September  2014, 34(9): 3437-3454. doi: 10.3934/dcds.2014.34.3437

On a generalized maximum principle for a transport-diffusion model with $\log$-modulated fractional dissipation

1. 

Division of Applied Mathematics, Brown University, 182 George Street, Providence, RI 02912

2. 

Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2

Received  September 2013 Revised  December 2013 Published  March 2014

We consider a transport-diffusion equation of the form $\partial_t \theta + v \cdot \nabla \theta + \nu \mathcal{A} \theta = 0$, where $v$ is a given time-dependent vector field on $\mathbb R^d$. The operator $\mathcal{A}$ represents log-modulated fractional dissipation: $\mathcal{A}=\frac {|\nabla|^{\gamma}}{\log^{\beta}(\lambda+|\nabla|)}$ and the parameters $\nu\ge 0$, $\beta\ge 0$, $0\le \gamma \le 2$, $\lambda>1$. We introduce a novel nonlocal decomposition of the operator $\mathcal{A}$ in terms of a weighted integral of the usual fractional operators $|\nabla|^{s}$, $0\le s \le \gamma$ plus a smooth remainder term which corresponds to an $L^1$ kernel. For a general vector field $v$ (possibly non-divergence-free) we prove a generalized $L^\infty$ maximum principle of the form $ \| \theta(t)\|_\infty \le e^{Ct} \| \theta_0 \|_{\infty}$ where the constant $C=C(\nu,\beta,\gamma)>0$. In the case $\text{div}(v)=0$ the same inequality holds for $\|\theta(t)\|_p$ with $1\le p \le \infty$. Under the additional assumption that $\theta_0\in L^2$, we show that $\|\theta(t)\|_p$ is uniformly bounded for $2\le p\le \infty$. At the cost of a possible exponential factor, this extends a recent result of Hmidi [7] to the full regime $d\ge 1$, $0\le \gamma \le 2$ and removes the incompressibility assumption in the $L^\infty$ case.
Citation: Hongjie Dong, Dong Li. On a generalized maximum principle for a transport-diffusion model with $\log$-modulated fractional dissipation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3437-3454. doi: 10.3934/dcds.2014.34.3437
References:
[1]

R. Askey, Radial Characteristic Functions,, University of Wisconsin-Madison, (1262). Google Scholar

[2]

A. Córdoba and D. Córdoba, A maximum principle applied to quasi-geostrophic equations,, Comm. Math. Phys., 249 (2004), 511. doi: 10.1007/s00220-004-1055-1. Google Scholar

[3]

C. H. Chan, M. Czubak and L. Silvestre, Eventual regularization of the slightly supercritical fractional Burgers equation,, Discrete Contin. Dyn. Syst., 27 (2010), 847. doi: 10.3934/dcds.2010.27.847. Google Scholar

[4]

H. Dong, D. Du and D. Li, Finite time singularities and global well-posedness for fractal Burgers equations,, Indiana Univ. Math. J., 58 (2009), 807. doi: 10.1512/iumj.2009.58.3505. Google Scholar

[5]

M. Dabkowski, A. Kiselev, L. Silvestre and V. Vicol, Global well-posedness of slightly supercritical active scalar equations,, Analysis and PDE, (). Google Scholar

[6]

M. Dabkowski, A. Kiselev and V. Vicol, Global well-posedness for a slightly supercritical surface quasi-geostrophic equation,, Nonlinearity, 25 (2012), 1525. doi: 10.1088/0951-7715/25/5/1525. Google Scholar

[7]

T. Hmidi, On a maximum principle and its application to the logarithmically critical Boussinesq system,, Anal. PDE, 4 (2011), 247. doi: 10.2140/apde.2011.4.247. Google Scholar

[8]

N. Ju, The maximum principle and the global attractor for the dissipative 2D quasi-geostrophic equations,, Comm. Math. Phys., 255 (2005), 161. doi: 10.1007/s00220-004-1256-7. Google Scholar

[9]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions,, Princeton Mathematical Series, (1970). Google Scholar

[10]

T. Tao, Global regularity for a logarithmically supercritical hyperdissipative Navier-Stokes equation,, Anal. PDE, 2 (2009), 361. doi: 10.2140/apde.2009.2.361. Google Scholar

[11]

J. Wu, Global regularity for a class of generalized magnetohydrodynamic equations,, J. Math. Fluid Mech., 13 (2011), 295. doi: 10.1007/s00021-009-0017-y. Google Scholar

show all references

References:
[1]

R. Askey, Radial Characteristic Functions,, University of Wisconsin-Madison, (1262). Google Scholar

[2]

A. Córdoba and D. Córdoba, A maximum principle applied to quasi-geostrophic equations,, Comm. Math. Phys., 249 (2004), 511. doi: 10.1007/s00220-004-1055-1. Google Scholar

[3]

C. H. Chan, M. Czubak and L. Silvestre, Eventual regularization of the slightly supercritical fractional Burgers equation,, Discrete Contin. Dyn. Syst., 27 (2010), 847. doi: 10.3934/dcds.2010.27.847. Google Scholar

[4]

H. Dong, D. Du and D. Li, Finite time singularities and global well-posedness for fractal Burgers equations,, Indiana Univ. Math. J., 58 (2009), 807. doi: 10.1512/iumj.2009.58.3505. Google Scholar

[5]

M. Dabkowski, A. Kiselev, L. Silvestre and V. Vicol, Global well-posedness of slightly supercritical active scalar equations,, Analysis and PDE, (). Google Scholar

[6]

M. Dabkowski, A. Kiselev and V. Vicol, Global well-posedness for a slightly supercritical surface quasi-geostrophic equation,, Nonlinearity, 25 (2012), 1525. doi: 10.1088/0951-7715/25/5/1525. Google Scholar

[7]

T. Hmidi, On a maximum principle and its application to the logarithmically critical Boussinesq system,, Anal. PDE, 4 (2011), 247. doi: 10.2140/apde.2011.4.247. Google Scholar

[8]

N. Ju, The maximum principle and the global attractor for the dissipative 2D quasi-geostrophic equations,, Comm. Math. Phys., 255 (2005), 161. doi: 10.1007/s00220-004-1256-7. Google Scholar

[9]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions,, Princeton Mathematical Series, (1970). Google Scholar

[10]

T. Tao, Global regularity for a logarithmically supercritical hyperdissipative Navier-Stokes equation,, Anal. PDE, 2 (2009), 361. doi: 10.2140/apde.2009.2.361. Google Scholar

[11]

J. Wu, Global regularity for a class of generalized magnetohydrodynamic equations,, J. Math. Fluid Mech., 13 (2011), 295. doi: 10.1007/s00021-009-0017-y. Google Scholar

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