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Journal of Differential Equations, ISSN 0022-0396, 12/2019, Volume 268, Issue 1, pp. 160 - 185
We show robustness of various truncated and subcritical approximations to the stochastic defocusing mass-critical nonlinear Schrödinger equation (NLS) in... 
MATHEMATICS | GLOBAL WELL-POSEDNESS | SCATTERING
Journal Article
Communications on Pure and Applied Mathematics, ISSN 0010-3640, 07/2016, Volume 69, Issue 7, pp. 1354 - 1396
For the physical vacuum free boundary problem with the sound speed being C1/2‐Hölder continuous near vacuum boundaries of the one‐dimensional compressible... 
MATHEMATICS | MATHEMATICS, APPLIED | WELL-POSEDNESS
Journal Article
Journal of Differential Equations, ISSN 0022-0396, 12/2015, Volume 259, Issue 11, pp. 5982 - 6008
In this paper, we obtain global existence and energy decay for 3D Hall-magnetohydrodynamics (Hall-MHD) system with −Δu and −ΔB. Besides the classical energy... 
Energy decay | Global well-posedness | Blow-up criteria | Hall-MHD system | MATHEMATICS | NAVIER-STOKES EQUATIONS | WELL-POSEDNESS | MAGNETOHYDRODYNAMICS | WEAK SOLUTIONS
Journal Article
Communications on Pure and Applied Mathematics, ISSN 0010-3640, 10/2015, Volume 68, Issue 10, pp. 1683 - 1741
We prove local existence and uniqueness for the two‐dimensional Prandtl system in weighted Sobolev spaces under Oleinik's monotonicity assumption. In... 
BOUNDARY-LAYER EQUATIONS | ANALYTIC SOLUTIONS | MATHEMATICS | MATHEMATICS, APPLIED | NAVIER-STOKES EQUATIONS | VANISHING VISCOSITY | WELL-POSEDNESS | HALF-SPACE | ZERO VISCOSITY LIMIT | ILL-POSEDNESS | EULER | INVISCID LIMIT
Journal Article
Journal of Differential Equations, ISSN 0022-0396, 01/2016, Volume 260, Issue 1, pp. 818 - 859
We propose a priori estimates for a weak solution to the derivative nonlinear Schrödinger equation (DNLS) on torus with small L2-norm datum in low regularity... 
Nonlinear Schrödinger equation | Well-posedness
Journal Article
Annales de l'Institut Henri Poincaré / Analyse non linéaire, ISSN 0294-1449, 03/2020, Volume 37, Issue 2, pp. 373 - 416
This paper is concerned with the Cauchy problem of the modified Kawahara equation (posed on T), which is well-known as a model of capillary-gravity waves in an... 
Unconditional uniqueness | Weak ill-posedness | Global well-posedness | Modified Kawahara equation | Initial value problem | KDV EQUATIONS | MATHEMATICS, APPLIED | GLOBAL EXISTENCE | STABILITY | CAUCHY-PROBLEM | SQUEEZING THEOREM | ILL-POSEDNESS | NONLINEAR SCHRODINGER-EQUATION | SOLITARY WAVES | CAPILLARY-GRAVITY WAVES | LOW REGULARITY
Journal Article
Nonlinearity, ISSN 0951-7715, 02/2012, Volume 25, Issue 2, pp. 449 - 479
This work studies the initial value problem for a Camassa-Holm type equation with cubic nonlinearities that has been recently discovered by Vladimir Novikov to... 
PEAKON | MATHEMATICS, APPLIED | NONUNIFORM DEPENDENCE | WELL-POSEDNESS | ILL-POSEDNESS | PHYSICS, MATHEMATICAL | INTEGRABLE EQUATION | Work study | Approximation | Fluid dynamics | Mathematical analysis | Nonlinearity | Initial value problems | Galerkin methods | Optimization
Journal Article
SIAM Journal on Mathematical Analysis, ISSN 0036-1410, 2018, Volume 50, Issue 3, pp. 3172 - 3209
The aim of this paper is to prove various ill-posedness and well-posedness results on the Cauchy problem associated to a class of fractional... 
Strichartz estimate | Burgers equation | KP equations | Ill-posedness | Dispersion | Well-posedness | EXISTENCE | MATHEMATICS, APPLIED | INSTABILITY | well-posedness | STABILITY | ill-posedness | SMOOTHING PROPERTIES | GLOBAL WELL-POSEDNESS | BURGERS | INTERNAL WAVES | INITIAL-VALUE PROBLEM | SOLITARY-WAVE SOLUTIONS | BLOW-UP | dispersion
Journal Article
Annals of Mathematics, ISSN 0003-486X, 5/2010, Volume 171, Issue 3, pp. 1903 - 1930
Motivated by the critical dissipative quasi-geostrophic equation, we prove that drift-diffusion equations with L² initial data and minimal assumptions on the... 
Cubes | Maximum principle | Mathematical theorems | Truncation | Cylinders | MATHEMATICS | REGULARITY | GLOBAL WELL-POSEDNESS | BEHAVIOR
Journal Article
Mathematische Nachrichten, ISSN 0025-584X, 02/2017, Volume 290, Issue 2-3, pp. 435 - 441
This paper addresses the problem of well‐posedness of non‐autonomous linear evolution equations ẋ=A(t)x in uniformly convex Banach spaces. We assume that... 
34K30 | 47D06 | 81Q99 | time‐dependent Hamiltonians | Schrödinger equations | Non‐autonomous linear evolution equations | well‐posedness | counterexamples to well‐posedness | counterexamples to well-posedness | Non-autonomous linear evolution equations | time-dependent Hamiltonians | well-posedness | MATHEMATICS | Schrodinger equations | TIME
Journal Article
Annales de l'Institut Henri Poincaré / Analyse non linéaire, ISSN 0294-1449, 07/2018, Volume 35, Issue 4, pp. 1119 - 1142
In this paper, we prove the well-posedness of the linearized Prandtl equation around a non-monotonic shear flow in Gevrey class 2−θ for any θ>0. This result is... 
Prandtl equation | Gevrey class | Well-posedness | EXISTENCE | SYSTEM | MATHEMATICS, APPLIED | MONOTONICITY | ILL-POSEDNESS | EULER | Information science
Journal Article
Journal of Functional Analysis, ISSN 0022-1236, 2011, Volume 260, Issue 4, pp. 1060 - 1085
This paper addresses well-posedness issues for the initial value problem (IVP) associated with the generalized Zakharov–Kuznetsov equation, namely, { u t + ∂ x... 
Local well-posedness | Global well-posedness | Zakharov–Kuznetsov equation | Zakharov-Kuznetsov equation | EXISTENCE | MATHEMATICS | CAUCHY-PROBLEM | ILL-POSEDNESS | MODIFIED KDV | KORTEWEG-DEVRIES
Journal Article
Transactions of the American Mathematical Society, ISSN 0002-9947, 02/2020, Volume 373, Issue 2, pp. 1043 - 1107
Consider the Hamiltonian abcd system in one dimension, with data posed in the energy space H^1\times H^1. This model, introduced by Bona, Chen, and Saut, is a... 
EXISTENCE | MATHEMATICS | AMPLITUDE LONG WAVES | LARGE TIME | STABILITY | BOUSSINESQ EQUATIONS | BREATHERS | SHARP WELL-POSEDNESS | ILL-POSEDNESS | MODEL-EQUATIONS | SCATTERING
Journal Article
Communications on Pure and Applied Analysis, ISSN 1534-0392, 11/2015, Volume 14, Issue 6, pp. 2265 - 2282
Let sigma is an element of (0, 1) with sigma not equal 1/2. We investigate the fractional nonlinear Schrodinger equation in R-d: i partial derivative(t)u +... 
Sobolev spaces | Ill-posedness | Local and global well-posedness | Fractional Schrödinger | MATHEMATICS | MATHEMATICS, APPLIED | WAVE EQUATION | Fractional Schrodinger | ill-posedness | BLOW-UP | GLOBAL WELL-POSEDNESS | SCATTERING
Journal Article
Journal of Differential Equations, ISSN 0022-0396, 10/2017, Volume 263, Issue 8, pp. 4658 - 4722
We establish the global well-posedness of the derivative nonlinear Schrödinger equation with periodic boundary condition in the Sobolev space H12, provided... 
Global well-posedness | Derivative nonlinear Schrödinger equation
Journal Article
Comptes rendus - Mathématique, ISSN 1631-073X, 05/2012, Volume 350, Issue 9-10, pp. 499 - 503
In this Note we study the generalized 2D Zakharov–Kuznetsov equations ∂tu+Δ∂xu+uk∂xu=0 for k⩾2. By an iterative method we prove the local well-posedness of... 
MATHEMATICS | WELL-POSEDNESS
Journal Article
Journal of Differential Equations, ISSN 0022-0396, 10/2018, Volume 265, Issue 8, pp. 3792 - 3840
We study the local and global solutions of the generalized derivative nonlinear Schrödinger equation i∂tu+Δu=P(u,u‾,∂xu,∂xu‾), where each monomial in P is of... 
Derivative nonlinear Schrödinger equations | Local well-posedness | Global well-posedness | EXISTENCE | MATHEMATICS | REGULARITY | SCATTERING | Derivative nonlinear Schrodinger equations
Journal Article
Journal of Mathematical Analysis and Applications, ISSN 0022-247X, 08/2019, Volume 476, Issue 2, pp. 391 - 425
We prove that the initial value problem associated to a nonlocal perturbation of the Benjamin-Ono equation is locally and globally well-posed in Sobolev spaces... 
Benjamin-Ono equation | Sobolev spaces | Local and global well-posedness | Weighted Sobolev spaces | MATHEMATICS | MATHEMATICS, APPLIED | ZAKHAROV-KUZNETSOV EQUATION | CAUCHY-PROBLEM | SHARP WELL-POSEDNESS
Journal Article
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