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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
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
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
Journal Article
SIAM Journal on Mathematical Analysis, ISSN 0036-1410, 2013, Volume 45, Issue 1, pp. 152 - 180
Journal Article
SIAM Journal on Applied Mathematics, ISSN 0036-1399, 1/2012, Volume 72, Issue 1, pp. 382 - 404
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
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
Journal Article
Mathematical Modelling of Natural Phenomena, ISSN 0973-5348, 01/2010, Volume 5, Issue 4, pp. 225 - 255
Journal Article
Fluid Dynamics Research, ISSN 0169-5983, 10/2015, Volume 47, Issue 6, pp. 1 - 22
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
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
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
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
Journal Article
by Do, Tam
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
Journal Article
Journal of Differential Equations, ISSN 0022-0396, 2017, Volume 263, Issue 9, pp. 6115 - 6142
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
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
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
Journal Article
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