Non-existence of some approximately self-similar singularities for the Landau, Vlasov-Poisson-Landau, and Boltzmann equations

Bedrossian, Jacob; Gualdani, Maria; Snelson, Stanley

doi:10.1090/tran/8568

Non-existence of some approximately self-similar singularities for the Landau, Vlasov-Poisson-Landau, and Boltzmann equations
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by Jacob Bedrossian, Maria Pia Gualdani and Stanley Snelson PDF

Trans. Amer. Math. Soc. 375 (2022), 2187-2216 Request permission

Abstract:

We consider the homogeneous and inhomogeneous Landau equation for very soft and Coulomb potentials and show that approximate Type I self-similar blow-up solutions do not exist under mild decay assumptions on the profile. We extend our analysis to the Vlasov-Poisson-Landau system and to the Boltzmann equation without angular cut-off.

References

R. Alexandre, L. Desvillettes, C. Villani, and B. Wennberg, Entropy dissipation and long-range interactions, Arch. Ration. Mech. Anal. 152 (2000), no. 4, 327–355. MR 1765272, DOI 10.1007/s002050000083
Radjesvarane Alexandre, Yoshinori Morimoto, Seiji Ukai, Chao-Jiang Xu, and Tong Yang, Regularizing effect and local existence for the non-cutoff Boltzmann equation, Arch. Ration. Mech. Anal. 198 (2010), no. 1, 39–123. MR 2679369, DOI 10.1007/s00205-010-0290-1
Radjesvarane Alexandre, Yoshinori Morimoto, Seiji Ukai, Chao-Jiang Xu, and Tong Yang, Bounded solutions of the Boltzmann equation in the whole space, Kinet. Relat. Models 4 (2011), no. 1, 17–40. MR 2765735, DOI 10.3934/krm.2011.4.17
R. Alexandre and C. Villani, On the Landau approximation in plasma physics, Ann. Inst. H. Poincaré C Anal. Non Linéaire 21 (2004), no. 1, 61–95 (English, with English and French summaries). MR 2037247, DOI 10.1016/S0294-1449(03)00030-1
A. A. Arsen′ev and N. V. Peskov, The existence of a generalized solution of Landau’s equation, Ž. Vyčisl. Mat i Mat. Fiz. 17 (1977), no. 4, 1063–1068, 1096 (Russian). MR 470442
G. Barenblatt and Y. B. Zel’Dovich. Self-similar solutions as intermediate asymptotics, Annual Review of Fluid Mechanics, 4 (1972), no. 1, 285–312.
Grigory Isaakovich Barenblatt, Scaling, self-similarity, and intermediate asymptotics, Cambridge Texts in Applied Mathematics, vol. 14, Cambridge University Press, Cambridge, 1996. With a foreword by Ya. B. Zeldovich. MR 1426127, DOI 10.1017/CBO9781107050242
Adrien Blanchet and Philippe Laurençot, Finite mass self-similar blowing-up solutions of a chemotaxis system with non-linear diffusion, Commun. Pure Appl. Anal. 11 (2012), no. 1, 47–60. MR 2833337, DOI 10.3934/cpaa.2012.11.47
T. Buckmaster, S. Shkoller, and V. Vicol, Formation of shocks for 2D isentropic compressible Euler, Comm. Pure Appl. Math. (2020).
T. Buckmaster, S. Shkoller, and V. Vicol, Formation of point shocks for 3D compressible Euler, Comm. Pure Appl. Math. (2021).
Stephen Cameron, Luis Silvestre, and Stanley Snelson, Global a priori estimates for the inhomogeneous Landau equation with moderately soft potentials, Ann. Inst. H. Poincaré C Anal. Non Linéaire 35 (2018), no. 3, 625–642 (English, with English and French summaries). MR 3778645, DOI 10.1016/j.anihpc.2017.07.001
Kleber Carrapatoso, Laurent Desvillettes, and Lingbing He, Estimates for the large time behavior of the Landau equation in the Coulomb case, Arch. Ration. Mech. Anal. 224 (2017), no. 2, 381–420. MR 3614751, DOI 10.1007/s00205-017-1078-3
K. Carrapatoso and S. Mischler, Landau equation for very soft and Coulomb potentials near Maxwellians, Ann. PDE 3 (2017), no. 1, Paper No. 1, 65. MR 3625186, DOI 10.1007/s40818-017-0021-0
Dongho Chae, Nonexistence of asymptotically self-similar singularities in the Euler and the Navier-Stokes equations, Math. Ann. 338 (2007), no. 2, 435–449. MR 2302070, DOI 10.1007/s00208-007-0082-6
Dongho Chae and Roman Shvydkoy, On formation of a locally self-similar collapse in the incompressible Euler equations, Arch. Ration. Mech. Anal. 209 (2013), no. 3, 999–1017. MR 3067830, DOI 10.1007/s00205-013-0630-z
Dongho Chae and Tai-Peng Tsai, Remark on Luo-Hou’s ansatz for a self-similar solution to the 3D Euler equations, J. Nonlinear Sci. 25 (2015), no. 1, 193–202. MR 3302128, DOI 10.1007/s00332-014-9225-6
Dongho Chae and Jörg Wolf, On the local type I conditions for the 3D Euler equations, Arch. Ration. Mech. Anal. 230 (2018), no. 2, 641–663. MR 3842055, DOI 10.1007/s00205-018-1254-0
S. Chaturvedi, Stability of vacuum for the Landau equation with hard potentials, Preprint, arXiv:2001.07208, 2029.
S. Chaturvedi, J. Luk, and T. T. Nguyen, The Vlasov–Poisson–Landau system in the weakly collisional regime, Preprint, arXiv:2104.05692, 2021.
Jiajie Chen, Thomas Y. Hou, and De Huang, On the finite time blowup of the De Gregorio model for the 3D Euler equations, Comm. Pure Appl. Math. 74 (2021), no. 6, 1282–1350. MR 4242826, DOI 10.1002/cpa.21991
Yemin Chen, Laurent Desvillettes, and Lingbing He, Smoothing effects for classical solutions of the full Landau equation, Arch. Ration. Mech. Anal. 193 (2009), no. 1, 21–55. MR 2506070, DOI 10.1007/s00205-009-0223-z
J. Chern and M. Gualdani, Uniqueness of higher integrable solution to the Landau equation with Coulomb interactions, Math. Res. Lett. (2021).
Demetrios Christodoulou, The formation of shocks in 3-dimensional fluids, EMS Monographs in Mathematics, European Mathematical Society (EMS), Zürich, 2007. MR 2284927, DOI 10.4171/031
Demetrios Christodoulou and Shuang Miao, Compressible flow and Euler’s equations, Surveys of Modern Mathematics, vol. 9, International Press, Somerville, MA; Higher Education Press, Beijing, 2014. MR 3288725
Charles Collot, Nonradial type II blow up for the energy-supercritical semilinear heat equation, Anal. PDE 10 (2017), no. 1, 127–252. MR 3611015, DOI 10.2140/apde.2017.10.127
C. Collot, T.-E. Ghoul, and N. Masmoudi, Singularity formation for Burgers equation with transverse viscosity. Preprint, arXiv:1803.07826, 2018.
C. Collot, T.-E. Ghoul, N. Masmoudi, and V. T. Nguyen, Refined description and stability for singular solutions of the 2D Keller-Segel system, Preprint, arXiv:1912.00721, 2019.
L. Desvillettes, Entropy dissipation estimates for the Landau equation in the Coulomb case and applications, J. Funct. Anal. 269 (2015), no. 5, 1359–1403. MR 3369941, DOI 10.1016/j.jfa.2015.05.009
L. Desvillettes, L.-B. He, and J.-C. Jiang, A new monotonicity formula for the spatially homogeneous Landau equation with Coulomb potential and its applications, Preprint, arXiv:2011.00386, 2020.
Renjun Duan and Hongjun Yu, The Vlasov-Poisson-Landau system near a local Maxwellian, Adv. Math. 362 (2020), 106956, 83. MR 4050580, DOI 10.1016/j.aim.2019.106956
J. Eggers and M. A. Fontelos, Singularities: formation, structure, and propagation, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2015. MR 3468749, DOI 10.1017/CBO9781316161692
T. M. Elgindi, Finite-time singularity formation for $C^{1,\alpha }$ solutions to the incompressible Euler equations on $\mathbb R^3$, Preprint, arXiv:1904.04795, 2019.
Tarek M. Elgindi and In-Jee Jeong, Finite-time singularity formation for strong solutions to the axi-symmetric 3D Euler equations, Ann. PDE 5 (2019), no. 2, Paper No. 16, 51. MR 4029562, DOI 10.1007/s40818-019-0071-6
Nicolas Fournier, Uniqueness of bounded solutions for the homogeneous Landau equation with a Coulomb potential, Comm. Math. Phys. 299 (2010), no. 3, 765–782. MR 2718931, DOI 10.1007/s00220-010-1113-9
T.-E. Ghoul and N. Masmoudi, Stability of infinite time blow up for the Patlak Keller Segel system, Preprint, arXiv:1610.00456, 2016.
F. Golse, M. Gualdani, C. Imbert, and A. Vasseur, Partial regularity in time for the space homogeneous Landau equation with Coulomb potential, Ann. Sci. Ec. Norm. Super. (4), To appear.
François Golse, Cyril Imbert, Clément Mouhot, and Alexis F. Vasseur, Harnack inequality for kinetic Fokker-Planck equations with rough coefficients and application to the Landau equation, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 19 (2019), no. 1, 253–295. MR 3923847
Philip T. Gressman and Robert M. Strain, Global classical solutions of the Boltzmann equation without angular cut-off, J. Amer. Math. Soc. 24 (2011), no. 3, 771–847. MR 2784329, DOI 10.1090/S0894-0347-2011-00697-8
Maria Pia Gualdani and Nestor Guillen, Estimates for radial solutions of the homogeneous Landau equation with Coulomb potential, Anal. PDE 9 (2016), no. 8, 1772–1809. MR 3599518, DOI 10.2140/apde.2016.9.1772
Maria Gualdani and Nestor Guillen, On $A_p$ weights and the Landau equation, Calc. Var. Partial Differential Equations 58 (2019), no. 1, Paper No. 17, 55. MR 3884792, DOI 10.1007/s00526-018-1451-6
Yan Guo, The Landau equation in a periodic box, Comm. Math. Phys. 231 (2002), no. 3, 391–434. MR 1946444, DOI 10.1007/s00220-002-0729-9
Yan Guo, The Vlasov-Poisson-Landau system in a periodic box, J. Amer. Math. Soc. 25 (2012), no. 3, 759–812. MR 2904573, DOI 10.1090/S0894-0347-2011-00722-4
Lingbing He, Well-posedness of spatially homogeneous Boltzmann equation with full-range interaction, Comm. Math. Phys. 312 (2012), no. 2, 447–476. MR 2917172, DOI 10.1007/s00220-012-1481-4
Lingbing He and Xiongfeng Yang, Well-posedness and asymptotics of grazing collisions limit of Boltzmann equation with Coulomb interaction, SIAM J. Math. Anal. 46 (2014), no. 6, 4104–4165. MR 3505177, DOI 10.1137/140965983
Christopher Henderson and Stanley Snelson, $C^\infty$ smoothing for weak solutions of the inhomogeneous Landau equation, Arch. Ration. Mech. Anal. 236 (2020), no. 1, 113–143. MR 4072211, DOI 10.1007/s00205-019-01465-7
Christopher Henderson, Stanley Snelson, and Andrei Tarfulea, Local solutions of the Landau equation with rough, slowly decaying initial data, Ann. Inst. H. Poincaré C Anal. Non Linéaire 37 (2020), no. 6, 1345–1377 (English, with English and French summaries). MR 4168919, DOI 10.1016/j.anihpc.2020.04.004
Christopher Henderson, Stanley Snelson, and Andrei Tarfulea, Local existence, lower mass bounds, and a new continuation criterion for the Landau equation, J. Differential Equations 266 (2019), no. 2-3, 1536–1577. MR 3906224, DOI 10.1016/j.jde.2018.08.005
Christopher Henderson, Stanley Snelson, and Andrei Tarfulea, Local well-posedness of the Boltzmann equation with polynomially decaying initial data, Kinet. Relat. Models 13 (2020), no. 4, 837–867. MR 4112183, DOI 10.3934/krm.2020029
Thomas Y. Hou and Pengfei Liu, Self-similar singularity of a 1D model for the 3D axisymmetric Euler equations, Res. Math. Sci. 2 (2015), Art. 5, 26. MR 3341705, DOI 10.1186/s40687-015-0021-1
Cyril Imbert, Clément Mouhot, and Luis Silvestre, Decay estimates for large velocities in the Boltzmann equation without cutoff, J. Éc. polytech. Math. 7 (2020), 143–184 (English, with English and French summaries). MR 4033752, DOI 10.5802/jep.113
Cyril Imbert, Clément Mouhot, and Luis Silvestre, Gaussian lower bounds for the Boltzmann equation without cutoff, SIAM J. Math. Anal. 52 (2020), no. 3, 2930–2944. MR 4112729, DOI 10.1137/19M1252375
Cyril Imbert and Luis Silvestre, The weak Harnack inequality for the Boltzmann equation without cut-off, J. Eur. Math. Soc. (JEMS) 22 (2020), no. 2, 507–592. MR 4049224, DOI 10.4171/jems/928
C. Imbert and L. Silvestre, Global regularity estimates for the Boltzmann equation without cut-off, J. Amer. Math. Soc. To appear.
Yoshikazu Giga and Robert V. Kohn, Asymptotically self-similar blow-up of semilinear heat equations, Comm. Pure Appl. Math. 38 (1985), no. 3, 297–319. MR 784476, DOI 10.1002/cpa.3160380304
Mohammed Lemou, Florian Méhats, and Pierre Raphael, The orbital stability of the ground states and the singularity formation for the gravitational Vlasov Poisson system, Arch. Ration. Mech. Anal. 189 (2008), no. 3, 425–468. MR 2424993, DOI 10.1007/s00205-008-0126-4
Mohammed Lemou, Florian Méhats, and Pierre Raphaël, Stable self-similar blow up dynamics for the three dimensional relativistic gravitational Vlasov-Poisson system, J. Amer. Math. Soc. 21 (2008), no. 4, 1019–1063. MR 2425179, DOI 10.1090/S0894-0347-07-00579-6
Shuangqian Liu and Xuan Ma, Regularizing effects for the classical solutions to the Landau equation in the whole space, J. Math. Anal. Appl. 417 (2014), no. 1, 123–143. MR 3191417, DOI 10.1016/j.jmaa.2014.03.006
Xuguang Lu and Clément Mouhot, On measure solutions of the Boltzmann equation, part I: moment production and stability estimates, J. Differential Equations 252 (2012), no. 4, 3305–3363. MR 2871802, DOI 10.1016/j.jde.2011.10.021
Jonathan Luk, Stability of vacuum for the Landau equation with moderately soft potentials, Ann. PDE 5 (2019), no. 1, Paper No. 11, 101. MR 3948345, DOI 10.1007/s40818-019-0067-2
G. Luo and T. Y. Hou, Potentially singular solutions of the 3D axisymmetric Euler equations, Proc. Nat. Acad. Sci. India 111 (2014), no. 36, 12968–12973.
Hiroshi Matano and Frank Merle, On nonexistence of type II blowup for a supercritical nonlinear heat equation, Comm. Pure Appl. Math. 57 (2004), no. 11, 1494–1541. MR 2077706, DOI 10.1002/cpa.20044
Frank Merle and Pierre Raphael, The blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation, Ann. of Math. (2) 161 (2005), no. 1, 157–222. MR 2150386, DOI 10.4007/annals.2005.161.157
F. Merle, P. Raphael, I. Rodnianski, and J. Szeftel, On smooth self similar solutions to the compressible Euler equations, Preprint, arXiv:1912.10998, 2019.
F. Merle, P. Raphael, I. Rodnianski, and J. Szeftel, On the implosion of a three dimensional compressible fluid, Preprint, arXiv:1912.11009, 2019.
Yoshinori Morimoto, Shuaikun Wang, and Tong Yang, Measure valued solutions to the spatially homogeneous Boltzmann equation without angular cutoff, J. Stat. Phys. 165 (2016), no. 5, 866–906. MR 3572500, DOI 10.1007/s10955-016-1655-0
J. Nečas, M. Růžička, and V. Šverák, On Leray’s self-similar solutions of the Navier-Stokes equations, Acta Math. 176 (1996), no. 2, 283–294. MR 1397564, DOI 10.1007/BF02551584
Pavol Quittner and Philippe Souplet, Superlinear parabolic problems, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser/Springer, Cham, 2019. Blow-up, global existence and steady states; Second edition of [ MR2346798]. MR 3967048, DOI 10.1007/978-3-030-18222-9
Pierre Raphaël and Rémi Schweyer, On the stability of critical chemotactic aggregation, Math. Ann. 359 (2014), no. 1-2, 267–377. MR 3201901, DOI 10.1007/s00208-013-1002-6
G. Seregin and V. Šverák, On type I singularities of the local axi-symmetric solutions of the Navier-Stokes equations, Comm. Partial Differential Equations 34 (2009), no. 1-3, 171–201. MR 2512858, DOI 10.1080/03605300802683687
Luis Silvestre, A new regularization mechanism for the Boltzmann equation without cut-off, Comm. Math. Phys. 348 (2016), no. 1, 69–100. MR 3551261, DOI 10.1007/s00220-016-2757-x
Luis Silvestre, Upper bounds for parabolic equations and the Landau equation, J. Differential Equations 262 (2017), no. 3, 3034–3055. MR 3582250, DOI 10.1016/j.jde.2016.11.010
Robert M. Strain and Keya Zhu, The Vlasov-Poisson-Landau system in $\Bbb {R}^3_x$, Arch. Ration. Mech. Anal. 210 (2013), no. 2, 615–671. MR 3101794, DOI 10.1007/s00205-013-0658-0
Tai-Peng Tsai, On Leray’s self-similar solutions of the Navier-Stokes equations satisfying local energy estimates, Arch. Rational Mech. Anal. 143 (1998), no. 1, 29–51. MR 1643650, DOI 10.1007/s002050050099
Cédric Villani, On the Cauchy problem for Landau equation: sequential stability, global existence, Adv. Differential Equations 1 (1996), no. 5, 793–816. MR 1392006