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Mathematische Nachrichten, ISSN 0025-584X, 02/2017, Volume 290, Issue 2-3, pp. 435 - 441
Journal Article
Journal of Mathematical Analysis and Applications, ISSN 0022-247X, 07/2019, Volume 475, Issue 2, pp. 1578 - 1595
Journal Article
Annals of Mathematics, ISSN 0003-486X, 7/2005, Volume 162, Issue 1, pp. 291 - 366
This article is concerned with local well-posedness of the Cauchy problem for second order quasilinear hyperbolic equations with rough initial data... 
Rectangles | Wave equations | Uniqueness | Cubes | Coordinate systems | Mathematics | Linear equations | Coefficients | Commutators | Wave packets | EXISTENCE | MATHEMATICS | 2ND-ORDER HYPERBOLIC OPERATORS | STRICHARTZ | COUNTEREXAMPLES | NONSMOOTH COEFFICIENTS | REGULARITY
Journal Article
Advances in Differential Equations, ISSN 1079-9389, 2012, Volume 17, Issue 3-4, pp. 267 - 306
We study the Cauchy problem for a quasilinear wave equation with low-regularity data. A space-time L-2 estimate for the variable coefficient wave equation... 
MATHEMATICS | COUNTEREXAMPLES | DIMENSIONS | MATHEMATICS, APPLIED | REGULARITY
Journal Article
Duke Mathematical Journal, ISSN 0012-7094, 03/2003, Volume 117, Issue 1, pp. 1 - 124
We improve recent results of H. Bahouri and J.-Y. Chemin and of D. Tataru concerning local well-posedness theory for quasilinear wave equations... 
MATHEMATICS | 2ND-ORDER HYPERBOLIC OPERATORS | STRICHARTZ | COUNTEREXAMPLES | NONSMOOTH COEFFICIENTS | REGULARITY | 35L15 | 35B30 | 35L70 | 58J45
Journal Article
Communications in Partial Differential Equations, ISSN 0360-5302, 01/2005, Volume 29, Issue 3-4, pp. 323 - 360
Journal Article
MATHEMATICAL RESEARCH LETTERS, ISSN 1073-2780, 2014, Volume 21, Issue 2, pp. 313 - 332
In this paper, we prove an optimal local well-posedness result for the 1+2 dimensional system of nonlinear wave equations (NLW... 
MATHEMATICS | COUNTEREXAMPLES | 3 SPACE DIMENSIONS | REGULARITY | SCHRODINGER-EQUATION | INITIAL DATA | LOCAL EXISTENCE | WELLPOSEDNESS
Journal Article
Forum Mathematicum, ISSN 0933-7741, 09/2018, Volume 30, Issue 5, pp. 1291 - 1307
...] and is proved by combining the methods in the former paper, the first author, Wang and Yokoyama’s paper [K. Hidano, C. Wang and K. Yokoyama, On almost global existence and local well posedness for some 3-D quasi-linear wave equations, Adv... 
35L52 | Keel–Smith–Sogge-type estimates | Elastic waves | 35Q74 | regularity | almost global existence | MATHEMATICS | LIFE-SPAN | COUNTEREXAMPLES | EXTERIOR | MATHEMATICS, APPLIED | EQUATIONS | Keel-Smith-Sogge-type estimates | LOCAL WELL-POSEDNESS
Journal Article
Bulletin of the Brazilian Mathematical Society, New Series, ISSN 1678-7544, 3/2016, Volume 47, Issue 1, pp. 241 - 253
Journal Article
Journal of Hyperbolic Differential Equations, ISSN 0219-8916, 03/2011, Volume 8, Issue 1, pp. 1 - 8
.... Local well-posedness is shown in the parameter range 2 >= r > 1, s > 1 + 2/r. For r = 2 this coincides with the result of Ponce and Sideris, which is optimal... 
generalized Bourgain spaces | local wellposedness | Nonlinear wave equation | Cauchy problem | EXISTENCE | COUNTEREXAMPLES | MATHEMATICS, APPLIED | REGULARITY | MODIFIED KDV EQUATION | LOCAL WELL-POSEDNESS | PHYSICS, MATHEMATICAL
Journal Article
Proceedings of the American Mathematical Society, ISSN 0002-9939, 08/2008, Volume 136, Issue 8, pp. 2881 - 2886
Explicit counterexamples to bilinear estimates related to the Benjamin-Ono equation in the periodic setting are calculated for functions of zero mean value. As... 
Zero | Mathematics | Counterexamples | Analyticity | Periodic Benjamin-Ono equation | Failure of bilinear estimates | MATHEMATICS | MATHEMATICS, APPLIED | BENJAMIN-ONO-EQUATION | GLOBAL WELL-POSEDNESS | periodic Benjamin-Ono equation | failure of bilinear estimates
Journal Article
Proceedings of the American Mathematical Society, ISSN 0002-9939, 7/2007, Volume 135, Issue 7, pp. 2037 - 2043
In this note we show that the well-posedness range p ∈ (1,2] for $L^{p}$ transmission boundary value problems for the Laplacian in the class of Lipschitz domains established by Escauriaza and Mitrea (2004) is sharp... 
Boundary value problems | Mathematical theorems | Mathematical integrals | Laplacians | Counterexamples | Mathematical functions | College mathematics
Journal Article
Transactions of the American Mathematical Society, ISSN 0002-9947, 11/2008, Volume 360, Issue 11, pp. 5805 - 5830
...=-1, and previous work by Colliander, Delort, Kenig and Staffilani, 2001, established local well-posedness for s > -3/4. 
Nonlinearity | Counterexamples | Function spaces | Parabolas | Cauchy problem | College mathematics | MATHEMATICS | DISPERSIVE EQUATIONS | ILL-POSEDNESS
Journal Article
International Mathematics Research Notices, ISSN 1073-7928, 2008, Volume 2008, Issue 1
We obtain a sharp local well-posedness result for the gradient nonlinear wave equation on a nonsmooth curved background... 
MATHEMATICS | WAVE | COUNTEREXAMPLES | LOCAL EXISTENCE
Journal Article
Mathematische Zeitschrift, ISSN 0025-5874, 6/2008, Volume 259, Issue 2, pp. 343 - 353
In this paper, we study the ill-posdness of the Cauchy problem for semilinear wave equation with very low regularity, where the nonlinear term depends on u and... 
35L05 | Mathematics, general | Mathematics | 35R25 | MATHEMATICS | COUNTEREXAMPLES | LOCAL EXISTENCE | Mathematics - Analysis of PDEs
Journal Article
Proceedings of the Royal Society of Edinburgh. Section A. Mathematics, ISSN 0308-2105, 12/2014, Volume 144, Issue 6, pp. 1245 - 1327
This paper is devoted to the problem of finite-dimensional reduction for parabolic partial differential equations. We give a detailed exposition of the... 
MATHEMATICS | COUNTEREXAMPLES | MATHEMATICS, APPLIED | NAVIER-STOKES EQUATIONS | PARABOLIC EQUATIONS | BACKWARD UNIQUENESS | REACTION-DIFFUSION SYSTEMS | PERSISTENCE | GLOBAL ATTRACTORS | DYNAMICS | LIPSCHITZ EMBEDDINGS | DIFFERENTIAL-EQUATIONS | Partial differential equations | Topological manifolds | Manifolds | Reduction | Dynamics | Dissipation | Projection | Spectra | Inertial
Journal Article
SIAM Journal on Numerical Analysis, ISSN 0036-1429, 4/1991, Volume 28, Issue 2, pp. 333 - 362
In this paper the approximation of the Stokes equation by spectral methods is studied. Attention here is focused on the boundary conditions that are assigned... 
Approximation | Real numbers | Boundary conditions | Spectroscopic analysis | Polynomials | Textual collocation | Mathematical functions | Counterexamples | Velocity | Spectral methods | STOKES EQUATIONS | MATHEMATICS, APPLIED | SOBOLEV SPACES | SPECTRAL METHODS | BOUNDARY CONDITIONS
Journal Article
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