Anal. 193 227–253.
• [14] Desvillettes, L. and Wennberg, B. (2004). Smoothness of the solution of the spatially homogeneous Boltzmann equation without cutoff. Comm. Partial Differential Equations 29 133–155.
• [15] Elmroth, T. (1983). Global boundedness of moments of solutions of the Boltzmann equation for forces of infinite range. Arch. Rational Mech. Anal. 82 1–12.
• [16] Fournier, N. (2000). Existence and regularity study for two-dimensional Kac equation without cutoff by a probabilistic approach. Ann. Appl. Probab. 10 434–462.
• [17] Fournier, N. (2001). Strict positivity of the solution to a 2-dimensional spatially homogeneous Boltzmann equation without cutoff. Ann. Inst. Henri Poincaré Probab. Stat. 37 481–502.
• [18] Fournier, N. (2006). Uniqueness for a class of spatially homogeneous Boltzmann equations without angular cutoff. J. Stat. Phys. 125 927–946.
• [19] Fournier, N. and Giet, J.-S. (2004). Exact simulation of nonlinear coagulation processes. Monte Carlo Methods Appl. 10 95–106.
• [20] Fournier, N. and Guérin, H. (2008). On the uniqueness for the spatially homogeneous Boltzmann equation with a strong angular singularity. J. Stat. Phys. 131 749–781.
• [21] Fournier, N. and Méléard, S. (2002). A stochastic particle numerical method for 3D Boltzmann equations without cutoff. Math. Comp. 71 583–604 (electronic).
• [22] 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.
• [23] Fournier, N. and Printems, J. (2010). Absolute continuity for some one-dimensional processes. Bernoulli 16 343–360.
• [24] 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.
• [25] 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.
• [26] Horowitz, J. and Karandikar, R. L. (1990). Martingale problems associated with the Boltzmann equation. In Seminar on Stochastic Processes, 1989 (San Diego, CA, 1989). Progress in Probability 18 75–122. Birkhäuser, Boston, MA.
• [27] Huo, Z., Morimoto, Y., Ukai, S. and Yang, T. (2008). Regularity of solutions for spatially homogeneous Boltzmann equation without angular cutoff. Kinet. Relat. Models 1 453–489.
• [28] Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes, 2nd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 288. Springer, Berlin.
• [29] Lu, X. and Mouhot, C. (2012). On measure solutions of the Boltzmann equation, part I: Moment production and stability estimates. J. Differential Equations 252 3305–3363.
• [30] Mouhot, C. (2005). Quantitative lower bounds for the full Boltzmann equation. I. Periodic boundary conditions. Comm. Partial Differential Equations 30 881–917.
• [31] Mouhot, C. and Villani, C. (2004). Regularity theory for the spatially homogeneous Boltzmann equation with cut-off. Arch. Ration. Mech. Anal. 173 169–212.
• [32] Pulvirenti, A. and Wennberg, B. (1997). A Maxwellian lower bound for solutions to the Boltzmann equation. Comm. Math. Phys. 183 145–160.
• [33] Runst, T. and Sickel, W. (1996). Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations. de Gruyter Series in Nonlinear Analysis and Applications 3. de Gruyter, Berlin.
• [34] Schilling, R. L., Sztonyk, P. and Wang, J. (2012). Coupling property and gradient estimates of Lévy processes via the symbol. Bernoulli 18 1128–1149.
• [35] Tanaka, H. (1978/79). Probabilistic treatment of the Boltzmann equation of Maxwellian molecules. Z. Wahrsch. Verw. Gebiete 46 67–105.
• [36] Toscani, G. and Villani, C. (1999). Probability metrics and uniqueness of the solution to the Boltzmann equation for a Maxwell gas. J. Stat. Phys. 94 619–637.
• [37] Villani, C. (1998). On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations. Arch. Rational Mech. Anal. 143 273–307.
• [38] Villani, C. (2002). A review of mathematical topics in collisional kinetic theory. In Handbook of Mathematical Fluid Dynamics, Vol. I 71–305. North-Holland, Amsterdam.
• [39] Zhang, X. and Zhang, X. (2006). Supports of measure solutions for spatially homogeneous Boltzmann equations. J. Stat. Phys. 124 485–495.