L2-well-posedness of 3D DIV-curl boundary value problems

Quarterly of Applied Mathematics

Volumn 63, Issue 3, 2005, Pages 479-508

  Auchmuty, Giles ^a   Alexander, James C ^b  

^a University of Houston   (United States)

^b CASE WESTERN RESERVE UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

APPROXIMATION THEORY; BOUNDARY CONDITIONS; DIFFERENTIAL EQUATIONS; ESTIMATION; FINITE ELEMENT METHOD; FLUID MECHANICS; FUNCTIONS; INTEGRAL EQUATIONS; MAXWELL EQUATIONS; VECTORS;

HODGE-WEYL DECOMPOSITION; INTEGRABILITY ASSUMPTIONS; ORTHOGONAL DECOMPOSITIONS; WELL-POSEDNESS;

BOUNDARY VALUE PROBLEMS;

EID: 27744475178     PISSN: 0033569X     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0033-569X-05-00972-5     Document Type: Article

References (20)

1. R. Abraham, J. E. Marsden and T. Ratiu, Manifolds, Tensor Analysis and Applications, 2nd. ed., Springer-Verlag, New York (1988). MR0960687 (89f:58001)
2. C. J. Amick, "Some remarks on Rellich's theorem and the Poincaré inequality," J. London Math. Soc. (2) 18 (1973), 81-93. MR0502660 (80a:46016)
3. G. Auchmuty, "Orthogonal decompositions and bases for three-dimensional vector fields," Numer. Funct. Anal. and Optimiz. 15 (1994), 445-488. MR1281557 (95j:58158)
4. G. Auchmuty, "Reconstruction of the velocity from the vorticity in three-dimensional fluid flows," Royal Soc. London Proc. Set A Math. Phys. Eng. Sci. A 454 (1998), 607-630. MR1638305 (99f:76032)
5. G. Auchmuty, "The Main Inequality of 3d vector analysis," Math Modelling and Methods in the Applied Sciences, 14 (2004), 79-103. MR2037781
6. 2 -well-posedness of planar div-curl systems," Arch Rat Mech & Anal, 160 (2001), 91-134. MR1864837 (2002i:35022)
7. P. Blanchard and E. Brüning, Variational Methods in Mathematical Physics, Springer-Verlag, Berlin (1992). MR1230382 (95b:58049)
8. J. Bolik and W. von Wahl, "Estimating ∇u in terms of div u, curl u, either u · ν or u × ν and the topology," Math. Methods in the Applied Sciences, 20 (1997), 737-744. MR1446207 (98d:35023)
9. R. Dautray and J. L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Volumes 1-6, Springer-Verlag, Berlin (1990). MR0969367 (89m:00001); MR1064315 (91h:00004a); MR1081946 (91h:00004b); MR1156075 (92k:00006); MR1295030 (95e:00001)
10. L. C. Evans, Partial Differential Equations AMS, Providence, (1998). MR1625845 (99e:35001)
11. L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton (1992). MR1158660 (93f:28001)
12. C. Foias and R. Temam, "Remarques sur les équations de Navier-Stokes stationnaires et les phénoménes successifs de bifurcation," Annali Scuola Normale Superiore-Pisa, (1978), 29-63. MR0481645 (58:1749)
13. K. O. Friedrichs, "Differential forms on Riemannian manifolds," Comm. Pure & Applied Math., 8 (1955), 551-590. MR0087763 (19:407a)
14. V. Girault and P. A. Raviart, Finite Element Methods for the Navier-Stokes Equations, Springer-Verlag, Berlin (1986). MR0553589 (81a:65100)
15. P. R. Kotiuga and P. P. Silvester, "Vector potential formulation for three-dimensional magnetostatics," J. Appl. Physics, 53 (1982), 8399-8401.
16. R. Picard, "On the boundary value problems of electro- and magnetostatics," Proc. Roy. Soc. Edinburgh, 92A (1982), 165-174. MR0667134 (83m:78006)
17. J. Saranen, "On generalized harmonic fields in domains with anisotropic nonhomogeneous media," J. Math. Anal. Appl., 88 (1982), 104-115. MR0661405 (84d:78004a)
18. J. Saranen, "On Electric and Magnetic Problems for Vector Fields in anisotropic nonhomogeneous media," J. Math. Anal. Appl., 91 (1983), 254-275. MR0688544 (85i:78004)
19. H. Weyl, "The method of orthogonal projection in potential theory," Duke Math J., 7 (1940), 411-444. MR0003331 (2:202a)
20. E. Zeidler, Nonlinear Functional Analysis and its Applications, III: Variational Methods and Optimization, Springer-Verlag, New York (1985). MR0768749 (90b:49005)