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mathematics - analysis of pdes (38) 38
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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
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
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
Journal Article
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
Journal Article
SIAM Journal on Mathematical Analysis, ISSN 0036-1410, 2014, Volume 46, Issue 2, pp. 1651 - 1680
...SIAM J. MATH. ANAL. c Vol. 46, No. 2, pp. 16511680 GLOBAL EXISTENCE FOR THE CONFINED MUSKAT PROBLEM RAFAEL GRANERO-BELINCHN Abstract. In this paper we show... 
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
Journal Article
Journal of Mathematical Fluid Mechanics, ISSN 1422-6928, 6/2019, Volume 21, Issue 2, pp. 1 - 31
.../s00021-019-0437-2 Fluid Mechanics On the Thin Film Muskat and the Thin Film Stokes Equations Gabriele Bruell and Rafael Granero-Belinch´ on Communicated by S. Shkoller Abstract... 
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
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
Journal Article
Nonlinear Analysis, ISSN 0362-546X, 10/2014, Volume 108, pp. 260 - 274
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
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
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
Journal Article
Journal of Mathematical Physics, ISSN 0022-2488, 04/2016, Volume 57, Issue 4, p. 41501
...JOURNAL OF MATHEMATICAL PHYSICS 57, 041501 (2016) On the generalized Buckley-Leverett equation Jan Burczak, 1,a) Rafael Granero-Belinchón, 2,b) and Garving K... 
MAXIMUM PRINCIPLE | HYPERBOLIC SYSTEMS | GLOBAL EXISTENCE | SINGULARITIES | TRANSPORT-EQUATION | BLOW-UP | PHYSICS, MATHEMATICAL | Diffusion | Entropy | Mathematics - Analysis of PDEs
Journal Article
Advances in Mathematics, ISSN 0001-8708, 01/2015, Volume 269, pp. 197 - 219
In this paper, we study transport equations with nonlocal velocity fields with rough initial data. We address the global existence of weak solutions of a one... 
Transport equation | Nonlocal velocity field | Entropy | Weak solution | SINGULARITIES | Non local velocity field | ONE-DIMENSIONAL MODEL | WELL-POSEDNESS | MATHEMATICS | FRONTS | QUASI-GEOSTROPHIC EQUATIONS | MUSKAT PROBLEM | REGULARITY | DIFFUSION | BLOW-UP | BREAKDOWN
Journal Article
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
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
Journal Article
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