PDF(565 KB)
Local Well-posedness for the Non-resistive MHD Equations in Sobolev Spaces
Ya Tao LI
Acta Mathematica Sinica, Chinese Series ›› 2020, Vol. 63 ›› Issue (4) : 335-348.
PDF(565 KB)
PDF(565 KB)
Local Well-posedness for the Non-resistive MHD Equations in Sobolev Spaces
This paper is concerned with the Cauchy problem of the non-resistive magnetohydrodynamics equations in Rd for d=2, 3. The local well-posedness in Sobolev space Hs-1×Hs for s > d/2 is obtained by establishing a commutator estimate.
non-resistive MHD equations / local wellposedness / commutator estimate {{custom_keyword}} /
[1] Bahouri H., Chemin J. Y., Danchin R., Fourier Analysis and Nonlinear Partial Differential Equations, Griondlerender Mathematischen Wissenschaften, Vol. 343, Springer, Heidelberg, 2011.
[2] Blömker D., Nolde D., Robinson J. C., Rigorous numerical verification of uniqueness and smoothness in a surface growth model, J. Math. Anal. Appl., 2015, 429(1):311-325.
[3] Bourgain J., Li D., Strong ill-posedness of the incompressible Euler equation in borderline Sobolev spaces, Invent. Math., 2015, 201(1):97-157.
[4] Cannone M., Ondelettes, Paraproduits et Navier-Stokes Nouveaux Essais, Diderot Éditeurs, Paris, 1995.
[5] Cannone M., Meyer Y., Planchon F., Solutions autosimilaires des équations de Navier-Stokes, Séminaire "Équations aux Dérivées Partielles" de l'École polytechnique, Expos é VIII, 1993-1994.
[6] Chemin J. Y., McCormick D. S., Robinson J. C., et al., Local existence for the non-resistive MHD equations in Besov spaces, Adv. Math., 2016, 286:1-31.
[7] Chen Q., Miao C., Zhang Z., The Beale-Kato-Majda criterion for the 3D magneto-hydrodynamics equations, Comm. Math. Phys., 2007, 275(3):861-872.
[8] Chen Q., Miao C., Zhang Z., On the regularity criterion of weak solution for the 3D viscous magnetohydrodynamics equations, Comm. Math. Phys., 2008, 284(3):919-930.
[9] Chen Q., Miao C., Zhang Z., Well-posedness in critical spaces for the compressible Navier-Stokes equations with density dependent viscosities, Rev. Mat. Iberoam., 2010, 26(3):915-946.
[10] Duvaut G., Lions J. L., Les Inéquations en Mécanique et en Physique, Dunod, Paris, 1972,
[11] Fefferman C., McCormick D., Robinson J. C., et al., Local existence for the non-resistive MHD equations in nearly optimal Sobolev spaces, Arch. Ration. Mech. Anal., 2017, 223(2):677-691.
[12] Fefferman C., McCormick D., Robinson J. C., et al., Higher order commutator estimates and local existence for the non-resistive MHD equations and related models, J. Funct. Anal., 2014, 267(4):1035-1056.
[13] Fujita H., Kato T., On the Navier-Stokes initial value problem I, Arch. Ration. Mech. Anal., 1964, 16:269-315.
[14] Jiu Q., Niu D., Mathematical results related to a two-dimensional magneto-hydrodynamic equations, Acta Math. Sci. Ser. B, Engl. Ed., 2006, 26(4):744-756.
[15] Koch H., Tataru D., Well-posedness for the Navier-Stokes equation, Advances in Math., 2001, 157:22-35.
[16] Li J., Tan W., Yin Z., Local existence and uniqueness for the non-resistive MHD equations in homogeneous Besov spaces, Adv. Math., 2017, 317:786-798.
[17] Lin F., Xu L., Zhang P., Global small solutions of 2-D incompressible MHD system, J. Differential Equations, 2015, 259(10):5440-5485.
[18] Miao C., Wu J., Zhang Z., Littlewood-Paley Theory and its Applications in Partial Differential Equations of Fluid Dynamics (in Chinese), Science Press, Beijing, 2012.
[19] Sermange M., Temam R., Some mathematical questions related to the MHD equations, Comm. Pure Appl. Math., 1983, 36(5):635-664.
[20] Wan R., On the uniqueness for the 2D MHD equations without magnetic diffusion, Nonlinear Anal. Real World Appl., 2016, 30:32-40.
PDF(565 KB)
/
| 〈 |
|
〉 |
