Wigner Measure and Semiclassical Limit s

of Nonlinear Schrodinger Equations

Courant Lecture Notes in Mathematics

Executive Editor Mai Shata h

Managing Editor Paul D. Monsour

Production Editor Reeva Goldsmith

Copy Editor Marc Nirenberg

Ping Zhang Academy of Mathematics and Systems Science, Chinese Academy of Sciences

17 Wigne r Measure and Semielassiea l Limits of Nonlinear Schrodinger Equation s

Courant Institute of Mathematical Science s New York University New York, New York

American Mathematical Societ y Providence, Rhode Island

http://dx.doi.org/10.1090/cln/017

2000 Mathematics Subject Classification. P r imar y 35Q40 , 35Q55 .

For addi t iona l informatio n an d upda te s o n thi s book , visi t w w w . a m s . o r g / b o o k p a g e s / c l n - 1 7

Library o f Congres s Cataloging-in-Publicatio n D a t a

Zhang, Ping , 1969 -Wigner measur e an d semiclassica l limit s o f nonlinea r Schrodinge r equation s / Pin g Zhang .

p. cm . — (Couran t lectur e note s ; 17 ) Includes bibliographica l references . ISBN 978-0-8218-4701- 5 (alk . paper ) 1. Schrodinge r equation . 2 . WK B approximation . 3 . Differentia l equations , Nonlinear .

4. Nonlinea r theories . I . Title .

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Contents

Preface vi i

Chapter 1 . Th e Classical WKB Method 1 1.1. Introductio n 1 1.2. WK B Approximation to the Linear Schrodinger Equation 2 1.3. Th e WKB Approximation to Nonlinear Schrodinger Equation 5 1.4. Reference s an d Remarks 1 0

Chapter 2. Wigne r Measure 1 1 2.1. Semiclassica l Pseudodifferential Operator s and the FBI Transform 1 1 2.2. Wigne r Measure 1 4 2.3. Semiclassica l Limit of the Linear Schrodinger Equation 3 0 2.4. Reference s an d Remarks 3 4

Chapter 3. Th e Limit from the One-Dimensional Schrodinger-Poisso n to Vlasov-Poisson Equations 3 7

3.1. Introductio n 3 7 3.2. Unifor m Estimates 4 2 3.3. Th e Limit to the Vlasov-Poisson Equations 4 6 3.4. Th e Main Theorem 7 3 3.5. Mixe d State Case 7 6 3.6. Reference s an d Remarks 7 9

Chapter 4. Semiclassica l Limit of Schrodinger-Poisson Equations 8 1 4.1. Introductio n 8 1 4.2. Preliminarie s 8 4 4.3. Modulate d Energy Estimate 8 9 4.4. Th e Main Theorem 9 5 4.5. Reference s an d Remarks 10 4

Chapter 5. Semiclassica l Limit of the Cubic Schrodinger Equation in an Exterior Domain 10 5

5.1. Introductio n 10 5 5.2. Loca l Existence of a Smooth Solution to (5.7)-(5.8) 10 9 5.3. Unifor m an d Modulated Energy Estimates 11 5 5.4. Semiclassica l Limit of the Cubic Schrodinger Equation 12 5 5.5. Reference s an d Remarks 12 7

vi CONTENT S

Chapter 6. Incompressibl e and Compressible Limits of Coupled Systems of Nonlinear Schrodinger Equations 12 9

6.1. Introductio n 12 9 6.2. Conservatio n Laws 13 1 6.3. Incompressibl e Limit of (6.1) 13 6 6.4. Compressibl e Limit of (6.2) 14 6 6.5. Reference s an d Remarks 15 2

Chapter 7. High-Frequenc y Limi t of the Helmholtz Equation 15 3 7.1. Introductio n 15 3 7.2. Unifor m Estimate for Solutions of (7.1) 15 4 7.3. Limi t to the Liouville Equation 15 9 7.4. Reference s an d Remarks 17 1

Appendix A. Globa l Solutions to (3.14) 17 3

Appendix B. Densenes s of Polynomials 17 7

Appendix C. Globa l Existence of a Solution to (5.1) 17 9

Appendix D. Globa l Smooth Solution to (6.1) 18 9

Bibliography 193

Preface

In fal l 2006 , I taught a course o n semielassiea l pseudodifferentia l operator s at the Couran t Institut e o f Ne w Yor k University. I n the semeste r tha t followed , I taugh t a cours e calle d "Wigne r measur e an d semielassiea l limit s o f nonlinea r Schrodinger equations " there. Th e main purpose o f the sprin g course was to ap-ply the theory of semielassieal pseudodifferential operator s to the study of various high-frequency limit s of equations from quantum mechanics. In particular, we pre-sented detailed explanations of Wigner measure and then tried to summarize some recent progress on how to use this tool to study various problems on semielassiea l limits of nonlinear Schrodinger equations.

Because there ar e already som e very nice books on semielassiea l pseudodif -ferential operators , such as those by D. Robert [84] and A. Martinez [72] , I mainly present the material from m y spring course in 2007.

This book is organized into seven chapters. Chapte r 1 deal s with the classical WKB method . Chapte r 2 deals wit h th e theor y o f Wigne r measure . Chapte r 3 deals wit h th e semielassiea l limi t fro m one-dimensiona l Schrodinger-Poisso n t o Vlasov-Poisson equations . Chapte r 4 deals wit h th e semielassiea l limit s o f mul-tidimensional Schrodinger-Poisso n equations . Chapte r 5 deals with semielassiea l limits of the cubic Schrodinger equation in an exterior domain . Chapte r 6 covers semielassieal limits of the coupled nonlinear Schrodinger system. In the last chap-ter, we shall digress a little bit from the main topic and talk about the applications of Wigner measure to the study of high-frequency limit s of Helmholtz equations. Finally, we have four appendices concerning the global existence of smooth solu-tions to various Schrodinger-type equations.

I would like to take this opportunity t o thank Professor Charle s M. Newman for invitin g m e t o visi t Couran t Institut e fro m th e fal l o f 200 5 t o th e sprin g o f 2007. It is my great pleasure to thank Prof. Fanghua Lin and Prof. Jalal Shatah for encouraging m e to write down these notes. I t is also my grea t pleasure to thank all o f m y collaborator s o n thi s topic : Profs . Fanghu a Lin , Taichi a Lin , Norber t J. Mauser, Xuepin g Wang , an d Yuxi Zheng. I should als o thank my class a t the Courant Institute: Diog o Arsenio, Hantaek Baik, Jose Diaz-Alban, David Gerard-Varet, Pierre Germain, Nam Le, Ian Tice, Yong Yu, and Zhifei Zhang, with special thanks to Zhifei Zhang for reading the draft versio n of this book. Finally , it was a pleasure to collaborate with Paul Monsour in his beautiful editin g work.

Part o f Chapte r 2 was als o presented b y th e autho r i n the summe r schoo l a t Fudan University an d Wuhan University , China , in the summer of 2007. I would

vii

viii PREFAC E

like to thank Profs. Jiaxing Hong and Hua Chen for their kind invitation and con-stant support . I woul d als o lik e t o acknowledg e th e suppor t fro m th e Nationa l Science Foundation of China under grants 1013105 0 and 10276036 , the National "973" Project, and the Chinese Academy of Sciences through an Innovation Grant.

Bibliography

[1] Agmon , S. , an d Hbrmander , L . Asymptotic propertie s o f solution s t o differential equation s with simple characteristics. /. Analyse Math. 30: 1-38 , 1976 .

[2] Alazard , T. , an d Carles , R . Semi-classica l limi t o f Schrbdinger-Poisso n equation s i n spac e dimension n > 3 . J. Differential Equations 233: 241-275, 2007.

[3] Alazard , T., and Carles, R. Supercritical geometric optics for nonlinear Schrodinger equations. Preprint, 2007. arXiv:0704.2488v2.

[4] Alazard , T. , an d Carles , R . WK B analysi s fo r th e Gross-Pitaevski i equatio n wit h non -trivial boundary condition s a t infinity. Ann. Inst. H. Poincare Anal. Non Lineaire, to appear . arXiv:0710.0816vl.

[5] Anton , R. Strichartz inequalities for Lipschitz metric on manifolds and nonlinear Schrodinger equation on domains. Preprint, 2005. arXiv:math/0512639vl.

[6] Ao , P. , an d Chui , S . T . Binarj|Bose-Einstein condensat e mixture s i n weakl y an d strongl y segregated phases. Phys. Rev. ^58(6): 4836-4840, 1998.

[7] Barros-Neto , J . An introduction to the theory of distributions. Pure and Applied Mathemat -ics, 14 . Dekker, New York, 1973.

[8] Bechouche , P., Mauser, N. J., and Poupaud, F. (Semi)-nonrelativistic limits of the Dirac equa-tion with external time-dependent electromagnetic field. Comm. Math. Phys. 197(2): 405^25, 1998.

[9] Bechouche , P., Mauser, N. J., and Poupaud, F. Semiclassical limit for the Schrodinger-Poisson equation in a crystal. Comm. Pure Appl. Math. 54(7): 851-890, 2001.

[10] Beira o d a Veiga , H . O n th e barotropi c motio n o f compressibl e perfec t fluids . Ann. Scuola Norm. Sup. Pisa CI. Set (4) 8(2): 317-351, 1981.

[11] Beira o d a Veiga , H . Data dependenc e i n th e mathematica l theor y o f compressibl e invisci d fluids. Arch. Rational Meek Anal. 119(2) : 109-127 , 1992.

[12] Benamou , J.-D. , Castella , R , Katsaounis , T. , an d Perthame, B . High frequency limi t o f th e Helmholtz equations. Rev. Mat. Iberoamericana 18(1) : 187-209 , 2002.

[13] Bony , J.-F . Mesures limite s pou r 1'equatio n de Helmholt z dan s l e ca s no n captif . Preprint , 2007. arXiv:0707.0829.

[14] Brenier , Y. Convergence of the Vlasov-Poisson system to the incompressible Euler equations. Comm. Partial Differential Equations 25(3-4): 737-754, 2000.

[15] Brezis , H., and Gallouet, T. Nonlinear Schrodinger evolution equations. Nonlinear Anal. 4(4): 677-681, 1980 .

[16] Brezzi , F., and Markowich, P. A. The three-dimensional Wigner-Poisson problem: existence , uniqueness and approximation. Math. Methods Appl. Sci. 14(1) : 35-61, 1991.

[17] Burq , N. Semi-classical estimates for the resolvent in nontrapping geometries. Int. Math. Res. Not. 2002(5): 221-241, 2002.

[18] Carles , R. Remarques sur les mesures de Wigner. C. R. Acad. Sci. Paris Ser. I Math. 332(11) : 981-984,2001.

[19] Carles , R . WKB analysi s fo r nonlinea r Schrodinge r equation s wit h potential . Comm. Math. Phys. 269(1): 195-221 , 2007.

[20] Carles , R. Semi-classical analysis for nonlinear Schrodinger equations. World Scientific, Lon-don, to appear.

193

194 BIBLIOGRAPH Y

[21] Castella , E L2 solution s to the Schrodinger-Poisson system : existence , uniqueness, time be-haviour, and smoothing effects. Math. Models Methods Appl. Sci. 7(8): 1051-1083 , 1997.

[22] Castella , F. The radiation condition at infinity for the high-frequency Helmholt z equation with source term: a wave packet approach. /. Funct. Anal. 223(1): 204-257, 2005.

[23] Castella , E , an d Jecko, T. Besov estimates i n the high-frequency Helmholt z equation , fo r a non-trapping and C2 potential . J. Differential Equations 228(2): 440-485, 2006.

[24] Castella , E, Perthame, B., and Runborg, O. High frequency limi t of the Helmholtz equation . II. Source on a general smooth manifold. Comm. Partial Differential Equations 27(3-4): 607-651,2002.

[25] Cazenave , T, an d Haraux, A. Introduction aux problemes d'evolution semi-lineaires. Mathe-matiques et Applications (Paris), 1 . Ellipses, Paris, 1990.

[26] Desjardins , B. , Lin , C.-K. , an d Tso , T.-C . Semiclassica l limi t o f th e derivativ e nonlinea r Schrodinger equation. Math. Models Methods Appl. Sci. 10(2) : 261-285, 2000.

[27] DiPerna , R . J. , an d Lions , P.-L . Solution s globale s d'equation s d u typ e Vlasov-Poisson . C. R. Acad. Sci. Paris Ser I Math. 307(12): 655-658, 1988.

[28] Dong , G. C. Nonlinear partial differential equations of second order. Translations of Mathe-matical Monographs, 95. American Mathematical Society , Providence, R.I., 1991.

[29] Esry , B . D. , an d Greene , C . H . Spontaneou s spatia l symmetr y breaking i n two-componen t Bose-Einstein condensates. Phys. Rev. A 59(2): 1457-1460 , 1999.

[30] Evans , L . C , an d Zworski , M . Lectures on semiclassical analysis. Versio n 0.3 . Availabl e online at: h t tp : / /math .berke ley .edu/~zworski / semic lass ica l .pd f

[31] Frisch , T. , Pomeau, Y. , and Rica, S . Transition to dissipation i n a model o f superflow . Phys. Rev. Lett. 69(11): 1644-1647 , 1992.

[32] Gerard , P . Mesure s semi-classique s e t onde s d e Bloch . Seminaire sur les Equations aux Derivees Partielles, 1990-1991, exp . 16. Ecole Polytechnique, Palaiseau, 1991.

[33] Gerard , P . Microlocal defec t measures . Comm. Partial Differential Equations 16(11) : 1761 -1794, 1991.

[34] Gerard , P. Remarques sur 1'analyse semi-classique de l'equation de Schrodinger non lineaire. Seminaire sur les Equations aux Derivees Partielles, 1992-1993, exp . 13 . Ecole Polytech -nique, Palaiseau, 1993.

[35] Gerard , P. The Cauchy problem for the Gross-Pitaevskii equation. Ann. Inst. H. Poincare Anal. Non Lineaire 23(5): 765-779, 2006.

[36] Gerard , P. , Markowich , P . A., Mauser , N . J. , an d Poupaud , F . Homogenization limit s an d Wigner transforms. Comm. Pure Appl. Math. 50(4): 323-379, 1997.

[37] Ginibre , J., and Velo, G. On a class of nonlinear Schrodinger equations with nonlocal interac-tion. Math. Z 170(2) : 109-136 , 1980.

[38] Ginibre , J., and Velo, G. On the global Cauchy problem for some nonlinear Schrodinger equa-tions. Ann. Inst. H. Poincare Anal. Non Lineaire 1(4): 309-323, 1984.

[39] Ginzburg , V . L., an d Pitaevskii , L . P . On th e theor y o f superfluidity . Soviet Physics. JETP 34(7): 858-861 , 1958.

[40] Grenier , E . Semiclassica l limi t o f th e nonlinea r Schrodinge r equatio n i n smal l time . Proc. Amer. Math. Soc. 126(2) : 523-530, 1998.

[41] Gross , E. P. Hydrodynamics of a superfluid condensate . /. Math. Phys. 4(2): 195-207 . [42] Hall , D. S., Matthews, M. R., Ensher, J. R., Wieman, C. E., and Cornell, E. A. The dynamics

of componen t separatio n i n a binary mixtur e o f Bose-Einstein condensates . Phys. Rev. Lett. 81: 1539-1542 , 1998 . arXiv:cond-mat/9804138vl.

[43] Hasan , Z. R., and Goble, D. F. Effect o f boundary condition s on finite Bose-Einstein assem -blies. Phys. Rev. A 10(2) : 618-624, 1974.

[44] He y wood, J. G. The Navier-Stokes equations: o n the existence, regularity an d decay of solu-tions. Indiana Univ. Math. J. 29(5): 639-681, 1980 .

[45] Hormander , L . Linear partial differential operators. Di e Grundlehre n de r mathematische n Wissenschaften, 116 . Academic, New York; Springer, Berlin-Gbttingen-Heidelberg, 1963 .

BIBLIOGRAPHY 195

[46] Hormander , L . The analysis of linear partial differential operators. I. Distribution theory and Fourier analysis. II. Differential operators with constant coefficients. III. Pseudodif-ferential operators. IV. Fourier integral operators. Grundlehre n de r mathematische n Wis -senschaften, 256 , 257, 274, 275. Springer, Berlin, 1983 , 1985. Hormander, L. Lectures on nonlinear hyperbolic differential equations. Mathematiques et Ap-plications (Berlin), 26. Springer, Berlin, 1997. Illner, R., Zweifel, P. R, and Lange, H. Global existence, uniqueness and asymptotic behaviour of solution s o f th e Wigner-Poisson an d Schrodinger-Poisso n systems . Math. Methods Appl. Sci. 17(5) : 349-376, 1994. Jin, S. , Levermore , C . D. , an d McLaughlin , D . W . The behavio r o f solution s o f th e NL S equation in the semiclassical limit. Singular limits of dispersive waves (Lyon, 1991), 235-255. NATO Advanced Science Institutes Series B: Physics, 320. Plenum, New York, 1994. Jin, S.; Levermore, C. D.; McLaughlin, D. W. The semiclassical limit of the defocusing NL S hierarchy. Comm. Pure Appl. Math. 52(5): 613-654, 1999. Josserand, C, an d Pomeau, Y. Nonlinear aspects of the theory of Bose-Einstein condensates . Nonlinearity 14(5) : R25-R62, 2001. Kato, T. Schrbdinger operators with singular potentials. Israel J. Math. 13 : 135-148, 1972. Kato, T. Perturbation theory for linear operators. Springer, Berlin, 1980. Landau, L. D., and Lifschitz, E. M. Lehrbuch der theoretischen Physik. III. Quantenmechanik. Akademie, Berlin, 1985. Landau, L. D., and Lifshitz, E . M. Course of theoretical physics. 6. Fluid mechanics. Perga-mon, London-New York , 1989. Lebeau, G. Regularite du probleme de Kelvin-Helmholtz pour 1'equation d'Euler 2d . ESAIM Control Optim. Calc. Van 8 : 801-825, 2002 (electronic). Lee, C. C, an d Lin, T.-C. Incompressible and compressible limits of two-component Gross -Pitaevskii equation s wit h rotatin g field s an d tra p potentials . J. Math. Phys. 49 : 043517-1 -043517-28,2008. Lin, F. H., and Lin, T.-C. Vortices in two-dimensional Bose-Einstei n condensates . Geometry and nonlinear partial differential equations (Hangzhou, 2001), 87-114 . AMS/IP Studie s i n Advanced Mathematics, 29. American Mathematical Society, Providence, R.I., 2002. Lin, F.-H., and Xin, J. X. On the incompressible fluid limi t and the vortex motion law of the nonlinear Schrbdinger equation. Comm. Math. Phys. 200(2): 249-274, 1999. Lin, E, and Zhang, P. On the hydrodynamic limi t of Ginzburg-Landau wav e vortices. Comm. Pure Appl. Math. 55(7): 831-856, 2002. Lin, E, an d Zhang, P . Semiclassical limi t of the Gross-Pitaevskii equatio n in an exterior do-main. Arch. Ration. Mech. Anal. 179(1) : 79-107, 2006. Lin, T.-C, and Zhang, P. Incompressible and compressible limits of coupled systems of non-linear Schrbdinger equations. Comm. Math. Phys. 266(2): 547-569, 2006. Lions, J.-L. Quelques methods de resolution des problemes aux limites non lineaires. Dunod; Gauthier-Villars, Paris, 1969. Lions, P.-L . Mathematical topics in fluid mechanics. Vol. 1. Incompressible models. Oxfor d Lecture Serie s in Mathematics an d Its Applications, 3 . Clarendon, Oxfor d Universit y Press , New York, 1996. Lions, P.-L., and Perthame, B. Propagation of moments an d regularity for the 3-dimensiona l Vlasov-Poisson system . Invent. Math. 105(2) : 415^30, 1991. Lions, P.-L., and Paul, T. Sur les mesures de Wigner. Rev. Mat. Iberoamericana 9(3): 553-618, 1993. Loffredo, M . I. , an d Morato , L . M . Self-consisten t hydrodynamica l mode l fo r H e I I nea r absolute zero in the framework of stochastic mechanics. Phys. Rev. B 35(4): 1742-1747 , 1987. Madelung, E. Quantentheorie in hydrodynamischer Form. Z. Physik 40: 322-326 , 1926. Majda, A . Compressible fluid flow and systems of conservation laws in several space vari-ables. Applied Mathematical Sciences , 53. Springer, New York, 1984.

196 BIBLIOGRAPH Y

[70] Markowich , P. A., and Mauser, N. J. The classical limit of a self-consistent quantum-Vlaso v equation in 3D. Math. Models Methods Appl. Sci. 3(1): 109-124 , 1993.

[71] Markowich , P . A., Ringhofer , C . A. , an d Schmeiser , C . Semiconductor equations. Springer , Vienna, 1990.

[72] Martinez , A. An introduction to semiclassical and microlocal analysis. Universitext. Springer, New York, 2002.

[73] Maslov , V . P. , an d Fedoriuk , M . V . Semiclassical approximation in quantum mechanics. Mathematical Physic s an d Applie d Mathematics , 7 . Contemporary Mathematics , 5 . Reidel , Dordrecht-Boston, 1981.

[74] Masmoudi, N., and Mauser, N. J. The self consistent Pauli equation. Monatsh. Math. 132(1) : 19-24,2001.

[75] Mourre , E. Absence of singular continuous spectrum for certain selfadjoint operators . Comm. Math. Phys. 78(3): 391-408, 1980/81.

[76] Mourre , E. Operateurs conjugues e t proprietes de propagation. Comm. in Math. Phys. 91(2): 279-300, 1983.

[77] Nirenberg , L . On elliptic partia l differentia l equations . Ann. Scuola Norm. Sup. Pisa (3) 13: 115-162,1959.

[78] Perez-Garcia , V. M., Konotop, V. V., and Brazhnyi, V. A. Feshbach resonance induced shock waves in Bose-Einstein condensates. Phys. Rev. Lett. 92(22): 2004. 220403-1-22043-4.

[79] Perthame, B., and Vega, L. Morrey-Campanato estimate s for Helmholt z equations . J. Fund. Anal. 164(2) : 340-355, 1999.

[80] Pfaffelmoser , K . Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data. /. Differential Equations 95(2): 281-303, 1992.

[81] Pitaevskii , L. , an d Stringari , S . Bose-Einstein condensation. Internationa l Serie s o f Mono -graphs on Physics, 116 . Clarendon, Oxford University Press, Oxford, 2003 .

[82] Puel , M. Convergence of the Schrodinger-Poisson syste m to the incompressible Eule r equa-tions. Comm. Partial Differential Equations 21 (\\-\2): 2311-2331 , 2002.

[83] Ranch , J. B., and Massey, F. J., III. Differentiability o f solutions to hyperbolic initial-boundary value problems. Trans. Amer. Math. Soc. 189: 303-318, 1974.

[84] Robert , D . Autour de Vapproximation semi-classique. Progres s i n Mathematics , 68 . Birkhouser, Boston, 1987.

[85] Schaeffer , J . Global existence of smoot h solution s to the Vlasov-Poisson syste m in three di-mensions. Comm. Partial Differential Equations 16(8-9) : 1313-1335 , 1991.

[86] Schochet , S . The compressible Euler equations in a bounded domain : existenc e of solution s and the incompressible limit. Comm. Math. Phys. 104(1) : 49-75, 1986.

[87] Segal , I. Non-linear semi-groups. Ann. of Math. (2) 78: 339-364, 1963. [88] Sideris , T . C . Formatio n o f singularitie s i n three-dimensiona l compressibl e fluids . Comm.

Math. Phys. 101(4) : 475^185, 1985. [89] Stein , E. M. Singular integrals and differentiability properties of functions. Princeto n Mathe-

matical Series, 30. Princeton University Press, Princeton, N.J., 1970. [90] Stein , E. M., and Weiss, G. Introduction to Fourier analysis on Euclidean spaces. Princeto n

Mathematical Series, 32. Princeton University Press, Princeton, N.J., 1971. [91] Sulem , C , an d Sulem , P . L. The nonlinear Schrodinger equation. Self-focusing and wave

collapse. Applied Mathematical Sciences, 139. Springer, New York, 1999. [92] Tartar , L . //-measures , a new approac h fo r studyin g homogenisation , oscillation s an d con-

centration effects i n partial differential equations . Proc. Roy. Soc. Edinburgh Sect. A 115(3-4) : 193-230, 1990.

[93] Taylor , M. E. Pseudodifferential operators and nonlinear PDE. Progress in Mathematics, 100. Birkhauser, Boston, 1991.

[94] Timmermans , E . Phas e separatio n o f Bose-Einstei n condensates . Phys. Rev. Lett. 81(26) : 5718-5721, 1998.

[95] Tsutsumi , M . O n smoot h solution s t o the initial-boundar y valu e problem fo r th e nonUnea r Schrodinger equation in two space dimensions. Nonlinear Anal. 13(9) : 1051-1056 , 1989.

BIBLIOGRAPHY 197

[96] Vol'pert , A. I. Spaces BV and quasilinear equations. Mat Sb. (N.S.) 73(115): 255-302, 1967; translation in Math. USSR Sb. 2(2): 225-267, 1967.

[97] Vol'pert , A. I., and Hudjaev, S . I. Analysis in classes of discontinuous functions and equations of mathematical physics. Mechanics : Analysis , 8. Nijhoff, Dordrecht , 1985.

[98] Wang , X . P . Time-decay o f scatterin g solution s an d resolven t estimate s fo r semiclassica l Schrodinger operators. J. Differential Equations 71(2): 348-395, 1988.

[99] Wang , X. P. Microlocal estimates of the Schrodinger equation in semi-classical limit . Partial differential equations and applications. Proceedings of the CIMPA School held in Lanzhou (2004), 265-307. Seminaires et Congres, 15. Societe Mathematique de France, Paris, 2007.

[100] Wang , X . P. , an d Zhang , P . High-frequency limi t o f th e Helmholt z equatio n wit h variabl e refraction index. /. Funct. Anal. 230(1): 116-168 , 2006.

[101] Wigner , E. On the quantum correction for thermodynamic equilibrium. Phys. Rev. 40(5): 742-759, 1932.

[102] Wu , S. Mathematical analysi s of vortex sheets . Comm. Pure Appl. Math. 59(8) : 1065-1206 , 2006.

[103] Xin , Z., and Zhang, P. On the weak solutions to a shallow water equation. Comm. Pure Appl. Math. 53(11): 1411-1433,2000 .

[104] Zhang , P . Semiclassica l limi t o f nonlinea r Schrodinge r equation . II . J. Partial Differential Equations 15(2) : 83-96, 2002.

[105] Zhang , P. Wigner measure and the semiclassical limit of Schrodinger-Poisson equations. SIAM J. Math. Anal. 34(3) : 700-718, 2002 (electronic).

[106] Zhang , P., and Qiu, Q. On the existence of almost global weak solution to multidimensiona l Vlasov-Poisson equation. Chinese Ann. Math. Ser. B 19(3) : 381-390, 1998.

[107] Zhang , P. , and Zheng , Y . Existence an d uniqueness o f solution s o f a n asymptoti c equatio n arising from a variational wave equation with general data. Arch. Ration. Mech. Anal. 155(1) : 49-83, 2000.

[108] Zhang , P., Zheng, Y., and Mauser, N. J. The limit from the Schrodinger-Poisson to the Vlasov-Poisson equations with general data in one dimension. Comm. Pure Appl. Math. 55(5): 582 -632, 2002.

[109] Zheng , Y . X., and Majda, A . Existence o f global weak solution s t o one-component Vlasov -Poisson and Fokker-Planck-Poisson system s in one space dimension with measures as initial data. Comm. Pure Appl. Math. 47(10): 1365-1401 , 1994.

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Titles i n Thi s Serie s

17 Pin g Zhang , Wigne r measur e an d semiclassica l limit s o f nonlinea r Schrodinge r equations , 2008

16 S . R . S . Varadhan , Stochasti c processes , 200 7

15 Emi l Art in , Algebr a wit h Galoi s theory , 200 7

14 Pete r D . Lax , Hyperboli c partia l differentia l equations , 200 6

13 Olive r Biihler , A brie f introductio n t o classical , statistical , an d quantu m mechanics , 200 6

12 Jiirge n Mose r an d Eduar d J . Zehnder , Note s o n dynamica l systems , 200 5

11 V . S . Varadarajan , Supersymmetr y fo r mathematicians : A n introduction , 200 4

10 Thierr y Cazenave , Semilinea r Schrodinge r equations , 200 3

9 Andre w Majda , Introductio n t o PDE s an d wave s fo r th e atmospher e an d ocean , 200 3

8 Fedo r Bogomolo v an d Tihomi r Petrov , Algebrai c curve s an d one-dimensiona l fields ,

2003

7 S . R . S . Varadhan , Probabilit y theory , 200 1

6 Loui s Nirenberg , Topic s i n nonlinea r functiona l analysis , 200 1

5 Emmanue l Hebey , Nonlinea r analysi s o n manifolds : Sobole v space s an d inequalities ,

2000

3 Perc y Deift , Orthogona l polynomial s an d rando m matrices : A Riemann-Hilber t

approach, 200 0

2 Jala l Shata h an d Michae l Struwe , Geometri c wav e equations , 200 0

1 Qin g Ha n an d Fanghu a Lin , Ellipti c partia l differentia l equations , 200 0