So, we have been looking at lists of really fundamental papers in PDE that could be read and reported on by grad students in a PDE course by asking all colleagues here at UNC and a few elsewhere. I am not sure that everything on this list is project ready so to speak, but I think it is a good start a really nice set of very fundamental papers in the field ... though obviously not complete!!! Please comment in your favorite fundamental PDE paper and hopefully the list will grow nicely!!

MR1555394 Leray, Jean Sur le mouvement d'un liquide visqueux emplissant l'espace. (French) Acta Math. 63 (1934), no. 1, 193–248, DML Item MR1234453 (94f:35121) Camassa, Roberto; Holm, Darryl D. An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71 (1993), no. 11, 1661–1664, 35Q51 (58F07 76B15 76B25)

MR1289328 (95h:35209) Pego, Robert L.; Weinstein, Michael I. Asymptotic stability of solitary waves. Comm. Math. Phys. 164(1994), no. 2, 305–349. (Reviewer: Peter L. Christiansen), 35Q53 (35B35 76B25)

MR0967634 (90d:35267) Jones, Christopher K. R. T. Instability of standing waves for nonlinear Schrödinger-type equations.Ergodic Theory Dynam. Systems 8$^*$ (1988), Charles Conley Memorial Issue, 119–138.(Reviewer: Rafael José Iório Jr.), 35Q20 (35B35 35J10 78A50)

MR1118699 (92j:35050) Crandall, Michael G.; Ishii, Hitoshi; Lions, Pierre-Louis User's guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. (N.S.) 27 (1992), no. 1, 1–67. (Reviewer: P. Szeptycki),35J60 (35B05 35D05 35G20)

MR0778974 (87e:49035b) Lions, P.-L. The concentration-compactness principle in the calculus of variations. The locally compact case. II. Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), no. 4, 223–283. (Reviewer: Gianfranco Bottaro), 49A50 (49A22)

MR0778970 (87e:49035a) Lions, P.-L. The concentration-compactness principle in the calculus of variations. The locally compact case. I. Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), no. 2, 109–145. (Reviewer: Gianfranco Bottaro), 49A50 (49A22)

MR0649347 (84h:35091a) Cheeger, Jeff; Taylor, Michael On the diffraction of waves by conical singularities. I. Comm. Pure Appl. Math. 35 (1982), no. 3, 275–331. (Reviewer: Vesselin M. Petkov), 35L10 (58G17 78A45)

MR0658471 (84b:58109) Cheeger, Jeff; Gromov, Mikhail; Taylor, Michael Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds. J. Differential Geom. 17 (1982), no. 1, 15–53. (Reviewer: Helga Baum), 58G30 (53C21)

MR0873380 (88b:35205) Sylvester, John; Uhlmann, Gunther A global uniqueness theorem for an inverse boundary value problem. Ann. of Math. (2) 125 (1987), no. 1, 153–169. (Reviewer: P. Szeptycki), 35R30 (86A20)

MR0673830 (84m:35097) Caffarelli, L.; Kohn, R.; Nirenberg, L. Partial regularity of suitable weak solutions of the Navier-Stokes equations. Comm. Pure Appl. Math. 35 (1982), no. 6, 771–831. (Reviewer: Tai Ping Liu), 35Q10 (76D05)

MR0097628 (20 #4096) Lax, Peter D. Asymptotic solutions of oscillatory initial value problems. Duke Math. J. 24 1957 627–646. (Reviewer: Yu Why Chen), 35.00

MR0454365 (56 #12616) Strauss, Walter A. Existence of solitary waves in higher dimensions. Comm. Math. Phys. 55 (1977), no. 2, 149–162. (Reviewer: Alan Jeffrey), 35L05 (81.35)

MR0336122 (49 #898) Gardner, Clifford S.; Greene, John M.; Kruskal, Martin D.; Miura, Robert M. Korteweg-deVries equation and generalization. VI. Methods for exact solution. Comm. Pure Appl. Math. 27 (1974), 97–133. (Reviewer: J. Smoller), 35Q99

MR0296602 (45 #5661) Fefferman, Charles The multiplier problem for the ball. Ann. of Math. (2) 94 (1971), 330–336. (Reviewer: R. Larsen), 42A18 (42A92 47B99)

MR0867665 (88d:35169) Floer, Andreas; Weinstein, Alan Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential. J. Funct. Anal. 69 (1986), no. 3, 397–408. (Reviewer: Pierre-Louis Lions), 35Q20 (58C15 58F05)

MR0708966 (85m:81040a) Simon, Barry Semiclassical analysis of low lying eigenvalues. I. Nondegenerate minima: asymptotic expansions. Ann. Inst. H. Poincaré Sect. A (N.S.) 38 (1983), no. 3, 295–308. (Reviewer: Jean Michel Combes), 81C12 (35P20)

MR0783974 (86i:35130) Weinstein, Michael I. Modulational stability of ground states of nonlinear Schrödinger equations. SIAM J. Math. Anal. 16 (1985), no. 3, 472–491. (Reviewer: Woodford W. Zachary), 35Q20 (78A45 82A45)

MR0820338 (87f:35023) Weinstein, Michael I. Lyapunov stability of ground states of nonlinear dispersive evolution equations.Comm. Pure Appl. Math. 39 (1986), no. 1, 51–67. (Reviewer: Roman Stankiewicz), 35B35 (35Q20)

MR1105875 (93d:35034) Journé, J.-L.; Soffer, A.; Sogge, C. D. Decay estimates for Schrödinger operators. Comm. Pure Appl. Math. 44 (1991), no. 5, 573–604. (Reviewer: P. Szeptycki), 35J10 (35B45 35P05)

MR1972492 (2004k:58046) Blue, P.; Soffer, A. Semilinear wave equations on the Schwarzschild manifold. I. Local decay estimates. Adv. Differential Equations 8 (2003), no. 5, 595–614, 58J45 (35Q75 83C57) NOTE: ORIGINAL VERSION CONTAINS SOME MISTAKES!!

MR1789924 (2001j:35180) Smith, Hart F.; Sogge, Christopher D. Global Strichartz estimates for nontrapping perturbations of the Laplacian. Comm. Partial Differential Equations 25 (2000), no. 11-12, 2171–2183. (Reviewer: Ya-Guang Wang), 35L15 (35B45 35L20)

MR2217314 (2007f:35201) Metcalfe, Jason; Sogge, Christopher D. Long-time existence of quasilinear wave equations exterior to star-shaped obstacles via energy methods. SIAM J. Math. Anal. 38 (2006), no. 1, 188–209 (electronic). (Reviewer: Nikos I. Karachalios), 35L70 (35B45)

MR1945285 (2003m:35167) Keel, Markus; Smith, Hart F.; Sogge, Christopher D. Almost global existence for some semilinear wave equations. Dedicated to the memory of Thomas H. Wolff. J. Anal. Math. 87 (2002), 265–279. (Reviewer: Albert J. Milani),35L70

MR0634248 (84a:35083) Gidas, B.; Ni, Wei Ming; Nirenberg, L. Symmetry of positive solutions of nonlinear elliptic equations in Rn.Mathematical analysis and applications, Part A, pp. 369–402, Adv. in Math. Suppl. Stud., 7a, Academic Press, New York-London,1981. (Reviewer: D. E. Edmunds), 35J60 (53C05 58G20)

MR0544879 (80h:35043) Gidas, B.; Ni, Wei Ming; Nirenberg, L. Symmetry and related properties via the maximum principle.Comm. Math. Phys. 68 (1979), no. 3, 209–243. (Reviewer: È. M. Saak), 35J25 (35B50)

http://math.berkeley.edu/%7Etataru/papers/nas.pdf http://math.berkeley.edu/%7Etataru/papers/phasespace.pdf