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

Tianwen Luo; Tao Tao; Liqun Zhang

doi:10.3934/dcds.2019230

Previous Article
Perturbative techniques for the construction of spike-layers
DCDS Home
This Issue
Next Article
A functional approach towards eigenvalue problems associated with incompressible flow

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
Full Text(HTML)
Related Papers
Under a Creative Commons license

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 & 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[4]	T. Buckmaster, C. De Lellis and L. Székelyhidi, Jr., Transporting microstructure and dissipative Euler flows, preprint, arXiv: 1302.2825. Google Scholar
[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. Google Scholar
[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. Google Scholar
[7]	T. Buckmaster, Shkoller and V. Vicol, Nonuniqueness of weak solutions to SQG equation, to appear in Comm. Pure Appl. Math. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[23]	P. Isett, Hölder continuous Euler flows in three dimensions with compact support in time, preprint, arXiv: 1211.4065. doi: 10.1515/9781400885428. Google Scholar
[24]	P. Isett, A proof of Onsager's conjecture, Ann. of. Math., 188 (2018), 871-963. doi: 10.4007/annals.2018.188.3.4. Google Scholar
[25]	P. Isett, On the endpoint regularity in Onsager's conjecture, preprint, arXiv: 1706.01549 Google Scholar
[26]	P. Isett and V. Vicol, H ölder continuous solutions of active scalar equations, Ann. of. PDE. doi: 10.1007/s40818-015-0002-0. Google Scholar
[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. Google Scholar
[28]	T. Luo, T. Tao and L. Zhang, Hölder continuous soltion of 2d Boussinesq equation with diffusive temperture, preprint, arXiv: 1901.10071. Google Scholar
[29]	T. Luo and Z. Xin, Hölder continuous solutions to the 3d Prandtl system, preprint, arXiv: 1804.04285. Google Scholar
[30]	X. Luo, Stationary solution and nonuniquenes of weak solution for the Navier-Stokes euation on high dimensions, preprint, arXiv: 1807.09318. Google Scholar
[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. Google Scholar
[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. Google Scholar
[33]	S. Modena and L. Székelyhidi, Jr., Non-Renormalized solution to the continuity equation, preprint, arXiv: 1806.09145. Google Scholar
[34]	L. Onsager, Statistical hydrodynamics, Nuovo Cimento, 9 (1949), 279-287. doi: 10.1007/BF02780991. Google Scholar
[35]	J. Pedlosky, Geophysical Fluid Dynamics, Springer, New York, 1987. Google Scholar
[36]	V. Scheffer, An inviscid flow with compact support in space-time, J. Geom. Anal., 3 (1993), 343-401. doi: 10.1007/BF02921318. Google Scholar
[37]	A. Shnirelman, Weak solution with decreasing energy of incompressible Euler equations, Comm. Math. Phys., 210 (2000), 541-603. doi: 10.1007/s002200050791. Google Scholar
[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. Google Scholar
[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. Google Scholar
[40]	R. Shvydkoy, Lectures on the Onsager conjecture, DCDS, Series S, 3 (2010), 473-496. doi: 10.3934/dcdss.2010.3.473. Google Scholar
[41]	L. Székelyhidi, Jr., From Isometric Embeddings to Turbulence, HCDTE lecture notes. Part Ⅱ. Nonlinear hyperbolic PDEs, dispersive and transport equations, 7: 63, 2012. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[4]	T. Buckmaster, C. De Lellis and L. Székelyhidi, Jr., Transporting microstructure and dissipative Euler flows, preprint, arXiv: 1302.2825. Google Scholar
[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. Google Scholar
[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. Google Scholar
[7]	T. Buckmaster, Shkoller and V. Vicol, Nonuniqueness of weak solutions to SQG equation, to appear in Comm. Pure Appl. Math. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[23]	P. Isett, Hölder continuous Euler flows in three dimensions with compact support in time, preprint, arXiv: 1211.4065. doi: 10.1515/9781400885428. Google Scholar
[24]	P. Isett, A proof of Onsager's conjecture, Ann. of. Math., 188 (2018), 871-963. doi: 10.4007/annals.2018.188.3.4. Google Scholar
[25]	P. Isett, On the endpoint regularity in Onsager's conjecture, preprint, arXiv: 1706.01549 Google Scholar
[26]	P. Isett and V. Vicol, H ölder continuous solutions of active scalar equations, Ann. of. PDE. doi: 10.1007/s40818-015-0002-0. Google Scholar
[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. Google Scholar
[28]	T. Luo, T. Tao and L. Zhang, Hölder continuous soltion of 2d Boussinesq equation with diffusive temperture, preprint, arXiv: 1901.10071. Google Scholar
[29]	T. Luo and Z. Xin, Hölder continuous solutions to the 3d Prandtl system, preprint, arXiv: 1804.04285. Google Scholar
[30]	X. Luo, Stationary solution and nonuniquenes of weak solution for the Navier-Stokes euation on high dimensions, preprint, arXiv: 1807.09318. Google Scholar
[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. Google Scholar
[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. Google Scholar
[33]	S. Modena and L. Székelyhidi, Jr., Non-Renormalized solution to the continuity equation, preprint, arXiv: 1806.09145. Google Scholar
[34]	L. Onsager, Statistical hydrodynamics, Nuovo Cimento, 9 (1949), 279-287. doi: 10.1007/BF02780991. Google Scholar
[35]	J. Pedlosky, Geophysical Fluid Dynamics, Springer, New York, 1987. Google Scholar
[36]	V. Scheffer, An inviscid flow with compact support in space-time, J. Geom. Anal., 3 (1993), 343-401. doi: 10.1007/BF02921318. Google Scholar
[37]	A. Shnirelman, Weak solution with decreasing energy of incompressible Euler equations, Comm. Math. Phys., 210 (2000), 541-603. doi: 10.1007/s002200050791. Google Scholar
[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. Google Scholar
[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. Google Scholar
[40]	R. Shvydkoy, Lectures on the Onsager conjecture, DCDS, Series S, 3 (2010), 473-496. doi: 10.3934/dcdss.2010.3.473. Google Scholar
[41]	L. Székelyhidi, Jr., From Isometric Embeddings to Turbulence, HCDTE lecture notes. Part Ⅱ. Nonlinear hyperbolic PDEs, dispersive and transport equations, 7: 63, 2012. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

[1]	Francesca Alessio, Piero Montecchiari, Andres Zuniga. Prescribed energy connecting orbits for gradient systems. Discrete & Continuous Dynamical Systems, 2019, 39 (8) : 4895-4928. doi: 10.3934/dcds.2019200
[2]	Jaeyoung Byeon, Sungwon Cho, Junsang Park. On the location of a peak point of a least energy solution for Hénon equation. Discrete & Continuous Dynamical Systems, 2011, 30 (4) : 1055-1081. doi: 10.3934/dcds.2011.30.1055
[3]	Hyungjin Huh. Towards the Chern-Simons-Higgs equation with finite energy. Discrete & Continuous Dynamical Systems, 2011, 30 (4) : 1145-1159. doi: 10.3934/dcds.2011.30.1145
[4]	Morched Boughariou. Closed orbits of Hamiltonian systems on non-compact prescribed energy surfaces. Discrete & Continuous Dynamical Systems, 2003, 9 (3) : 603-616. doi: 10.3934/dcds.2003.9.603
[5]	Mitsuru Shibayama. Periodic solutions for a prescribed-energy problem of singular Hamiltonian systems. Discrete & Continuous Dynamical Systems, 2017, 37 (5) : 2705-2715. doi: 10.3934/dcds.2017116
[6]	J. F. Toland. Energy-minimising parallel flows with prescribed vorticity distribution. Discrete & Continuous Dynamical Systems, 2014, 34 (8) : 3193-3210. doi: 10.3934/dcds.2014.34.3193
[7]	Futoshi Takahashi. On the number of maximum points of least energy solution to a two-dimensional Hénon equation with large exponent. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1237-1241. doi: 10.3934/cpaa.2013.12.1237
[8]	Anthony Suen. Existence and a blow-up criterion of solution to the 3D compressible Navier-Stokes-Poisson equations with finite energy. Discrete & Continuous Dynamical Systems, 2020, 40 (3) : 1775-1798. doi: 10.3934/dcds.2020093
[9]	Jerry L. Bona, Zoran Grujić, Henrik Kalisch. A KdV-type Boussinesq system: From the energy level to analytic spaces. Discrete & Continuous Dynamical Systems, 2010, 26 (4) : 1121-1139. doi: 10.3934/dcds.2010.26.1121
[10]	Yanbing Yang, Runzhang Xu. Nonlinear wave equation with both strongly and weakly damped terms: Supercritical initial energy finite time blow up. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1351-1358. doi: 10.3934/cpaa.2019065
[11]	Xiaoqiang Dai, Chao Yang, Shaobin Huang, Tao Yu, Yuanran Zhu. Finite time blow-up for a wave equation with dynamic boundary condition at critical and high energy levels in control systems. Electronic Research Archive, 2020, 28 (1) : 91-102. doi: 10.3934/era.2020006
[12]	Daomin Cao, Hang Li. High energy solutions of the Choquard equation. Discrete & Continuous Dynamical Systems, 2018, 38 (6) : 3023-3032. doi: 10.3934/dcds.2018129
[13]	Kazumasa Fujiwara, Shuji Machihara, Tohru Ozawa. Remark on a semirelativistic equation in the energy space. Conference Publications, 2015, 2015 (special) : 473-478. doi: 10.3934/proc.2015.0473
[14]	Shaoyong Lai, Yong Hong Wu. The asymptotic solution of the Cauchy problem for a generalized Boussinesq equation. Discrete & Continuous Dynamical Systems - B, 2003, 3 (3) : 401-408. doi: 10.3934/dcdsb.2003.3.401
[15]	Nicola Guglielmi, László Hatvani. On small oscillations of mechanical systems with time-dependent kinetic and potential energy. Discrete & Continuous Dynamical Systems, 2008, 20 (4) : 911-926. doi: 10.3934/dcds.2008.20.911
[16]	Sergio Grillo, Leandro Salomone, Marcela Zuccalli. Explicit solutions of the kinetic and potential matching conditions of the energy shaping method. Journal of Geometric Mechanics, 2021, 13 (4) : 629-646. doi: 10.3934/jgm.2021022
[17]	Chiara Corsato, Franco Obersnel, Pierpaolo Omari, Sabrina Rivetti. On the lower and upper solution method for the prescribed mean curvature equation in Minkowski space. Conference Publications, 2013, 2013 (special) : 159-169. doi: 10.3934/proc.2013.2013.159
[18]	Hongyu Ye. Positive high energy solution for Kirchhoff equation in $\mathbb{R}^{3}$ with superlinear nonlinearities via Nehari-Pohožaev manifold. Discrete & Continuous Dynamical Systems, 2015, 35 (8) : 3857-3877. doi: 10.3934/dcds.2015.35.3857
[19]	Manuel del Pino, Monica Musso, Juncheng Wei, Yifu Zhou. Type Ⅱ finite time blow-up for the energy critical heat equation in $ \mathbb{R}^4 $. Discrete & Continuous Dynamical Systems, 2020, 40 (6) : 3327-3355. doi: 10.3934/dcds.2020052
[20]	Shanshan Guo, Zhong Tan. Energy dissipation for weak solutions of incompressible liquid crystal flows. Kinetic & Related Models, 2015, 8 (4) : 691-706. doi: 10.3934/krm.2015.8.691

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors