Nonlinear Differential Equations and Applications NoDEA, ISSN 1021-9722, 12/2011, Volume 18, Issue 6, pp. 707 - 735

In this paper we consider the following 2D Boussinesq–Navier–Stokes systems $${\begin{array}{lll}\partial_t u + u \cdot \nabla u + \nabla p = - \nu |D|^\alpha...

Para-differential calculus | Regularization effect | Global well-posedness | Analysis | 35Q35 | 35B33 | 76D03 | Boussinesq system | Mathematics | 76D05 | MATHEMATICS, APPLIED | MAXIMUM PRINCIPLE | Mathematics - Analysis of PDEs

Para-differential calculus | Regularization effect | Global well-posedness | Analysis | 35Q35 | 35B33 | 76D03 | Boussinesq system | Mathematics | 76D05 | MATHEMATICS, APPLIED | MAXIMUM PRINCIPLE | Mathematics - Analysis of PDEs

Journal Article

Communications on Pure and Applied Mathematics, ISSN 0010-3640, 01/2015, Volume 68, Issue 1, pp. 61 - 111

An important problem in gas and fluid dynamics is to understand the behavior of vacuum states, namely the behavior of the system in the presence of a vacuum....

MATHEMATICS | WATER-WAVES | VISCOSITY METHOD | MATHEMATICS, APPLIED | NAVIER-STOKES EQUATIONS | FLUID-DYNAMICS | HYPERBOLIC SYSTEMS | ISENTROPIC GAS-DYNAMICS | FREE-BOUNDARY | GLOBAL SMOOTH SOLUTIONS | SURFACE-TENSION | ELLIPTICAL OPERATORS

MATHEMATICS | WATER-WAVES | VISCOSITY METHOD | MATHEMATICS, APPLIED | NAVIER-STOKES EQUATIONS | FLUID-DYNAMICS | HYPERBOLIC SYSTEMS | ISENTROPIC GAS-DYNAMICS | FREE-BOUNDARY | GLOBAL SMOOTH SOLUTIONS | SURFACE-TENSION | ELLIPTICAL OPERATORS

Journal Article

Journal de mathématiques pures et appliquées, ISSN 0021-7824, 2009, Volume 91, Issue 6, pp. 583 - 597

We prove that the Korteweg–de Vries initial-value problem is globally well-posed in H − 3 / 4 ( R ) and the modified Korteweg–de Vries initial-value problem is...

Low regularity | Global well-posedness | Korteweg–de Vries equation | Korteweg-de Vries equation | MATHEMATICS | MATHEMATICS, APPLIED | REGULARITY | DISPERSIVE EQUATIONS | ILL-POSEDNESS | KDV | SCATTERING

Low regularity | Global well-posedness | Korteweg–de Vries equation | Korteweg-de Vries equation | MATHEMATICS | MATHEMATICS, APPLIED | REGULARITY | DISPERSIVE EQUATIONS | ILL-POSEDNESS | KDV | SCATTERING

Journal Article

Journal of Differential Equations, ISSN 0022-0396, 09/2015, Volume 259, Issue 5, pp. 1722 - 1742

This paper is concerned with an initial–boundary value problem of the incompressible Navier–Stokes equations with density-dependent viscosity in a smooth...

Density-dependent viscosity | Global existence | Incompressible Navier–Stokes equations | Strong solutions | Incompressible Navier-Stokes equations | EXISTENCE | FLUIDS | MATHEMATICS | BOUNDARY-VALUE PROBLEM | SOLVABILITY | FLOWS | Fluid dynamics

Density-dependent viscosity | Global existence | Incompressible Navier–Stokes equations | Strong solutions | Incompressible Navier-Stokes equations | EXISTENCE | FLUIDS | MATHEMATICS | BOUNDARY-VALUE PROBLEM | SOLVABILITY | FLOWS | Fluid dynamics

Journal Article

Journal of Mathematical Analysis and Applications, ISSN 0022-247X, 11/2016, Volume 443, Issue 2, pp. 1142 - 1157

We prove global well-posedness for the 3D Klein–Gordon equation with a concentrated nonlinearity.

Nonlinear interaction | Point oscillators | Global well-posedness | Klein–Gordon equation | Klein-Gordon equation | MATHEMATICS | MATHEMATICS, APPLIED | SOLITARY WAVES | FIELD | SCHRODINGER-EQUATION | OSCILLATOR | Mathematics - Analysis of PDEs

Nonlinear interaction | Point oscillators | Global well-posedness | Klein–Gordon equation | Klein-Gordon equation | MATHEMATICS | MATHEMATICS, APPLIED | SOLITARY WAVES | FIELD | SCHRODINGER-EQUATION | OSCILLATOR | Mathematics - Analysis of PDEs

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

Derivative nonlinear Schrödinger equations | Local well-posedness | Global well-posedness | EXISTENCE | MATHEMATICS | REGULARITY | SCATTERING | Derivative nonlinear Schrodinger equations

Journal Article

Journal of Differential Equations, ISSN 0022-0396, 2011, Volume 251, Issue 12, pp. 3381 - 3402

We consider the defocusing energy-critical nonlinear Schrödinger equation of fourth order i u t + Δ 2 u = − | u | 8 d − 4 u . We prove that any finite energy...

Defocusing | Fourth order Schrödinger equations | Global well-posedness | Scattering | Energy-critical | Fourth order schrödinger equations | MATHEMATICS | Fourth order Schrodinger equations | RADIAL DATA | BLOW-UP

Defocusing | Fourth order Schrödinger equations | Global well-posedness | Scattering | Energy-critical | Fourth order schrödinger equations | MATHEMATICS | Fourth order Schrodinger equations | RADIAL DATA | BLOW-UP

Journal Article

Journal of Differential Equations, ISSN 0022-0396, 01/2018, Volume 264, Issue 2, pp. 1080 - 1118

We consider the inertial Qian–Sheng model of liquid crystals which couples a hyperbolic-type equation involving a second-order material derivative with a...

Nematic liquid crystal fluids | Navier–Stokes equations | Global wellposedness | EXISTENCE | MATHEMATICS | DE-GENNES THEORY | EQUATIONS | Navier-Stokes equations

Nematic liquid crystal fluids | Navier–Stokes equations | Global wellposedness | EXISTENCE | MATHEMATICS | DE-GENNES THEORY | EQUATIONS | Navier-Stokes equations

Journal Article

Duke Mathematical Journal, ISSN 0012-7094, 2015, Volume 164, Issue 6, pp. 973 - 1040

We prove that the critical Maxwell-Klein-Gordon equation on R4+1 is globally well-posed for smooth initial data which are small in the energy norm. This...

SPACE | MATHEMATICS | WAVE MAPS | REGULARITY | YANG-MILLS EQUATIONS | LOCAL EXISTENCE | CRITICAL SOBOLEV NORM | SCATTERING | Mathematics - Analysis of PDEs | critical MKG | 70S15 | global well-posedness | 35L70 | critical dispersive equations

SPACE | MATHEMATICS | WAVE MAPS | REGULARITY | YANG-MILLS EQUATIONS | LOCAL EXISTENCE | CRITICAL SOBOLEV NORM | SCATTERING | Mathematics - Analysis of PDEs | critical MKG | 70S15 | global well-posedness | 35L70 | critical dispersive equations

Journal Article

10.
Full Text
MHD Boundary Layers Theory in Sobolev Spaces Without Monotonicity I: Well‐Posedness Theory

Communications on Pure and Applied Mathematics, ISSN 0010-3640, 01/2019, Volume 72, Issue 1, pp. 63 - 121

We study the well‐posedness theory for the MHD boundary layer. The boundary layer equations are governed by the Prandtl‐type equations that are derived from...

ANALYTIC SOLUTIONS | SYSTEM | MATHEMATICS | MATHEMATICS, APPLIED | GLOBAL EXISTENCE | PRANDTL EQUATIONS | NAVIER-STOKES EQUATION | HALF-SPACE | ZERO VISCOSITY LIMIT | ILL-POSEDNESS | EULER | FLOW | Boundary layer

ANALYTIC SOLUTIONS | SYSTEM | MATHEMATICS | MATHEMATICS, APPLIED | GLOBAL EXISTENCE | PRANDTL EQUATIONS | NAVIER-STOKES EQUATION | HALF-SPACE | ZERO VISCOSITY LIMIT | ILL-POSEDNESS | EULER | FLOW | Boundary layer

Journal Article

Nonlinear Analysis: Real World Applications, ISSN 1468-1218, 10/2017, Volume 37, pp. 249 - 286

The purpose of this paper is to study well-posedness of the initial value problem (IVP) for the inhomogeneous nonlinear Schrödinger equation (INLS)...

Local well-posedness | Global well-posedness | Inhomogeneous nonlinear Schrödinger equation | EXISTENCE | Inhomogeneous nonlinear | MATHEMATICS, APPLIED | WAVES | Schrodinger equation | SOLITONS | STABILITY | CAUCHY-PROBLEM | BLOW-UP SOLUTIONS | SCATTERING

Local well-posedness | Global well-posedness | Inhomogeneous nonlinear Schrödinger equation | EXISTENCE | Inhomogeneous nonlinear | MATHEMATICS, APPLIED | WAVES | Schrodinger equation | SOLITONS | STABILITY | CAUCHY-PROBLEM | BLOW-UP SOLUTIONS | SCATTERING

Journal Article

Advances in Mathematics, ISSN 0001-8708, 04/2019, Volume 347, pp. 619 - 676

We consider the Cauchy problem for the defocusing cubic nonlinear Schrödinger equation in four space dimensions and establish almost sure local well-posedness...

Almost sure scattering | Nonlinear Schrödinger equation | Almost sure well-posedness | Random initial data | MATHEMATICS | BENJAMIN-ONO-EQUATION | INVARIANT-MEASURES | WAVE EQUATION | GLOBAL EXISTENCE | MAPS | REGULARITY | Nonlinear Schrodinger equation | DATA CAUCHY-THEORY

Almost sure scattering | Nonlinear Schrödinger equation | Almost sure well-posedness | Random initial data | MATHEMATICS | BENJAMIN-ONO-EQUATION | INVARIANT-MEASURES | WAVE EQUATION | GLOBAL EXISTENCE | MAPS | REGULARITY | Nonlinear Schrodinger equation | DATA CAUCHY-THEORY

Journal Article

Journal of Differential Equations, ISSN 0022-0396, 07/2017, Volume 263, Issue 2, pp. 1419 - 1450

We prove a Prodi–Serrin-type global regularity condition for the three-dimensional Magnetohydrodynamic-Boussinesq system (3D MHD-Boussinesq) without thermal...

Magnetohydrodynamic equations | Boussinesq equations | Partial viscosity | Regularity | Prodi–Serrin | Inviscid | EXISTENCE | ONE VELOCITY | GLOBAL REGULARITY | BLOW-UP CRITERION | GRADIENT | UNIQUENESS | MATHEMATICS | NAVIER-STOKES EQUATIONS | THEOREMS | PARTIAL DISSIPATION | Prodi-Serrin | FLOWS

Magnetohydrodynamic equations | Boussinesq equations | Partial viscosity | Regularity | Prodi–Serrin | Inviscid | EXISTENCE | ONE VELOCITY | GLOBAL REGULARITY | BLOW-UP CRITERION | GRADIENT | UNIQUENESS | MATHEMATICS | NAVIER-STOKES EQUATIONS | THEOREMS | PARTIAL DISSIPATION | Prodi-Serrin | FLOWS

Journal Article

Nonlinear Analysis, ISSN 0362-546X, 08/2018, Volume 173, pp. 164 - 179

We consider the nonlinear stability of the Timoshenko–Cattaneo system in the one-dimensional whole space. The Timoshenko system consists of two coupled wave...

Timoshenko systems | Regularity-loss | Cattaneo’s law | Global existence | Decay estimate | Cattaneo's law | EXISTENCE | MATHEMATICS, APPLIED | 2ND SOUND | PROPERTY | STABILITY | TRANSVERSE VIBRATIONS | MATHEMATICS | MEMORY | BARS

Timoshenko systems | Regularity-loss | Cattaneo’s law | Global existence | Decay estimate | Cattaneo's law | EXISTENCE | MATHEMATICS, APPLIED | 2ND SOUND | PROPERTY | STABILITY | TRANSVERSE VIBRATIONS | MATHEMATICS | MEMORY | BARS

Journal Article

Journal of Differential Equations, ISSN 0022-0396, 2012, Volume 252, Issue 3, pp. 2698 - 2724

We prove the global well-posedness for the 3-D micropolar fluid system in the critical Besov spaces by making a suitable transformation of the solutions and...

Highly oscillating | Littlewood–Paley decomposition | Global well-posedness | Besov space | Micropolar fluid | Littlewood-Paley decomposition | EXISTENCE | MATHEMATICS | THEOREM | EQUATIONS | UNIQUENESS | Mathematics - Analysis of PDEs

Highly oscillating | Littlewood–Paley decomposition | Global well-posedness | Besov space | Micropolar fluid | Littlewood-Paley decomposition | EXISTENCE | MATHEMATICS | THEOREM | EQUATIONS | UNIQUENESS | Mathematics - Analysis of PDEs

Journal Article

Nonlinear Analysis, ISSN 0362-546X, 09/2016, Volume 142, pp. 112 - 133

This paper is concerned with a new integrable multi-component Camassa–Holm system, which has been proven to be integrable and has peakon solutions. This system...

Local well-posedness | Global existence | Integrable multi-component Camassa–Holm system | Blow-up | Integrable multi-component | CamassaHolm system | MATHEMATICS, APPLIED | STABILITY | TRAJECTORIES | CAUCHY-PROBLEM | SHALLOW-WATER EQUATION | MATHEMATICS | BOUNDARY VALUE-PROBLEMS | SOLITONS | Camassa-Holm system | WEAK SOLUTIONS | WAVE-BREAKING | Nonlinearity | Criteria | Fluid dynamics | Partial differential equations | Graphical user interfaces

Local well-posedness | Global existence | Integrable multi-component Camassa–Holm system | Blow-up | Integrable multi-component | CamassaHolm system | MATHEMATICS, APPLIED | STABILITY | TRAJECTORIES | CAUCHY-PROBLEM | SHALLOW-WATER EQUATION | MATHEMATICS | BOUNDARY VALUE-PROBLEMS | SOLITONS | Camassa-Holm system | WEAK SOLUTIONS | WAVE-BREAKING | Nonlinearity | Criteria | Fluid dynamics | Partial differential equations | Graphical user interfaces

Journal Article

ZAMM ‐ Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, ISSN 0044-2267, 06/2019, Volume 99, Issue 6, pp. e201700306 - n/a

In this paper, we mainly study the Cauchy problem of the tropical climate model in negative‐order Besov spaces. By using the iterative scheme and compactness...

global existence | 35Q35 | well‐posedness | 76D03 | 35Q30 | Besov spaces | tropical climate model | EXISTENCE | MATHEMATICS, APPLIED | MECHANICS | well-posedness | EQUATIONS | CRITICAL SPACES | Climate | Analysis | Climate models | Function space | Iterative methods | Cauchy problem

global existence | 35Q35 | well‐posedness | 76D03 | 35Q30 | Besov spaces | tropical climate model | EXISTENCE | MATHEMATICS, APPLIED | MECHANICS | well-posedness | EQUATIONS | CRITICAL SPACES | Climate | Analysis | Climate models | Function space | Iterative methods | Cauchy problem

Journal Article

Journal of Functional Analysis, ISSN 0022-1236, 2006, Volume 233, Issue 1, pp. 60 - 91

We investigate well-posedness in classes of discontinuous functions for the nonlinear and third order dispersive Degasperis–Procesi equation (DP) ∂ t u - ∂ txx...

Entropy condition | Weak solution | Shallow water equation | Hyperbolic equation | Uniqueness | Integrable equation | Discontinuous solution | Existence | BREAKING | discontinuous solution | integrable equation | existence | CAUCHY-PROBLEM | hyperbolic equation | SHALLOW-WATER EQUATION | MATHEMATICS | entropy conditiom | weak solution | CAMASSA-HOLM | WAVES | uniqueness | KORTEWEG-DE-VRIES | shallow water equation | GLOBAL WEAK SOLUTIONS | SCATTERING

Entropy condition | Weak solution | Shallow water equation | Hyperbolic equation | Uniqueness | Integrable equation | Discontinuous solution | Existence | BREAKING | discontinuous solution | integrable equation | existence | CAUCHY-PROBLEM | hyperbolic equation | SHALLOW-WATER EQUATION | MATHEMATICS | entropy conditiom | weak solution | CAMASSA-HOLM | WAVES | uniqueness | KORTEWEG-DE-VRIES | shallow water equation | GLOBAL WEAK SOLUTIONS | SCATTERING

Journal Article