Journal of Mathematical Analysis and Applications, ISSN 0022-247X, 05/2017, Volume 449, Issue 1, pp. 872 - 883

We study a hyperbolic–parabolic model of chemotaxis related to tumor angiogenesis in dimensions one and two. We consider diffusions given by the fractional...

Global classical solutions | Chemotaxis | Hyperbolic–parabolic system | EXISTENCE | MATHEMATICS | MATHEMATICS, APPLIED | NONLINEAR STABILITY | BEHAVIOR | SENSITIVITY | MODEL | Hyperbolic-parabolic system | TRAVELING-WAVES | Analysis of PDEs | Mathematics

Global classical solutions | Chemotaxis | Hyperbolic–parabolic system | EXISTENCE | MATHEMATICS | MATHEMATICS, APPLIED | NONLINEAR STABILITY | BEHAVIOR | SENSITIVITY | MODEL | Hyperbolic-parabolic system | TRAVELING-WAVES | Analysis of PDEs | Mathematics

Journal Article

Journal of Mathematical Fluid Mechanics, ISSN 1422-6928, 6/2019, Volume 21, Issue 2, pp. 1 - 31

The present paper is concerned with the analysis of two strongly coupled systems of degenerate parabolic partial differential equations arising in multiphase...

two-phase thin film approximation | Fluid- and Aerodynamics | 76B03 | 35K25 | Physics | moving interfaces | stokes flow | Mathematical Methods in Physics | free-boundary problems | Classical and Continuum Physics | Muskat problem | 35Q35 | 35D30 | 35R35 | EXISTENCE | FLUIDS | PHYSICS, FLUIDS & PLASMAS | STABILITY | WELL-POSEDNESS | DRIVEN | GRAVITY | FLOW | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | DYNAMICS | PARABOLICITY | GLOBAL WEAK SOLUTIONS

two-phase thin film approximation | Fluid- and Aerodynamics | 76B03 | 35K25 | Physics | moving interfaces | stokes flow | Mathematical Methods in Physics | free-boundary problems | Classical and Continuum Physics | Muskat problem | 35Q35 | 35D30 | 35R35 | EXISTENCE | FLUIDS | PHYSICS, FLUIDS & PLASMAS | STABILITY | WELL-POSEDNESS | DRIVEN | GRAVITY | FLOW | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | DYNAMICS | PARABOLICITY | GLOBAL WEAK SOLUTIONS

Journal Article

Mathematical Models and Methods in Applied Sciences, ISSN 0218-2025, 01/2016, Volume 26, Issue 1, pp. 111 - 160

We study a doubly parabolic Keller-Segel system in one spatial dimension, with diffusions given by fractional Laplacians. We obtain several local and global...

Doubly parabolic Keller-Segel system | logistic damping | fractional diffusions | spatio-temporal chaos | existence of attractor | global-in-time solutions | AGGREGATION EQUATIONS | MATHEMATICS, APPLIED | GLOBAL EXISTENCE | CHEMOTAXIS SYSTEM | WELL-POSEDNESS | NONLINEAR DIFFUSION | DEGENERATE DIFFUSION | LOGISTIC SOURCE | MUSKAT PROBLEM | BLOW-UP | FRACTIONAL DIFFUSION | Mathematics - Analysis of PDEs

Doubly parabolic Keller-Segel system | logistic damping | fractional diffusions | spatio-temporal chaos | existence of attractor | global-in-time solutions | AGGREGATION EQUATIONS | MATHEMATICS, APPLIED | GLOBAL EXISTENCE | CHEMOTAXIS SYSTEM | WELL-POSEDNESS | NONLINEAR DIFFUSION | DEGENERATE DIFFUSION | LOGISTIC SOURCE | MUSKAT PROBLEM | BLOW-UP | FRACTIONAL DIFFUSION | Mathematics - Analysis of PDEs

Journal Article

Nonlinearity, ISSN 0951-7715, 04/2015, Volume 28, Issue 4, pp. 1103 - 1133

We study a nonlocal equation, analogous to the Kuramoto-Sivashinsky equation, in which short waves are stabilized by a possibly fractional diffusion of order...

Kuramoto-Sivashinsky equation | spatial chaos | attractor | MATHEMATICS, APPLIED | MAXIMUM PRINCIPLE | STABILITY | PHYSICS, MATHEMATICAL | ANALYTICITY | WAVES | MUSKAT PROBLEM | REGULARITY | BOUNDS | PROPAGATION | Chaos theory | Mathematical analysis | Uniqueness | Traveling waves | Nonlinearity | Mathematical models | Diffusion | Short wave | Mathematics - Analysis of PDEs

Kuramoto-Sivashinsky equation | spatial chaos | attractor | MATHEMATICS, APPLIED | MAXIMUM PRINCIPLE | STABILITY | PHYSICS, MATHEMATICAL | ANALYTICITY | WAVES | MUSKAT PROBLEM | REGULARITY | BOUNDS | PROPAGATION | Chaos theory | Mathematical analysis | Uniqueness | Traveling waves | Nonlinearity | Mathematical models | Diffusion | Short wave | Mathematics - Analysis of PDEs

Journal Article

Physica D: Nonlinear Phenomena, ISSN 0167-2789, 11/2013, Volume 262, pp. 71 - 82

This work studies a simplified model of the gravitational instability of an initially homogeneous infinite medium, represented by Td, based on the...

Patlak–Keller–Segel model | Fractional calculus | Gravitational collapse | Well-posedness | Instant analyticity | Blow-up | Patlak-Keller-Segel model | MATHEMATICS, APPLIED | PHYSICS, MULTIDISCIPLINARY | CHEMOTAXIS | STABILITY | EQUATIONS | PHYSICS, MATHEMATICAL | R-2 | PARABOLIC-ELLIPTIC SYSTEM | DIFFUSION | KELLER-SEGEL MODEL | AGGREGATION | BREAKDOWN | Approximation | Computer simulation | Gravitational instability | Sobolev space | Mathematical analysis | Mathematical models | Derivatives | Acceleration

Patlak–Keller–Segel model | Fractional calculus | Gravitational collapse | Well-posedness | Instant analyticity | Blow-up | Patlak-Keller-Segel model | MATHEMATICS, APPLIED | PHYSICS, MULTIDISCIPLINARY | CHEMOTAXIS | STABILITY | EQUATIONS | PHYSICS, MATHEMATICAL | R-2 | PARABOLIC-ELLIPTIC SYSTEM | DIFFUSION | KELLER-SEGEL MODEL | AGGREGATION | BREAKDOWN | Approximation | Computer simulation | Gravitational instability | Sobolev space | Mathematical analysis | Mathematical models | Derivatives | Acceleration

Journal Article

Nonlinearity, ISSN 0951-7715, 02/2015, Volume 28, Issue 2, pp. 435 - 461

In this paper we study a model of an interface between two fluids in a porous medium. For this model we prove several local and global well-posedness results...

one-dimensional model | Muskat problem | porous medium | MATHEMATICS, APPLIED | MAXIMUM PRINCIPLE | GLOBAL EXISTENCE | DYNAMICS | WELL-POSEDNESS | HELE-SHAW | MODEL | PHYSICS, MATHEMATICAL | Porous media | Fluids | Computational fluid dynamics | Fluid flow | Nonlinearity | Mathematical models | Boundaries | Mathematics - Analysis of PDEs

one-dimensional model | Muskat problem | porous medium | MATHEMATICS, APPLIED | MAXIMUM PRINCIPLE | GLOBAL EXISTENCE | DYNAMICS | WELL-POSEDNESS | HELE-SHAW | MODEL | PHYSICS, MATHEMATICAL | Porous media | Fluids | Computational fluid dynamics | Fluid flow | Nonlinearity | Mathematical models | Boundaries | Mathematics - Analysis of PDEs

Journal Article

Communications in Mathematical Sciences, ISSN 1539-6746, 2014, Volume 12, Issue 3, pp. 423 - 455

We study the evolution of the interface given by two incompressible fluids with different densities in the porous strip. This problem is known as the Muskat...

Hele-Shaw cell | Ill-posedness | Muskat problem | Maximum principle | Darcy's law | Blow-up | Well-posedness | FLUIDS | MATHEMATICS, APPLIED | GLOBAL EXISTENCE | well-posedness | ill-posedness | HELE-SHAW | maximum principle | FLOW | blow-up | DYNAMICS

Hele-Shaw cell | Ill-posedness | Muskat problem | Maximum principle | Darcy's law | Blow-up | Well-posedness | FLUIDS | MATHEMATICS, APPLIED | GLOBAL EXISTENCE | well-posedness | ill-posedness | HELE-SHAW | maximum principle | FLOW | blow-up | DYNAMICS

Journal Article

Journal of Differential Equations, ISSN 0022-0396, 02/2017, Volume 262, Issue 4, pp. 3250 - 3283

We introduce new lower bounds for the fractional Fisher information. Equipped with these bounds we study a hyperbolic–parabolic model of chemotaxis and prove...

Hyperbolic–parabolic system | Weak solutions | Fisher information | SMOOTH SOLUTIONS | FUNCTIONAL INEQUALITIES | GLOBAL EXISTENCE | WELL-POSEDNESS | TRAVELING-WAVES | MATHEMATICS | NONLOCAL VELOCITY | Hyperbolic parabolic system | CRITICAL MASS | NONLINEAR STABILITY | DIFFUSION | KELLER-SEGEL MODEL | Analysis of PDEs | Mathematics

Hyperbolic–parabolic system | Weak solutions | Fisher information | SMOOTH SOLUTIONS | FUNCTIONAL INEQUALITIES | GLOBAL EXISTENCE | WELL-POSEDNESS | TRAVELING-WAVES | MATHEMATICS | NONLOCAL VELOCITY | Hyperbolic parabolic system | CRITICAL MASS | NONLINEAR STABILITY | DIFFUSION | KELLER-SEGEL MODEL | Analysis of PDEs | Mathematics

Journal Article

SIAM Journal on Mathematical Analysis, ISSN 0036-1410, 2014, Volume 46, Issue 2, pp. 1651 - 1680

In this paper we show global existence of the Lipschitz continuous solution for the stable Muskat problem with finite depth (confined) and initial data...

Inhomogeneus Muskat problem | Darcy's law | Well-posedness | FLUIDS | MATHEMATICS, APPLIED | well-posedness | DYNAMICS | inhomogeneus Muskat problem | HELE-SHAW | FLOW | Maximum principle | Amplitudes | Case depth | Mathematical analysis

Inhomogeneus Muskat problem | Darcy's law | Well-posedness | FLUIDS | MATHEMATICS, APPLIED | well-posedness | DYNAMICS | inhomogeneus Muskat problem | HELE-SHAW | FLOW | Maximum principle | Amplitudes | Case depth | Mathematical analysis

Journal Article

Advances in Mathematics, ISSN 0001-8708, 06/2016, Volume 295, pp. 334 - 367

We study the global existence of solutions to a one-dimensional drift–diffusion equation with logistic term, generalizing the classical parabolic–elliptic...

Drift–diffusion equation | Global existence | Nonlocal diffusion | Drift-diffusion equation | MAXIMUM PRINCIPLE | FINITE-TIME SINGULARITIES | WELL-POSEDNESS | MODEL | MATHEMATICS | KELLER-SEGEL SYSTEM | NONLOCAL VELOCITY | MUSKAT PROBLEM | TRANSPORT-EQUATION | BLOW-UP | FRACTIONAL DIFFUSION | Mathematics - Analysis of PDEs

Drift–diffusion equation | Global existence | Nonlocal diffusion | Drift-diffusion equation | MAXIMUM PRINCIPLE | FINITE-TIME SINGULARITIES | WELL-POSEDNESS | MODEL | MATHEMATICS | KELLER-SEGEL SYSTEM | NONLOCAL VELOCITY | MUSKAT PROBLEM | TRANSPORT-EQUATION | BLOW-UP | FRACTIONAL DIFFUSION | Mathematics - Analysis of PDEs

Journal Article

Annales Henri Poincaré, ISSN 1424-0637, 12/2016, Volume 17, Issue 12, pp. 3473 - 3498

In this note, we study a fractional Poisson–Nernst–Planck equation modeling a semiconductor device. We prove several decay estimates for the Lebesgue and...

Mathematical Methods in Physics | Theoretical, Mathematical and Computational Physics | Quantum Physics | Dynamical Systems and Ergodic Theory | Classical and Quantum Gravitation, Relativity Theory | Physics | Elementary Particles, Quantum Field Theory | NERNST-PLANCK | TRANSPORT-EQUATIONS | GLOBAL EXISTENCE | PHYSICS, MULTIDISCIPLINARY | LARGE-TIME BEHAVIOR | WELL-POSEDNESS | PHYSICS, MATHEMATICAL | ASYMPTOTIC-BEHAVIOR | QUASI-GEOSTROPHIC EQUATIONS | KELLER-SEGEL MODEL | FRACTIONAL DIFFUSION | PLANCK-POISSON SYSTEM | PHYSICS, PARTICLES & FIELDS | Mathematics - Analysis of PDEs

Mathematical Methods in Physics | Theoretical, Mathematical and Computational Physics | Quantum Physics | Dynamical Systems and Ergodic Theory | Classical and Quantum Gravitation, Relativity Theory | Physics | Elementary Particles, Quantum Field Theory | NERNST-PLANCK | TRANSPORT-EQUATIONS | GLOBAL EXISTENCE | PHYSICS, MULTIDISCIPLINARY | LARGE-TIME BEHAVIOR | WELL-POSEDNESS | PHYSICS, MATHEMATICAL | ASYMPTOTIC-BEHAVIOR | QUASI-GEOSTROPHIC EQUATIONS | KELLER-SEGEL MODEL | FRACTIONAL DIFFUSION | PLANCK-POISSON SYSTEM | PHYSICS, PARTICLES & FIELDS | Mathematics - Analysis of PDEs

Journal Article

Nonlinear Analysis, ISSN 0362-546X, 10/2014, Volume 108, pp. 260 - 274

In this paper we study an aggregation equation with a general nonlocal flux. We study the local well-posedness and some conditions ensuring global existence....

Aggregation | Global existence | Nonlinear parabolic partial differential equation | Numerical simulations | MATHEMATICS, APPLIED | FUNCTIONAL INEQUALITIES | CHEMOTAXIS | MATHEMATICS | DEGENERATE DIFFUSION | PARTIAL-DIFFERENTIAL-EQUATIONS | CRITICAL MASS | R-2 | PARABOLIC-ELLIPTIC SYSTEM | KELLER-SEGEL MODEL | FRACTIONAL DIFFUSION | Flux | Nonlinearity | Mathematical models | Agglomeration | Convexity | Mathematical analysis

Aggregation | Global existence | Nonlinear parabolic partial differential equation | Numerical simulations | MATHEMATICS, APPLIED | FUNCTIONAL INEQUALITIES | CHEMOTAXIS | MATHEMATICS | DEGENERATE DIFFUSION | PARTIAL-DIFFERENTIAL-EQUATIONS | CRITICAL MASS | R-2 | PARABOLIC-ELLIPTIC SYSTEM | KELLER-SEGEL MODEL | FRACTIONAL DIFFUSION | Flux | Nonlinearity | Mathematical models | Agglomeration | Convexity | Mathematical analysis

Journal Article

Journal of Differential Equations, ISSN 0022-0396, 11/2017, Volume 263, Issue 9, pp. 6115 - 6142

We consider a two dimensional parabolic–elliptic Keller–Segel equation with a fractional diffusion of order α∈(0,2) and a logistic term. In the case of an...

Logistic source | Keller–Segel system | Global-in-time smoothness | Fractional dissipation | Active scalar equations | Nonlocal maximum principle | EXISTENCE | AGGREGATION EQUATION | INSTABILITY | CHEMOTAXIS | MODEL | DIFFUSION EQUATION | GLOBAL WELL-POSEDNESS | MATHEMATICS | Keller-Segel system | REGULARITY | DYNAMICS | QUASI-GEOSTROPHIC EQUATION

Logistic source | Keller–Segel system | Global-in-time smoothness | Fractional dissipation | Active scalar equations | Nonlocal maximum principle | EXISTENCE | AGGREGATION EQUATION | INSTABILITY | CHEMOTAXIS | MODEL | DIFFUSION EQUATION | GLOBAL WELL-POSEDNESS | MATHEMATICS | Keller-Segel system | REGULARITY | DYNAMICS | QUASI-GEOSTROPHIC EQUATION

Journal Article

Transactions of the American Mathematical Society, ISSN 0002-9947, 04/2019, Volume 372, Issue 4, pp. 2255 - 2286

We first prove local-in-time well-posedness for the Muskat problem, modeling fluid flow in a two-dimensional inhomogeneous porous media. The permeability of...

MATHEMATICS | FINITE-TIME SPLASH | HELE-SHAW FLOW | GLOBAL EXISTENCE | SINGULARITIES | SOLVABILITY | DYNAMICS | ABSENCE

MATHEMATICS | FINITE-TIME SPLASH | HELE-SHAW FLOW | GLOBAL EXISTENCE | SINGULARITIES | SOLVABILITY | DYNAMICS | ABSENCE

Journal Article

Physica D: Nonlinear Phenomena, ISSN 0167-2789, 05/2019, Volume 392, pp. 1 - 16

We provide rigorous asymptotic models for the free boundary Darcy and Forchheimer problem under the assumption of weak nonlinear interaction, in a regime in...

Moving interfaces | Darcy law | Free-boundary problems | Muskat problem | Forchheimer flow | FLUIDS | MATHEMATICS, APPLIED | PHYSICS, MULTIDISCIPLINARY | STABILITY | EQUATIONS | WELL-POSEDNESS | PHYSICS, MATHEMATICAL | WATER-WAVES | EVOLUTION | DYNAMICS | SURFACE GRAVITY-WAVES

Moving interfaces | Darcy law | Free-boundary problems | Muskat problem | Forchheimer flow | FLUIDS | MATHEMATICS, APPLIED | PHYSICS, MULTIDISCIPLINARY | STABILITY | EQUATIONS | WELL-POSEDNESS | PHYSICS, MATHEMATICAL | WATER-WAVES | EVOLUTION | DYNAMICS | SURFACE GRAVITY-WAVES

Journal Article

Topological Methods in Nonlinear Analysis, ISSN 1230-3429, 03/2016, Volume 47, Issue 1, pp. 369 - 387

A semilinear version of parabolic -elliptic Keller -Segel system with the critical nonlocal diffusion is considered in one space dimension. We show boundedness...

Critical dissipation | Logistic dampening | Bounded solutions | Fractional Keller-Segel | SYSTEM | FUNCTIONAL INEQUALITIES | critical dissipation | GLOBAL EXISTENCE | SINGULARITIES | bounded solutions | logistic dampening | MATHEMATICS | DEGENERATE DIFFUSION | LOGISTIC SOURCE | CRITICAL MASS | KELLER-SEGEL MODEL | EQUATION | AGGREGATION

Critical dissipation | Logistic dampening | Bounded solutions | Fractional Keller-Segel | SYSTEM | FUNCTIONAL INEQUALITIES | critical dissipation | GLOBAL EXISTENCE | SINGULARITIES | bounded solutions | logistic dampening | MATHEMATICS | DEGENERATE DIFFUSION | LOGISTIC SOURCE | CRITICAL MASS | KELLER-SEGEL MODEL | EQUATION | AGGREGATION

Journal Article

SIAM Journal on Applied Mathematics, ISSN 0036-1399, 01/2019, Volume 79, Issue 6, pp. 2530 - 2550

Journal Article

Journal of Differential Equations, ISSN 0022-0396, 06/2020, Volume 268, Issue 12, pp. 7582 - 7608

This paper studies the existence and asymptotic behavior of global weak solutions for a thin film equation with insoluble surfactant under the influence of...

Thin film equations | System of quasilinear parabolic equations | Surfactant | Global weak solutions | Degenerate equations | Decay rates | EXISTENCE | FINITE-ELEMENT APPROXIMATION | TIME | NONNEGATIVE SOLUTIONS | TRAVELING-WAVES | MATHEMATICS | VISCOUS FILMS | MUSKAT PROBLEM | FINGERING PHENOMENA | DROPLET

Thin film equations | System of quasilinear parabolic equations | Surfactant | Global weak solutions | Degenerate equations | Decay rates | EXISTENCE | FINITE-ELEMENT APPROXIMATION | TIME | NONNEGATIVE SOLUTIONS | TRAVELING-WAVES | MATHEMATICS | VISCOUS FILMS | MUSKAT PROBLEM | FINGERING PHENOMENA | DROPLET

Journal Article

Nonlinearity, ISSN 0951-7715, 2014, Volume 27, Issue 6, pp. 1471 - 1498

We exhibit a family of graphs that develop turning singularities (i.e. their Lipschitz seminorm blows up and they cease to be a graph, passing from the stable...

Darcys law | inhomogeneous Muskat problem | turning | blow-up | computerassisted | water waves | singularity | FLUIDS | MATHEMATICS, APPLIED | DIFFERENT DENSITIES | GLOBAL EXISTENCE | WELL-POSEDNESS | HELE-SHAW | PHYSICS, MATHEMATICAL | FLOW | WATER-WAVES | DYNAMICS | Darcy's law | Infinity | Singularities | Mathematical analysis | Proving | Turning | Graphs | Boundary conditions | Permeability | Mathematics - Analysis of PDEs

Darcys law | inhomogeneous Muskat problem | turning | blow-up | computerassisted | water waves | singularity | FLUIDS | MATHEMATICS, APPLIED | DIFFERENT DENSITIES | GLOBAL EXISTENCE | WELL-POSEDNESS | HELE-SHAW | PHYSICS, MATHEMATICAL | FLOW | WATER-WAVES | DYNAMICS | Darcy's law | Infinity | Singularities | Mathematical analysis | Proving | Turning | Graphs | Boundary conditions | Permeability | Mathematics - Analysis of PDEs

Journal Article

Journal of Mathematical Physics, ISSN 0022-2488, 04/2016, Volume 57, Issue 4, p. 41501

In this paper we study the generalized Buckley-Leverett equation with nonlocal regularizing terms. One of these regularizing terms is diffusive, while the...

MAXIMUM PRINCIPLE | HYPERBOLIC SYSTEMS | GLOBAL EXISTENCE | SINGULARITIES | TRANSPORT-EQUATION | BLOW-UP | PHYSICS, MATHEMATICAL | Diffusion | Entropy | Mathematics - Analysis of PDEs

MAXIMUM PRINCIPLE | HYPERBOLIC SYSTEMS | GLOBAL EXISTENCE | SINGULARITIES | TRANSPORT-EQUATION | BLOW-UP | PHYSICS, MATHEMATICAL | Diffusion | Entropy | Mathematics - Analysis of PDEs

Journal Article

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