Nonlinear Analysis, ISSN 0362-546X, 02/2019, Volume 179, pp. 254 - 269

... of the combined regularities of the coefficients and the scalar. This is motivated by the case of active scalar equations, where the transporting vector field has the same regularity as the transported scalar. We use commutator estimates similar to those of Constantin–E–Titi in the context of Onsager’s conjecture, but we require novel arguments to handle the case of Lp norms when p≠2.

Active scalar equation | Onsager’s conjecture | Renormalization | Onsager's conjecture | DISSIPATION | NONUNIQUENESS | MATHEMATICS, APPLIED | INCOMPRESSIBLE EULER | ENERGY-CONSERVATION | ONSAGERS CONJECTURE | UNIQUENESS | MATHEMATICS | SETS | WEAK SOLUTIONS | Nonlinear equations | Algebra | Mathematical analysis | Norms | Mathematical models | Fields (mathematics) | Regularity

Active scalar equation | Onsager’s conjecture | Renormalization | Onsager's conjecture | DISSIPATION | NONUNIQUENESS | MATHEMATICS, APPLIED | INCOMPRESSIBLE EULER | ENERGY-CONSERVATION | ONSAGERS CONJECTURE | UNIQUENESS | MATHEMATICS | SETS | WEAK SOLUTIONS | Nonlinear equations | Algebra | Mathematical analysis | Norms | Mathematical models | Fields (mathematics) | Regularity

Journal Article

Monatshefte für Mathematik, ISSN 1436-5081, 2014, Volume 175, Issue 4, pp. 491 - 509

We are concerned with a family of dissipative active scalar equation with velocity fields coupled via multiplier operators that can be of positive-order...

35C06 | 42B35 | Global well-posedness | Self-similar solutions | 35Q35 | Asymptotic behavior | 35A01 | 35R11 | Mathematics, general | Mathematics | 35B40 | Active scalar equations | MAXIMUM PRINCIPLE | FINITE-TIME SINGULARITIES | MODEL | ASYMPTOTIC-BEHAVIOR | MATHEMATICS | NAVIER-STOKES EQUATIONS | REGULARITY | QUASI-GEOSTROPHIC EQUATION | DIFFUSION | CONTINUUM

35C06 | 42B35 | Global well-posedness | Self-similar solutions | 35Q35 | Asymptotic behavior | 35A01 | 35R11 | Mathematics, general | Mathematics | 35B40 | Active scalar equations | MAXIMUM PRINCIPLE | FINITE-TIME SINGULARITIES | MODEL | ASYMPTOTIC-BEHAVIOR | MATHEMATICS | NAVIER-STOKES EQUATIONS | REGULARITY | QUASI-GEOSTROPHIC EQUATION | DIFFUSION | CONTINUUM

Journal Article

Analysis and PDE, ISSN 2157-5045, 2014, Volume 7, Issue 1, pp. 43 - 72

The paper is devoted to the study of slightly supercritical active scalars with nonlocal diffusion...

Active scalars | Global regularity | Finite time blow-up | Surface quasigeostrophic equation | Burgers equation | Nonlocal dissipation | Nonlocal maximum principle | Supercritical dissipation | SQG equation | MATHEMATICS, APPLIED | supercritical dissipation | active scalars | MAXIMUM-PRINCIPLES | FRACTAL BURGERS EQUATIONS | MATHEMATICS | surface quasigeostrophic equation | nonlocal dissipation | nonlocal maximum principle | REGULARITY | global regularity | QUASI-GEOSTROPHIC EQUATION | DIFFUSION | finite time blow-up | Mathematics - Analysis of PDEs

Active scalars | Global regularity | Finite time blow-up | Surface quasigeostrophic equation | Burgers equation | Nonlocal dissipation | Nonlocal maximum principle | Supercritical dissipation | SQG equation | MATHEMATICS, APPLIED | supercritical dissipation | active scalars | MAXIMUM-PRINCIPLES | FRACTAL BURGERS EQUATIONS | MATHEMATICS | surface quasigeostrophic equation | nonlocal dissipation | nonlocal maximum principle | REGULARITY | global regularity | QUASI-GEOSTROPHIC EQUATION | DIFFUSION | finite time blow-up | Mathematics - Analysis of PDEs

Journal Article

Advances in Mathematics, ISSN 0001-8708, 2011, Volume 227, Issue 5, pp. 1806 - 1826

Active scalars appear in many problems of fluid dynamics. The most common examples of active scalar equations are 2D Euler, Burgers, and 2D surface quasi-geostrophic equations...

Fluid mechanics | Active scalar | Maximum principle | Surface quasigeostrophic equation | Regularity of solutions | VELOCITY | GLOBAL WELL-POSEDNESS | MATHEMATICS | REGULARITY | QUASI-GEOSTROPHIC EQUATION | TRANSPORT-EQUATION | BURGERS-EQUATION | BLOW-UP | Fluid dynamics

Fluid mechanics | Active scalar | Maximum principle | Surface quasigeostrophic equation | Regularity of solutions | VELOCITY | GLOBAL WELL-POSEDNESS | MATHEMATICS | REGULARITY | QUASI-GEOSTROPHIC EQUATION | TRANSPORT-EQUATION | BURGERS-EQUATION | BLOW-UP | Fluid dynamics

Journal Article

SIAM Journal on Mathematical Analysis, ISSN 0036-1410, 2013, Volume 45, Issue 1, pp. 152 - 180

We analyze operator splitting methods applied to scalar equations with a nonlinear advection operator and a linear (local or nonlocal...

Fractional diffusion | Active scalar equation | Operator splitting | KdV equation | Kawahara equation | Aggregation equation | Burgers equation | Quasi-geostrophic equation | Convergence | quasi-geostrophic equation | MATHEMATICS, APPLIED | fractional diffusion | convergence | aggregation equation | BLOW-UP | MODEL | active scalar equation | operator splitting | Operators | Splitting | Advection | Mathematical analysis | Surface chemistry | Scalars | Dispersions | Diffusion

Fractional diffusion | Active scalar equation | Operator splitting | KdV equation | Kawahara equation | Aggregation equation | Burgers equation | Quasi-geostrophic equation | Convergence | quasi-geostrophic equation | MATHEMATICS, APPLIED | fractional diffusion | convergence | aggregation equation | BLOW-UP | MODEL | active scalar equation | operator splitting | Operators | Splitting | Advection | Mathematical analysis | Surface chemistry | Scalars | Dispersions | Diffusion

Journal Article

SIAM Journal on Applied Mathematics, ISSN 0036-1399, 1/2012, Volume 72, Issue 1, pp. 382 - 404

In this paper we derive evolution equations for the two-dimensional active scalar problem when the solution is supported on one-dimensional curves...

Aggregation | Spiral arms | Simulations | Vortex sheets | Kelvin Helmholtz instability | Vorticity | Scalars | Mathematics | Velocity distribution | Modeling | Biological swarming | Birkhoff-Rott equation | Vortex density functions | Kelvin- Helmholtz instability | Active scalar problems | AGGREGATION EQUATION | MATHEMATICS, APPLIED | active scalar problems | biological swarming | Kelvin-Helmholtz instability | VORTEX METHOD | CONVERGENCE | BLOW-UP | vortex density functions

Aggregation | Spiral arms | Simulations | Vortex sheets | Kelvin Helmholtz instability | Vorticity | Scalars | Mathematics | Velocity distribution | Modeling | Biological swarming | Birkhoff-Rott equation | Vortex density functions | Kelvin- Helmholtz instability | Active scalar problems | AGGREGATION EQUATION | MATHEMATICS, APPLIED | active scalar problems | biological swarming | Kelvin-Helmholtz instability | VORTEX METHOD | CONVERGENCE | BLOW-UP | vortex density functions

Journal Article

Communications on pure and applied mathematics, ISSN 0010-3640, 2012, Volume 65, Issue 8, pp. 1037 - 1066

This paper establishes several existence and uniqueness results for two families of active scalar equations with velocity fields determined by the scalars through very singular integrals...

EXISTENCE | MATHEMATICS | MATHEMATICS, APPLIED | SPACES | GLOBAL WELL-POSEDNESS

EXISTENCE | MATHEMATICS | MATHEMATICS, APPLIED | SPACES | GLOBAL WELL-POSEDNESS

Journal Article

Publicacions Matematiques, ISSN 0214-1493, 2016, Volume 60, Issue 2, pp. 525 - 550

We consider a family of dissipative active scalar equations outside the L-2-space...

Active scalar equations, global well-posedness, decay of solutions, sym-metry, critical spaces | MAXIMUM PRINCIPLE | global well-posedness | decay of solutions | SELF-SIMILAR SOLUTIONS | ASYMPTOTIC-BEHAVIOR | MATHEMATICS | NONLOCAL VELOCITY | symmetry | REGULARITY | critical spaces | NAVIER-STOKES | QUASI-GEOSTROPHIC EQUATION | TRANSPORT-EQUATION | MORREY SPACES | BLOW-UP | Active scalar equations

Active scalar equations, global well-posedness, decay of solutions, sym-metry, critical spaces | MAXIMUM PRINCIPLE | global well-posedness | decay of solutions | SELF-SIMILAR SOLUTIONS | ASYMPTOTIC-BEHAVIOR | MATHEMATICS | NONLOCAL VELOCITY | symmetry | REGULARITY | critical spaces | NAVIER-STOKES | QUASI-GEOSTROPHIC EQUATION | TRANSPORT-EQUATION | MORREY SPACES | BLOW-UP | Active scalar equations

Journal Article

Journal of Mathematical Fluid Mechanics, ISSN 1422-6928, 6/2013, Volume 15, Issue 2, pp. 415 - 423

We address the problem of inviscid limits for a class of active scalar equations, where the drift velocity u and the active scalar θ...

Mathematical Methods in Physics | Inviscid limits | Weak solutions | Fluid- and Aerodynamics | Primary 35Q35 | Secondary 76W05 | Active scalar equations | Physics | Classical Continuum Physics | MATHEMATICS, APPLIED | MECHANICS | REGULARITY | SPACES | PHYSICS, FLUIDS & PLASMAS | POSEDNESS

Mathematical Methods in Physics | Inviscid limits | Weak solutions | Fluid- and Aerodynamics | Primary 35Q35 | Secondary 76W05 | Active scalar equations | Physics | Classical Continuum Physics | MATHEMATICS, APPLIED | MECHANICS | REGULARITY | SPACES | PHYSICS, FLUIDS & PLASMAS | POSEDNESS

Journal Article

Mathematical Modelling of Natural Phenomena, ISSN 0973-5348, 01/2010, Volume 5, Issue 4, pp. 225 - 255

We review some recent results for a class of fluid mechanics equations called active scalars, with fractional dissipation...

finite time blow up | global regularity | active scalars | nonlocal maximum principle | MATHEMATICS, APPLIED | MAXIMUM PRINCIPLE | SINGULARITIES | BEHAVIOR | INEQUALITY | GLOBAL WELL-POSEDNESS | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MATHEMATICAL & COMPUTATIONAL BIOLOGY | QUASI-GEOSTROPHIC EQUATION | TRANSPORT-EQUATION

finite time blow up | global regularity | active scalars | nonlocal maximum principle | MATHEMATICS, APPLIED | MAXIMUM PRINCIPLE | SINGULARITIES | BEHAVIOR | INEQUALITY | GLOBAL WELL-POSEDNESS | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MATHEMATICAL & COMPUTATIONAL BIOLOGY | QUASI-GEOSTROPHIC EQUATION | TRANSPORT-EQUATION

Journal Article

Fluid Dynamics Research, ISSN 0169-5983, 10/2015, Volume 47, Issue 6, pp. 1 - 22

... equation for the local mass fraction. The coefficient of viscosity thus behaves as an active scalar...

Variable-viscosity | spectral methods | viscous shock | Burgers flow | active scalar | WAVES | MECHANICS | PHYSICS, FLUIDS & PLASMAS | TURBULENCE | Burgers' flow | MODEL | variable-viscosity | EQUATION | Viscosity | Advection-diffusion equation | Propagation | Energy dissipation | Small scale | Scalars | Sine waves | Mathematical models

Variable-viscosity | spectral methods | viscous shock | Burgers flow | active scalar | WAVES | MECHANICS | PHYSICS, FLUIDS & PLASMAS | TURBULENCE | Burgers' flow | MODEL | variable-viscosity | EQUATION | Viscosity | Advection-diffusion equation | Propagation | Energy dissipation | Small scale | Scalars | Sine waves | Mathematical models

Journal Article

数学年刊：B辑英文版, ISSN 0252-9599, 2017, Volume 38, Issue 1, pp. 281 - 292

The author reviews some results about nonlocal advection-diffusion equations based on lower bounds for the fractional Laplacian.

Advection | Nonlocal | 35Q35 | Mathematics, general | Mathematics | 35Q86 | Applications of Mathematics | Diffusion | Fractional Laplacian | MATHEMATICS | ACTIVE SCALAR | REGULARITY | DYNAMICS | QUASI-GEOSTROPHIC EQUATION | GLOBAL WELL-POSEDNESS | MAXIMUM-PRINCIPLES

Advection | Nonlocal | 35Q35 | Mathematics, general | Mathematics | 35Q86 | Applications of Mathematics | Diffusion | Fractional Laplacian | MATHEMATICS | ACTIVE SCALAR | REGULARITY | DYNAMICS | QUASI-GEOSTROPHIC EQUATION | GLOBAL WELL-POSEDNESS | MAXIMUM-PRINCIPLES

Journal Article

Journal of mathematical physics, ISSN 1089-7658, 2012, Volume 53, Issue 11, p. 115602

This paper considers a family of active scalar equations with transport velocities which are more singular by a derivative of order β...

flow through porous media | NONLINEAR INSTABILITY | ACTIVE SCALAR | PHYSICS, MATHEMATICAL | QUASI-GEOSTROPHIC EQUATIONS | transport processes | Porous media | Scalars | Derivatives | Ill posed problems | Transport | Mathematical analysis | Mathematics - Analysis of PDEs

flow through porous media | NONLINEAR INSTABILITY | ACTIVE SCALAR | PHYSICS, MATHEMATICAL | QUASI-GEOSTROPHIC EQUATIONS | transport processes | Porous media | Scalars | Derivatives | Ill posed problems | Transport | Mathematical analysis | Mathematics - Analysis of PDEs

Journal Article

Communications on Pure and Applied Mathematics, ISSN 0010-3640, 10/2008, Volume 61, Issue 10, pp. 1331 - 1346

Inspired by recent developments in Berdina‐like models for turbulence, we propose an inviscid regularization for the surface quasi‐geostrophic (SQG) equations...

MATHEMATICS | MATHEMATICS, APPLIED | ACTIVE SCALAR | FLOW

MATHEMATICS | MATHEMATICS, APPLIED | ACTIVE SCALAR | FLOW

Journal Article

Mathematical Methods in the Applied Sciences, ISSN 0170-4214, 03/2018, Volume 41, Issue 4, pp. 1642 - 1652

We are concerned with a family of dissipative active scalar equation on R2. By using similar methods from the previous paper of Y. Giga et al...

almost periodic initial data | active scalar equations | frequency sets | NAVIER-STOKES | MATHEMATICS, APPLIED

almost periodic initial data | active scalar equations | frequency sets | NAVIER-STOKES | MATHEMATICS, APPLIED

Journal Article

Journal of Differential Equations, ISSN 0022-0396, 2014, Volume 256, Issue 9, pp. 3166 - 3178

We study a 1D transport equation with nonlocal velocity. First, we prove eventual regularization of the viscous regularization when dissipation is in the supercritical range with non-negative initial data...

Active scalar | Nonlocal velocity | Modulus of continuity | Supercritical dissipation | MATHEMATICS | QUASI-GEOSTROPHIC EQUATION | GLOBAL WELL-POSEDNESS

Active scalar | Nonlocal velocity | Modulus of continuity | Supercritical dissipation | MATHEMATICS | QUASI-GEOSTROPHIC EQUATION | GLOBAL WELL-POSEDNESS

Journal Article

Journal of Differential Equations, ISSN 0022-0396, 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...

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

Journal of mathematical fluid mechanics, ISSN 1422-6952, 2019, Volume 21, Issue 4, pp. 1 - 25

This paper considers a family of non-diffusive active scalar equations where a viscosity type parameter enters the equations via the constitutive law that relates the drift velocity with the scalar field...

vanishing viscosity limit | Mathematical Methods in Physics | Fluid- and Aerodynamics | Classical and Continuum Physics | 35Q35 | 76D03 | Gevrey-class solutions | Active scalar equations | 76W05 | Physics | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | INSTABILITY | PHYSICS, FLUIDS & PLASMAS | DIFFUSION | GLOBAL WELL-POSEDNESS | UNIQUENESS

vanishing viscosity limit | Mathematical Methods in Physics | Fluid- and Aerodynamics | Classical and Continuum Physics | 35Q35 | 76D03 | Gevrey-class solutions | Active scalar equations | 76W05 | Physics | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | INSTABILITY | PHYSICS, FLUIDS & PLASMAS | DIFFUSION | GLOBAL WELL-POSEDNESS | UNIQUENESS

Journal Article

Physical Review D, ISSN 2470-0010, 10/2017, Volume 96, Issue 8

Continuum spectrum from black hole accretion disc holds enormous information regarding the strong gravity regime around the black hole and hence about the...

DISK-ACCRETION | TENSOR | QUASARS | EVOLUTION | ACTIVE GALACTIC NUCLEI | MODELS | EINSTEIN | ENERGY-DISTRIBUTIONS | ASTRONOMY & ASTROPHYSICS | FIELD EQUATIONS | MODIFIED GRAVITY | PHYSICS, PARTICLES & FIELDS

DISK-ACCRETION | TENSOR | QUASARS | EVOLUTION | ACTIVE GALACTIC NUCLEI | MODELS | EINSTEIN | ENERGY-DISTRIBUTIONS | ASTRONOMY & ASTROPHYSICS | FIELD EQUATIONS | MODIFIED GRAVITY | PHYSICS, PARTICLES & FIELDS

Journal Article

Communications on pure and applied mathematics, ISSN 1097-0312, 2005, Volume 58, Issue 6, pp. 821 - 866

We consider the problem of the evolution of sharp fronts for the surface quasi‐geostrophic (QG) equation. This problem is the analogue to the vortex patch problem for the two...

MATHEMATICS | MATHEMATICS, APPLIED | ACTIVE SCALAR | MODEL | FLOW

MATHEMATICS | MATHEMATICS, APPLIED | ACTIVE SCALAR | MODEL | FLOW

Journal Article

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