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A functional approach towards eigenvalue problems associated with incompressible flow
Finite energy weak solutions of 2d Boussinesq equations with diffusive temperature
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 |
We show the existence of finite kinetic energy solution with prescribed kinetic energy to the 2d Boussinesq equations with diffusive temperature on torus.
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. 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. |
[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. |
[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. 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. |
[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 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. |
[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. |
[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. Google Scholar |
[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. 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. |
[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. 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. |
[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. 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. |
[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. |
[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. 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. |
[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 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. |
[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. |
[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. Google Scholar |
[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. 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. |
[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. 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. |
[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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