{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,3,26]],"date-time":"2025-03-26T08:00:41Z","timestamp":1742976041752,"version":"3.40.3"},"publisher-location":"Cham","reference-count":29,"publisher":"Springer International Publishing","isbn-type":[{"type":"print","value":"9783030238537"},{"type":"electronic","value":"9783030238544"}],"license":[{"start":{"date-parts":[[2019,1,1]],"date-time":"2019-01-01T00:00:00Z","timestamp":1546300800000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2019]]},"DOI":"10.1007\/978-3-030-23854-4_8","type":"book-chapter","created":{"date-parts":[[2019,10,15]],"date-time":"2019-10-15T21:20:36Z","timestamp":1571174436000},"page":"173-190","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Applications of Parabolic Dirac Operators to the Instationary Viscous MHD Equations on Conformally Flat Manifolds"],"prefix":"10.1007","author":[{"given":"Paula","family":"Cerejeiras","sequence":"first","affiliation":[]},{"given":"Uwe","family":"K\u00e4hler","sequence":"additional","affiliation":[]},{"given":"R. S\u00f6ren","family":"Krau\u00dfhar","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2019,10,16]]},"reference":[{"issue":"5","key":"8_CR1","doi-asserted-by":"publisher","first-page":"429","DOI":"10.1080\/17476930500481400","volume":"51","author":"S Bernstein","year":"2006","unstructured":"S. Bernstein, Factorization of the nonlinear Schr\u00f6dinger equation and applications. Complex Var. Elliptic Equ. 51(5), 429\u2013452 (2006)","journal-title":"Complex Var. Elliptic Equ."},{"key":"8_CR2","volume-title":"Weak and strong solutions of equations of compressible magnetohydrodynamics, in Handbook of Mathematical Analysis in Mechanics of Viscous Fluids","author":"X Blanc","year":"2016","unstructured":"X. Blanc, B. Ducomet, Weak and strong solutions of equations of compressible magnetohydrodynamics, in Handbook of Mathematical Analysis in Mechanics of Viscous Fluids (Springer, Cham, 2016)"},{"key":"8_CR3","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1515\/crelle.2010.042","volume":"643","author":"E Bulla","year":"2010","unstructured":"E. Bulla, D. Constales, R.S. Krau\u00dfhar, J. Ryan, Dirac type operators for arithmetic subgroups of generalized modular groups. J. Reine Angew. Math. 643, 1\u201319 (2010)","journal-title":"J. Reine Angew. Math."},{"key":"8_CR4","first-page":"161","volume-title":"Handbook of Mathematical Fluid Dynamics, vol. 3","author":"M Cannone","year":"2004","unstructured":"M. Cannone, Harmonic analysis tools for solving the incompressible Navier-Stokes equations, in Handbook of Mathematical Fluid Dynamics, vol. 3, ed. by S. Friedlander, D. Serre (Elsevier, Amsterdam, 2004), pp. 161\u2013244"},{"issue":"1","key":"8_CR5","doi-asserted-by":"publisher","first-page":"81","DOI":"10.1002\/(SICI)1099-1476(20000110)23:1<81::AID-MMA105>3.0.CO;2-Z","volume":"23","author":"P Cerejeiras","year":"2000","unstructured":"P. Cerejeiras, U. K\u00e4hler, Elliptic boundary value problems of fluid dynamics over unbounded domains. Math. Methods Appl. Sci. 23(1), 81\u2013101 (2000)","journal-title":"Math. Methods Appl. Sci."},{"issue":"4","key":"8_CR6","doi-asserted-by":"publisher","first-page":"535","DOI":"10.1002\/mma.1050","volume":"32","author":"P Cerejeiras","year":"2009","unstructured":"P. Cerejeiras, N. Vieira, Regularization of the non-stationary Schr\u00f6dinger operator. Math. Methods Appl. Sci. 32(4), 535\u2013555 (2009)","journal-title":"Math. Methods Appl. Sci."},{"issue":"14","key":"8_CR7","doi-asserted-by":"publisher","first-page":"1715","DOI":"10.1002\/mma.634","volume":"28","author":"P Cerejeiras","year":"2005","unstructured":"P. Cerejeiras, U. K\u00e4hler, F. Sommen, Parabolic Dirac operators and the Navier-Stokes equations over time-varying domains. Math. Methods Appl. Sci. 28(14), 1715\u20131724 (2005)","journal-title":"Math. Methods Appl. Sci."},{"key":"8_CR8","doi-asserted-by":"crossref","unstructured":"P. Cerejeiras, U. K\u00e4hler, R.S. Krau\u00dfhar, Some applications of parabolic Dirac operators to the instationary Navier-Stokes problem on conformally flat cylinders and tori \u211d 3 $$\\mathbb {R}^3$$ , in Clifford Analysis and Related Topics, ed. by P. Cerejeiras et al. (Springer, New York, 2018)","DOI":"10.1007\/978-3-030-00049-3_2"},{"key":"8_CR9","doi-asserted-by":"publisher","first-page":"919","DOI":"10.1007\/s00220-008-0545-y","volume":"284","author":"Q Chen","year":"2008","unstructured":"Q. Chen, C. Miao, Z. Zhang, On the regularity criterion of weak solution for the 3D viscous magneto-hydrodynamics equations. Comm. Math. Phys. 284, 919\u2013930 (2008)","journal-title":"Comm. Math. Phys."},{"issue":"6","key":"8_CR10","doi-asserted-by":"publisher","first-page":"735","DOI":"10.1002\/mma.941","volume":"31","author":"D Constales","year":"2008","unstructured":"D. Constales, R.S. Krau\u00dfhar, On the Navier-Stokes equation with Free Convection in three dimensional triangular channels. Math. Methods Appl. Sci. 31(6), 735\u2013751 (2008)","journal-title":"Math. Methods Appl. Sci."},{"issue":"16","key":"8_CR11","doi-asserted-by":"publisher","first-page":"2050","DOI":"10.1002\/mma.1122","volume":"32","author":"D Constales","year":"2009","unstructured":"D. Constales, R.S. Krau\u00dfhar, Multiperiodic eigensolutions to the Dirac operator and applications to the generalized Helmholtz equation on flat cylinders and on the n-torus. Math. Methods Appl. Sci. 32(16), 2050\u20132070 (2009)","journal-title":"Math. Methods Appl. Sci."},{"key":"8_CR12","doi-asserted-by":"publisher","DOI":"10.1007\/978-94-011-2922-0","volume-title":"Clifford Algebra and Spinor Valued Functions","author":"R Delanghe","year":"1992","unstructured":"R. Delanghe, F. Sommen, V. Sou\u010dek, Clifford Algebra and Spinor Valued Functions (Kluwer, Dortrecht, 1992)"},{"issue":"6","key":"8_CR13","doi-asserted-by":"publisher","first-page":"577","DOI":"10.1080\/10236190600637965","volume":"12","author":"N Faustino","year":"2006","unstructured":"N. Faustino, K. G\u00fcrlebeck, A. Hommel, U. K\u00e4hler, Difference potentials for the Navier-Stokes equations in unbounded domains. J. Differ. Equ. Appl. 12(6), 577\u2013595 (2006)","journal-title":"J. Differ. Equ. Appl."},{"issue":"12","key":"8_CR14","doi-asserted-by":"crossref","first-page":"1496","DOI":"10.1002\/mma.1263","volume":"32","author":"S Gala","year":"2010","unstructured":"S. Gala, Extension criterion on regularity for weak solutions to the 3D MHD equations. Math. Methods Appl. Sci. 32(12), 1496\u20131503 (2010)","journal-title":"Math. Methods Appl. Sci."},{"key":"8_CR15","doi-asserted-by":"publisher","first-page":"571","DOI":"10.1016\/j.jmaa.2014.09.003","volume":"425","author":"Y Ge","year":"2015","unstructured":"Y. Ge, S. Shao, Global solution of 3D incompressible magnetohydrodyanamic equations with finite energy. J. Math. Anal. Appl. 425, 571\u2013578 (2015)","journal-title":"J. Math. Anal. Appl."},{"key":"8_CR16","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781139195560","volume-title":"Advanced Magnetohydrodynamics: With Applications to Laboratory and Astrophysical Plasmas","author":"H Goedbloed","year":"2010","unstructured":"H. Goedbloed, S. Poedts, Advanced Magnetohydrodynamics: With Applications to Laboratory and Astrophysical Plasmas (Cambridge University Press, Cambridge, 2010)"},{"issue":"194","key":"8_CR17","doi-asserted-by":"publisher","first-page":"523","DOI":"10.1090\/S0025-5718-1991-1066834-0","volume":"56","author":"M Gunzburger","year":"1991","unstructured":"M. Gunzburger, A. Meir, J. Peterson, On the existence, uniqueness and finite element approximation of the equations of stationary, incompressible magnetohydrodynamics. Math. Comput. 56(194), 523\u2013563 (1991)","journal-title":"Math. Comput."},{"key":"8_CR18","doi-asserted-by":"crossref","unstructured":"K. G\u00fcrlebeck, W. Spr\u00f6\u00dfig, Quaternionic Analysis and Elliptic Boundary Value Problems (Birkh\u00e4user, Basel, 1990)","DOI":"10.1007\/978-3-0348-7295-9"},{"key":"8_CR19","doi-asserted-by":"crossref","unstructured":"R.S. Krau\u00dfhar, On the incompressible viscous MHD equations and explicit solution formulas for some three dimensional radially symmetric domains, in Hypercomplex Analysis and Applications, ed. by I. Sabadini, F. Sommen. Trends in Mathematics (Birkh\u00e4user, Basel, 2011), pp. 125\u2013137","DOI":"10.1007\/978-3-0346-0246-4_9"},{"key":"8_CR20","unstructured":"V. Kravchenko, Applied Quaternionic Analysis. Research and Exposition in Mathematics, vol. 28 (Heldermann Verlag, Lemgo, 2003)"},{"key":"8_CR21","doi-asserted-by":"publisher","first-page":"916","DOI":"10.2307\/1969587","volume":"50","author":"NH Kuiper","year":"1949","unstructured":"N.H. Kuiper, On conformally flat spaces in the large. Ann. Math. 50, 916\u2013924 (1949)","journal-title":"Ann. Math."},{"key":"8_CR22","volume-title":"Spin Geometry","author":"HB Lawson","year":"1989","unstructured":"H.B. Lawson, M.-L. Michelsohn, Spin Geometry (Princeton University Press, New York, 1989)"},{"key":"8_CR23","doi-asserted-by":"publisher","first-page":"3202","DOI":"10.1016\/j.jde.2015.04.017","volume":"259","author":"Z Lei","year":"2015","unstructured":"Z. Lei, On axially symmetric incompressible magnetohydrodynamics in three dimensions. J. Diff. Equ. 259, 3202\u20133215 (2015)","journal-title":"J. Diff. Equ."},{"issue":"1","key":"8_CR24","doi-asserted-by":"publisher","first-page":"53","DOI":"10.1002\/mma.1026","volume":"32","author":"C Miao","year":"2010","unstructured":"C. Miao, B. Yuan, On well-posedness of the Cauchy problem for MHD systems in Besov spaces. Math. Methods Appl. Sci. 32(1), 53\u201376 (2010)","journal-title":"Math. Methods Appl. Sci."},{"key":"8_CR25","doi-asserted-by":"publisher","first-page":"358","DOI":"10.1016\/j.jmmm.2017.05.014","volume":"439","author":"S Rashidi","year":"2017","unstructured":"S. Rashidi, J.A. Esfahani, M. Maskaniyan, Applications of magnetohydrodynamics in biological systems-a review on the numerical studies. J. Magn. Magn. Mater. 439, 358\u2013372 (2017)","journal-title":"J. Magn. Magn. Mater."},{"key":"8_CR26","first-page":"635","volume":"6","author":"M Sermagne","year":"1983","unstructured":"M. Sermagne, R. Temam, Some mathematical questions related to the MHD equations. Commun. Pure Appl. Math. 6, 635\u2013664 (1983)","journal-title":"Commun. Pure Appl. Math."},{"key":"8_CR27","doi-asserted-by":"publisher","first-page":"445","DOI":"10.1090\/S0894-0347-99-00290-8","volume":"12","author":"S Wu","year":"1999","unstructured":"S. Wu, Well-posedness in Sobolev spaces of the full water wave problem in 3D. J. Am. Math. Soc. 12, 445\u2013495 (1999)","journal-title":"J. Am. Math. Soc."},{"key":"8_CR28","doi-asserted-by":"crossref","unstructured":"E. Zeidler, Nonlinear Functional Analysis and ist Applications \u2013 IV. Applications to Mathematical Physics (Springer, Berlin, 1988)","DOI":"10.1007\/978-1-4612-4566-7_12"},{"key":"8_CR29","doi-asserted-by":"publisher","first-page":"1359","DOI":"10.1016\/j.jde.2016.10.016","volume":"262","author":"X Zhai","year":"2017","unstructured":"X. Zhai, Z. Yin, Global well-posedness for the 3D incompressible inhomogeneous Navier-Stokes equations and MHD equations. J. Diff. Equ. 262, 1359\u20131412 (2017)","journal-title":"J. Diff. Equ."}],"container-title":["Trends in Mathematics","Topics in Clifford Analysis"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/978-3-030-23854-4_8","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,1,24]],"date-time":"2021-01-24T22:25:43Z","timestamp":1611527143000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/978-3-030-23854-4_8"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019]]},"ISBN":["9783030238537","9783030238544"],"references-count":29,"URL":"https:\/\/doi.org\/10.1007\/978-3-030-23854-4_8","relation":{},"ISSN":["2297-0215","2297-024X"],"issn-type":[{"type":"print","value":"2297-0215"},{"type":"electronic","value":"2297-024X"}],"subject":[],"published":{"date-parts":[[2019]]},"assertion":[{"value":"16 October 2019","order":1,"name":"first_online","label":"First Online","group":{"name":"ChapterHistory","label":"Chapter History"}}]}}