MARIA SCHONBEK Professor of Mathematics Licenciatura, University of Buenos Aires Ph.D., University of Michigan Publications My Page

Phone: 831-459-4657 Office: Baskin 353A Office Hours: By Appt. Email: schonbek_at_ucsc_edu

RESEARCH INTERESTS

Maria Schonbek's research focuses on non-linear partial differential equations derived from models in fluid dynamics. The primary questions she is concerned with are related to the qualitative behavior of solutions, specifically questions concerning long-time behavior of solutions.

Understanding the behavior of solutions to equations for physical models is a first step in understanding the behavior of the world we live in.

In her research, Professor Schonbek has tried to make interdisciplinary connections since she believes that such work is fundamental to the progress of science.

SELECTED PUBLICATIONS

Schonbek, M.: Estimates for the pressure and the Fourier transform of solutions and derivatives to the Navier-Stokes equations. Forthcoming.

M. Schonbek, T. Schonbek, and E. Suli: Large-time behavior of solutions to the Magneto Hydrodynamics equations. Mathematische Annalen 304 (1996) 717-756.

M. Cannone, F. Planchon and M. Schonbek: Solutions to the incompressible Navier-Stokes equations in the half space. Submitted.

Malek, Necas, Pocorny and Schonbek: On the possible singular solutions to the Navier-Stokes equations. Mathematische Nachrichten (1999) 97-114.

M. Schonbek and G. K. Vallis: Energy decay of solutions of the Planetary geostrophic equations. Accepted for publication JMAA (1999).

M. Schonbek: Asymptotic behavior of solutions to the three-dimensional Navier-Stokes equations. Indiana Univ. J. 41(2) (1992).

M. Schonbek: Lower bounds of decay rates for solutions to the Navier-Stokes equations. J. Amer. Math. Soc. 4(3):423Ð449, (1991).

M. Schonbek and T. P. Schonbek: On the boundedness and decay of moments of solutions to the Navier-Stokes equations. Submitted.

M. Schonbek, J. Bona, and B. Amick: Decay of solutions of some nonlinear wave equations. J. Differential Equations 81(1):1Ð49, (1989).

M. Schonbek: L2 decay for weak solutions of the Navier-Stokes equations. Arch. Rational Mech. 88 (3):209Ð222, (1986).

M. Schonbek: Large-time behavior of solutions to the Navier-Stokes equations. Comm. Partial Differential Equations 11(7):733Ð763, (1986).

PUBLICATIONS

1. Boundary value problems for the Fitzhugh-Nagumo equations. Journal of Differential Equations 30 (1), pp. 119-47, (1978).

2. Some results on the Fitzhugh-Nagumo equations, Research Notes in Mathematics, No. 14, Editors: W. E. Fitzgibbon and H. F. Walker, (Pitman, 1977).

3. Decay of solutions to parabolic conservation laws, Communications in Partial Differential Equations 7 (1), pp.449-73, (1980).

4. Decay of solutions to the Korteweg-deVries-Burger equation and to parabolic conservation laws, Nonlinear Partial Differential Equations in Engineering and Applied Science, vol. 54, Editors: R. L. Sternberg, A. Kalinowski and J. Papadakis, (Marcel Dekker, 1980). 10. L2 decay for weak solutions of the Navier-Stokes equations, Archive of Rational Mechanics 88 (3), pp. 209-22, (1985).

5. Apriori estimates of higher order derivatives of solutions of the Fitzhugh-Nagumo equations, Journal of Mathematical Applications 82 (2), pp. 553-65, (1981).

6. Existence of solutions to the Boussinesq system of equations, Journal of Differential Equations 42 (3), (1981), pp. 325-51.

7. Convergence of solutions to nonlinear dispersion equations, Communications in Partial Differential Equations 7 (8), (1982), pp. 959-1000.

8. Existence of solution to singular conservation laws, SIAM Journal of Mathematical Analysis 15 (6), pp. 1125-39, (1984).

9. Second order conservative schemes and the entropy condition, Mathematics of Computations Vol. 10, No. 169, pp. 31-38, (Jan. 1985).

10. L2 decay for weak solutions of the Navier-Stokes equations, Archive of Rational Mechanics 88 (3), pp. 209-22, (1985).

11. Some results on the traveling wave solutions of the Korteweg-deVries-Burger equation, (with J. Bona) Royal Society of Edinburgh, 101A, pp. 207-26, (1985).

12. Large time behavior of derivatives and Lp norms of solutions to Navier Stokes equations, Preprint.

13. Uniform decay rates for parabolic conservation laws, Journal of Nonlinear Analysis Vol. 10, No. 9, pp. 943-56, (1986).

14. Large time behavior of solutions to the Navier-Stokes equations, Communications in Partial Differential Equations 11 (7), pp. 733-763, (1986).

15. Applications to the theory of compensated compactness, Oscillation theory, computation, and methods of compensated compactness (Minneapolis, Minn, 1985), 289-294 (IMA) Vol. Math. Appl2, Springer, New York, (1986).

16. Nonlinear geometric optics and conservation laws, Proceedings of the Symposium Year on material instabilities in continuous mechanics. Heriot-Watt University. Edited by John Ball, (1986).

17. Some aspects on compressible and incompressible gas dynamics, Proceedings of the Conference on Nonlinear Partial Differential Equations and Applications at the University Palazzo dei Gesuiti, L'Aquila, Italy, (1986).

18. A canonical system of integro-differential equations arising in resonant wave asymptotics, (with A. Majda and R. Rosales) Studies in Applied Mathematics, (79), no. 3 205-262, (1988).

19. Asymptotic properties for long waves which incorporate dissipation, (with J. Bona and C. Amick), Journal of Differential Equations 81, no 1, 1-49, (1989).

20. Lower bounds of decay rates for the solutions to Navier-Stokes equations, Journal of the American Mathematical Society, (July 1991).

21. Asymptotic behavior of solutions to the three dimensional Navier-Stokes equations, Indiana University Journal, Vol. 41, no. 2 (1992).

22. Some results on the asymptotic behavior of solutions to Navier Stokes. Proceedings of the Oberwolfach Conference on Navier-Stokes equations, Springer-Verlag (NOTE: these proceedings are reviewed), (1991).

23. Models for propagation of bores I. Two-dimensional theory. (Previously titled: Long wave models with initial data corresponding to bore propagation), (with J. Bona and S. Rajopadhye) Differential Integral Equations, 7, no.3-4 699-734, (1994).

24. Large-time behavior of solutions to the Navier-Stokes equations in H m spaces, Comm. in P.D.E, 20 (1& 2), pp. 103-117 (1995).

25. Estimates for the pressure and the Fourier transform of solutions and derivatives to the Navier-Stokes equations. Indiana Math Journal, vol. 43 #2, (1994).

26. The Fourier splitting method, Advances in Geometric Analysis and Continuum Mechanics, (269-274), Internatl. Press, Cambridge, Ma. (1995).

27. Asymptotic behavior of solutions to the Korteweg de Vries-Burger System (with S. Rajopadhye, Analyse Nonlineare, vol. 12, no.44, pp. 425-457 (1995).

28. Decay results for Solutions to the Magneto Hydrodynamics equations, (with T.P. Schonbek and E. Suli) Mathematical Analysis of Phenomena in Fluids and Plasma Dynamics. Editors: K. Asano, A. Tani, S. Ukai. RIMS Kokyuroku 914, 98-102, (1995).

29. Large-time behavior of solutions to the Magneto-Hydrodynamics equations (with T. P. Schonbek and E. SŸli). Mathematische Annalen, 304, no.4 717-756, (1996).

30. On the decay of higher order norms of the solutions of Navier-Stokes equations, (joint with M. Wiegner) Proc. Royal Society of Edinburgh Sect. A 126, no.3, 677-685, (1996).

31. Asymptotic behavior of solutions to the Navier-Stokes equations with slowly decaying external force, (with S. Rajopadhye and T. Ogawa), J. Functional Analysis 144, no. 2. 325-358, (1997).

31. Asymptotic behavior of solutions to the Navier-Stokes equations with slowly decaying external force, (with S. Rajopadhye and T. Ogawa), J. Functional Analysis 144, no. 2. 325-358, (1997).

32. Decay of Solutions to non-oscillating Magneto Hydrodynamics equations. Proceedings of Oberwolfach Conference on Navier Stokes (Springer Verlag Proceedings), Theory of the Navier-Stokes Equations. Editors Heywood, Masuda, Rautmann, Solonikov. Series on advances in Math for Applied Sciences- Vol. 47 (1998). (NOTE: these proceedings are reviewed).

33. On the possible singular solutions to the Navier-Stokes equations (joint with Malek, Necas, Pocorny), Mathematische Nachrichten, 199, (1999) 97-114.

34. Asymptotic decay to a generalized Boussinesq System (with S. Rajopadhye and M. Wiegner), J. of Dynamics and Differential Equations vol. 11, #4, (1999).

35. Energy decay of solutions of the Planetary geostrophic equations, joint with G.K. Vallis, J. Math. Ann. and Appl., 234, 457-481 (1999).

36. On the boundedness and decay of moments of solutions to the Navier-Stokes equations, joint with T.P. Schonbek, Advances in Differential Equations, Vol. 5, #7-9, July-September (2000).

37. Strong solutions to the incompressible Navier-Stokes equations in the half space. Joint with M. Cannone and F. Planchon. Comm. in PDE, 25 (5&6), 903-924, (2000).

38. On the Decay of Solutions to the Navier-Stokes Equations. Applied Nonlinear Analysis, Editors: A. Sequeira, H. Beirao da Vega, J. H. Videman, Kluwer Academic/ Plenum Publisher. (NOTE: The papers in this book are reviewed). (1999).

39. Pointwise decay of solutions and higher derivatives to Navier-Stokes equations, (joint with Amrouche, Girault and T. Schonbek), SIAM Journal of Analysis, Vol. 31, (4), (2000).

40. Geodesic flow on Diffeomorphism of the Circle, Grassmanians and the Geometry of Periodic KdV equations, joint with A.Todorov, J. Zubelli, Adv. Theor. Math. Physics, 4,1027-1090, (2000).

41. Total variation decay of solutions to the Navier-Stokes equations. Methods and Applications of Analysis, Vol7, No 3,555-564, (2000).

42. Decay of the total variation and Hardy norms to parabolic conservation laws, joint with SŸli, Nonlinear Analysis, 45, 515-528, (2001).

43. Nonexistence of Singular Pseudo-self-similar Solutions of the Navier-Stokes System, joint with J. Miller, M. O'Leary, Mathematiche Annalen, 319, 809-815, (2001).

44. On Optimal Decay Rates for Weak Solutions to the Navier-Stokes Equations in IRn, joint with T. Miyakawa. Mathematica Bohemica, Vol. 126, No 2, 257-263, (2001).

LECTURE NOTES

1. Problems related to solutions to the Navier-Stokes equations. Fachbereich Mathematik, Bericht 203, Nov. 1998.

SUBMITTED PAPERS

1. On zero mass solutions of viscous conservation laws. Submitted to Communications n Partial Differential equations.

2. Fitzhugh-Nagumo Revisited, Joint with T. Kostova and R. Ravindran. Submitted to Mathematical Biosciences.

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