Finite energy weak solutions of 2d Boussinesq equations with diffusive temperature

Tianwen Luo; Tao Tao; Liqun Zhang

doi:10.3934/dcds.2019230

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June 2020, 40(6): 3737-3765. doi: 10.3934/dcds.2019230

Finite energy weak solutions of 2d Boussinesq equations with diffusive temperature

Tianwen Luo ^1, , Tao Tao ^2, and Liqun Zhang ^3,,

1.	Yau Mathematical Science center, Tsinghua University, Beijing 100084, China
2.	School of Mathematics Sciences, Shandong University, Jinan 250100, China
3.	Academy of Mathematic and System Science, CAS; School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100190, China

^* Corresponding author: Liqun Zhang

Received December 2018 Revised March 2019 Published June 2019

Fund Project: The first author is supported in part by NSFC Grants 11601258.6. The second author is supported by the fundamental research funds of Shandong university under Grant 11140078614006. The third author is partially supported by the Chinese NSF under Grant 11471320 and 11631008

Abstract
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We show the existence of finite kinetic energy solution with prescribed kinetic energy to the 2d Boussinesq equations with diffusive temperature on torus.

Keywords: Finite energy weak solution, Boussinesq equation, prescribed kinetic energy.

Mathematics Subject Classification: 35Q30, 76D03.

Citation: Tianwen Luo, Tao Tao, Liqun Zhang. Finite energy weak solutions of 2d Boussinesq equations with diffusive temperature. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3737-3765. doi: 10.3934/dcds.2019230

References:

[1]	T. Buckmaster, Onsager's conjecture almost everywhere in time, Comm. Math. Phys., 333 (2015), 1175-1198. doi: 10.1007/s00220-014-2262-z.
[2]	T. Buckmaster, M. Colombo and V. Vicol, Wild solutions of the Navier-Stokes equations whose singular sets in time have Hausdorff dimension strictly less than 1, preprint, arXiv: 1809.00600.
[3]	T. Buckmaster, C. De Lellis, P. Isett and L. Székelyhidi Jr., Anomalous dissipation for 1/5-Hölder Euler flows, Ann. of. Math., 182 (2015), 127-172. doi: 10.4007/annals.2015.182.1.3.
[4]	T. Buckmaster, C. De Lellis and L. Székelyhidi, Jr., Transporting microstructure and dissipative Euler flows, preprint, arXiv: 1302.2825.
[5]	T. Buckmaster, C. De Lellis and L. Székelyhidi Jr., Dissipative Euler flows with Onsager-critical spatial regularity, Comm. Pure Appl. Math., 69 (2016), 1613-1670. doi: 10.1002/cpa.21586.
[6]	T. Buckmaster, C. De Lellis, L. Székelyhidi Jr. and V. Vicol, Onsager conjecture for admissible weak solution, Comm. Pure Appl. Math., 72 (2019), 229-274. doi: 10.1002/cpa.21781.
[7]	T. Buckmaster, Shkoller and V. Vicol, Nonuniqueness of weak solutions to SQG equation, to appear in Comm. Pure Appl. Math.
[8]	T. Buckmaster and V. Vicol, Nonuniqueness of weak solutions to Navier-Stokes equation, Ann. of Math., 189 (2019), 101-144. doi: 10.4007/annals.2019.189.1.3.
[9]	D. Chae, Global regularity for the 2-D Boussinesq equation with partial viscous terms, Adv. Math., 203 (2006), 497-513. doi: 10.1016/j.aim.2005.05.001.
[10]	A. Cheskidov, P. Constantin, S. Friedlander and R. Shvydkoy, Energy conservation and Onsager's conjecture for the Euler equations, Nonlinearity, 21 (2008), 1233-1252. doi: 10.1088/0951-7715/21/6/005.
[11]	A. Choffrut, H-principles for the incompressible Euler equations, Arch. Ration. Mech. Anal., 210 (2013), 133-163. doi: 10.1007/s00205-013-0639-3.
[12]	P. Constantin, E. W and E. S. Titi, Onsager's conjecture on the energy conservation for solutions of Euler's equation, Comm. Math. Phys., 165 (1994), 207-209. doi: 10.1007/BF02099744.
[13]	S. Daneri, Cauchy problem for dissipative Hölder solutions to the incompressible Euler equations, Comm. Math. Phy., 329 (2014), 745-786. doi: 10.1007/s00220-014-1973-5.
[14]	S. Daneri and L. Székelyhidi Jr., Non-uniqueness and h-principle for Hölder continuous weak solution of Euler equation, Arch. Ration. Mech. Anal., 224 (2017), 471-514. doi: 10.1007/s00205-017-1081-8.
[15]	C. De Lellis and L. Székelyhidi Jr., The Euler equation as a differential inclusion, Ann. of Math., 170 (2009), 1417-1436. doi: 10.4007/annals.2009.170.1417.
[16]	C. De Lellis and L. Székelyhidi Jr., On admissibility criteria for weak solutions of the Euler equations, Arch. Ration. Mech. Anal., 195 (2010), 225-260. doi: 10.1007/s00205-008-0201-x.
[17]	C. De Lellis and L. Székelyhidi Jr., Dissipative continuous Euler flows, Invent. Math., 193 (2013), 377-407. doi: 10.1007/s00222-012-0429-9.
[18]	C. De Lellis and L. Székelyhidi Jr., Dissipative Euler flows and Onsager's conjecture, Jour. Eur. Math. Soc., 16 (2014), 1467-1505. doi: 10.4171/JEMS/466.
[19]	J. Duchon and R. Raoul, Inertial energy dissipation for weak solutions of incompressible Euler and Navier-Stokes equations, Nonlinearity, 13 (2000), 249-255. doi: 10.1088/0951-7715/13/1/312.
[20]	T. Hou and C. Li, Global well-posedness of the viscous Boussinesq equations, DCDS, Series A, 12 (2005), 1-12. doi: 10.3934/dcds.2005.12.1.
[21]	P. Isett and S.-J. Oh, A heat flow approach to Onsager's conjecture for the Euler equations on manifolds, Trans. Amer. Math. Soc., 368 (2016), 6519-6537. doi: 10.1090/tran/6549.
[22]	P. Isett and S.-J. Oh, On nonperiodic Euler flows with Hölder regularity, Arch. Ration. Mech. Anal., 221 (2016), 725-804. doi: 10.1007/s00205-016-0973-3.
[23]	P. Isett, Hölder continuous Euler flows in three dimensions with compact support in time, preprint, arXiv: 1211.4065. doi: 10.1515/9781400885428.
[24]	P. Isett, A proof of Onsager's conjecture, Ann. of. Math., 188 (2018), 871-963. doi: 10.4007/annals.2018.188.3.4.
[25]	P. Isett, On the endpoint regularity in Onsager's conjecture, preprint, arXiv: 1706.01549
[26]	P. Isett and V. Vicol, H ölder continuous solutions of active scalar equations, Ann. of. PDE. doi: 10.1007/s40818-015-0002-0.
[27]	T. Luo and Titi, Non-uniqueness of Weak Solutions to Hyperviscous Navier-Stokes Equations - On Sharpness of J.-L. Lions Exponent, preprint, arXiv: 1808.07595.
[28]	T. Luo, T. Tao and L. Zhang, Hölder continuous soltion of 2d Boussinesq equation with diffusive temperture, preprint, arXiv: 1901.10071.
[29]	T. Luo and Z. Xin, Hölder continuous solutions to the 3d Prandtl system, preprint, arXiv: 1804.04285.
[30]	X. Luo, Stationary solution and nonuniquenes of weak solution for the Navier-Stokes euation on high dimensions, preprint, arXiv: 1807.09318.
[31]	A. Majda, Introduction to PDEs and Waves for the Atmosphere and Ocean, Courant Lecture Notes in Mathematics, Vol. 9. AMS/CIMS, 2003. doi: 10.1090/cln/009.
[32]	S. Modena and L. Székelyhidi, Jr., Non-uniqueness for the transport equation with Sobolev vector fields, to appear in Ann. PDE. doi: 10.1007/s40818-018-0056-x.
[33]	S. Modena and L. Székelyhidi, Jr., Non-Renormalized solution to the continuity equation, preprint, arXiv: 1806.09145.
[34]	L. Onsager, Statistical hydrodynamics, Nuovo Cimento, 9 (1949), 279-287. doi: 10.1007/BF02780991.
[35]	J. Pedlosky, Geophysical Fluid Dynamics, Springer, New York, 1987.
[36]	V. Scheffer, An inviscid flow with compact support in space-time, J. Geom. Anal., 3 (1993), 343-401. doi: 10.1007/BF02921318.
[37]	A. Shnirelman, Weak solution with decreasing energy of incompressible Euler equations, Comm. Math. Phys., 210 (2000), 541-603. doi: 10.1007/s002200050791.
[38]	A. Shnirelman, On the nonuniqueness of weak solution of Euler equation, Comm. Pure Appl. Math., 50 (1997), 1261-1286. doi: 10.1002/(SICI)1097-0312(199712)50:12<1261::AID-CPA3>3.0.CO;2-6.
[39]	R. Shvydkoy, Convex integration for a class of active scalar equations, J. Amer. Math. Soc., 24 (2011), 1159-1174. doi: 10.1090/S0894-0347-2011-00705-4.
[40]	R. Shvydkoy, Lectures on the Onsager conjecture, DCDS, Series S, 3 (2010), 473-496. doi: 10.3934/dcdss.2010.3.473.
[41]	L. Székelyhidi, Jr., From Isometric Embeddings to Turbulence, HCDTE lecture notes. Part Ⅱ. Nonlinear hyperbolic PDEs, dispersive and transport equations, 7: 63, 2012.
[42]	T. Tao and L. Zhang, On the continuous periodic weak solution of Boussinesq equations, SIAM, J. Math. Anal., 50 (2018), 1120-1162. doi: 10.1137/17M1115526.
[43]	T. Tao and L. Zhang, Hölder continuous solution of Boussinesq equations with compact support, J. Funct. Anal., 272 (2017), 4334-4402. doi: 10.1016/j.jfa.2017.01.013.
[44]	T. Tao and L. Zhang, Hölder continuous periodic solution of Boussinesq equations with partial viscosity, Calc. Var. Partial Differential Equations. doi: 10.1007/s00526-018-1337-7.

show all references

References:

[1]	T. Buckmaster, Onsager's conjecture almost everywhere in time, Comm. Math. Phys., 333 (2015), 1175-1198. doi: 10.1007/s00220-014-2262-z.
[2]	T. Buckmaster, M. Colombo and V. Vicol, Wild solutions of the Navier-Stokes equations whose singular sets in time have Hausdorff dimension strictly less than 1, preprint, arXiv: 1809.00600.
[3]	T. Buckmaster, C. De Lellis, P. Isett and L. Székelyhidi Jr., Anomalous dissipation for 1/5-Hölder Euler flows, Ann. of. Math., 182 (2015), 127-172. doi: 10.4007/annals.2015.182.1.3.
[4]	T. Buckmaster, C. De Lellis and L. Székelyhidi, Jr., Transporting microstructure and dissipative Euler flows, preprint, arXiv: 1302.2825.
[5]	T. Buckmaster, C. De Lellis and L. Székelyhidi Jr., Dissipative Euler flows with Onsager-critical spatial regularity, Comm. Pure Appl. Math., 69 (2016), 1613-1670. doi: 10.1002/cpa.21586.
[6]	T. Buckmaster, C. De Lellis, L. Székelyhidi Jr. and V. Vicol, Onsager conjecture for admissible weak solution, Comm. Pure Appl. Math., 72 (2019), 229-274. doi: 10.1002/cpa.21781.
[7]	T. Buckmaster, Shkoller and V. Vicol, Nonuniqueness of weak solutions to SQG equation, to appear in Comm. Pure Appl. Math.
[8]	T. Buckmaster and V. Vicol, Nonuniqueness of weak solutions to Navier-Stokes equation, Ann. of Math., 189 (2019), 101-144. doi: 10.4007/annals.2019.189.1.3.
[9]	D. Chae, Global regularity for the 2-D Boussinesq equation with partial viscous terms, Adv. Math., 203 (2006), 497-513. doi: 10.1016/j.aim.2005.05.001.
[10]	A. Cheskidov, P. Constantin, S. Friedlander and R. Shvydkoy, Energy conservation and Onsager's conjecture for the Euler equations, Nonlinearity, 21 (2008), 1233-1252. doi: 10.1088/0951-7715/21/6/005.
[11]	A. Choffrut, H-principles for the incompressible Euler equations, Arch. Ration. Mech. Anal., 210 (2013), 133-163. doi: 10.1007/s00205-013-0639-3.
[12]	P. Constantin, E. W and E. S. Titi, Onsager's conjecture on the energy conservation for solutions of Euler's equation, Comm. Math. Phys., 165 (1994), 207-209. doi: 10.1007/BF02099744.
[13]	S. Daneri, Cauchy problem for dissipative Hölder solutions to the incompressible Euler equations, Comm. Math. Phy., 329 (2014), 745-786. doi: 10.1007/s00220-014-1973-5.
[14]	S. Daneri and L. Székelyhidi Jr., Non-uniqueness and h-principle for Hölder continuous weak solution of Euler equation, Arch. Ration. Mech. Anal., 224 (2017), 471-514. doi: 10.1007/s00205-017-1081-8.
[15]	C. De Lellis and L. Székelyhidi Jr., The Euler equation as a differential inclusion, Ann. of Math., 170 (2009), 1417-1436. doi: 10.4007/annals.2009.170.1417.
[16]	C. De Lellis and L. Székelyhidi Jr., On admissibility criteria for weak solutions of the Euler equations, Arch. Ration. Mech. Anal., 195 (2010), 225-260. doi: 10.1007/s00205-008-0201-x.
[17]	C. De Lellis and L. Székelyhidi Jr., Dissipative continuous Euler flows, Invent. Math., 193 (2013), 377-407. doi: 10.1007/s00222-012-0429-9.
[18]	C. De Lellis and L. Székelyhidi Jr., Dissipative Euler flows and Onsager's conjecture, Jour. Eur. Math. Soc., 16 (2014), 1467-1505. doi: 10.4171/JEMS/466.
[19]	J. Duchon and R. Raoul, Inertial energy dissipation for weak solutions of incompressible Euler and Navier-Stokes equations, Nonlinearity, 13 (2000), 249-255. doi: 10.1088/0951-7715/13/1/312.
[20]	T. Hou and C. Li, Global well-posedness of the viscous Boussinesq equations, DCDS, Series A, 12 (2005), 1-12. doi: 10.3934/dcds.2005.12.1.
[21]	P. Isett and S.-J. Oh, A heat flow approach to Onsager's conjecture for the Euler equations on manifolds, Trans. Amer. Math. Soc., 368 (2016), 6519-6537. doi: 10.1090/tran/6549.
[22]	P. Isett and S.-J. Oh, On nonperiodic Euler flows with Hölder regularity, Arch. Ration. Mech. Anal., 221 (2016), 725-804. doi: 10.1007/s00205-016-0973-3.
[23]	P. Isett, Hölder continuous Euler flows in three dimensions with compact support in time, preprint, arXiv: 1211.4065. doi: 10.1515/9781400885428.
[24]	P. Isett, A proof of Onsager's conjecture, Ann. of. Math., 188 (2018), 871-963. doi: 10.4007/annals.2018.188.3.4.
[25]	P. Isett, On the endpoint regularity in Onsager's conjecture, preprint, arXiv: 1706.01549
[26]	P. Isett and V. Vicol, H ölder continuous solutions of active scalar equations, Ann. of. PDE. doi: 10.1007/s40818-015-0002-0.
[27]	T. Luo and Titi, Non-uniqueness of Weak Solutions to Hyperviscous Navier-Stokes Equations - On Sharpness of J.-L. Lions Exponent, preprint, arXiv: 1808.07595.
[28]	T. Luo, T. Tao and L. Zhang, Hölder continuous soltion of 2d Boussinesq equation with diffusive temperture, preprint, arXiv: 1901.10071.
[29]	T. Luo and Z. Xin, Hölder continuous solutions to the 3d Prandtl system, preprint, arXiv: 1804.04285.
[30]	X. Luo, Stationary solution and nonuniquenes of weak solution for the Navier-Stokes euation on high dimensions, preprint, arXiv: 1807.09318.
[31]	A. Majda, Introduction to PDEs and Waves for the Atmosphere and Ocean, Courant Lecture Notes in Mathematics, Vol. 9. AMS/CIMS, 2003. doi: 10.1090/cln/009.
[32]	S. Modena and L. Székelyhidi, Jr., Non-uniqueness for the transport equation with Sobolev vector fields, to appear in Ann. PDE. doi: 10.1007/s40818-018-0056-x.
[33]	S. Modena and L. Székelyhidi, Jr., Non-Renormalized solution to the continuity equation, preprint, arXiv: 1806.09145.
[34]	L. Onsager, Statistical hydrodynamics, Nuovo Cimento, 9 (1949), 279-287. doi: 10.1007/BF02780991.
[35]	J. Pedlosky, Geophysical Fluid Dynamics, Springer, New York, 1987.
[36]	V. Scheffer, An inviscid flow with compact support in space-time, J. Geom. Anal., 3 (1993), 343-401. doi: 10.1007/BF02921318.
[37]	A. Shnirelman, Weak solution with decreasing energy of incompressible Euler equations, Comm. Math. Phys., 210 (2000), 541-603. doi: 10.1007/s002200050791.
[38]	A. Shnirelman, On the nonuniqueness of weak solution of Euler equation, Comm. Pure Appl. Math., 50 (1997), 1261-1286. doi: 10.1002/(SICI)1097-0312(199712)50:12<1261::AID-CPA3>3.0.CO;2-6.
[39]	R. Shvydkoy, Convex integration for a class of active scalar equations, J. Amer. Math. Soc., 24 (2011), 1159-1174. doi: 10.1090/S0894-0347-2011-00705-4.
[40]	R. Shvydkoy, Lectures on the Onsager conjecture, DCDS, Series S, 3 (2010), 473-496. doi: 10.3934/dcdss.2010.3.473.
[41]	L. Székelyhidi, Jr., From Isometric Embeddings to Turbulence, HCDTE lecture notes. Part Ⅱ. Nonlinear hyperbolic PDEs, dispersive and transport equations, 7: 63, 2012.
[42]	T. Tao and L. Zhang, On the continuous periodic weak solution of Boussinesq equations, SIAM, J. Math. Anal., 50 (2018), 1120-1162. doi: 10.1137/17M1115526.
[43]	T. Tao and L. Zhang, Hölder continuous solution of Boussinesq equations with compact support, J. Funct. Anal., 272 (2017), 4334-4402. doi: 10.1016/j.jfa.2017.01.013.
[44]	T. Tao and L. Zhang, Hölder continuous periodic solution of Boussinesq equations with partial viscosity, Calc. Var. Partial Differential Equations. doi: 10.1007/s00526-018-1337-7.

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Finite energy weak solutions of 2d Boussinesq equations with diffusive temperature

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