# American Institute of Mathematical Sciences

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 and Continuous Dynamical Systems, 2014, 34 (9) : 3437-3454. doi: 10.3934/dcds.2014.34.3437
##### References:
 [1] R. Askey, Radial Characteristic Functions, University of Wisconsin-Madison, Mathematics Research Center, 1262, 1973. [2] A. Córdoba and D. Córdoba, A maximum principle applied to quasi-geostrophic equations, Comm. Math. Phys., 249 (2004), 511-528. doi: 10.1007/s00220-004-1055-1. [3] C. H. Chan, M. Czubak and L. Silvestre, Eventual regularization of the slightly supercritical fractional Burgers equation, Discrete Contin. Dyn. Syst., 27 (2010), 847-861. doi: 10.3934/dcds.2010.27.847. [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), 807-821. doi: 10.1512/iumj.2009.58.3505. [5] M. Dabkowski, A. Kiselev, L. Silvestre and V. Vicol, Global well-posedness of slightly supercritical active scalar equations,, Analysis and PDE, (). [6] M. Dabkowski, A. Kiselev and V. Vicol, Global well-posedness for a slightly supercritical surface quasi-geostrophic equation, Nonlinearity, 25 (2012), 1525-1535. doi: 10.1088/0951-7715/25/5/1525. [7] T. Hmidi, On a maximum principle and its application to the logarithmically critical Boussinesq system, Anal. PDE, 4 (2011), 247-284. doi: 10.2140/apde.2011.4.247. [8] N. Ju, The maximum principle and the global attractor for the dissipative 2D quasi-geostrophic equations, Comm. Math. Phys., 255 (2005), 161-181. doi: 10.1007/s00220-004-1256-7. [9] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30 Princeton University Press, Princeton, N.J. 1970. [10] T. Tao, Global regularity for a logarithmically supercritical hyperdissipative Navier-Stokes equation, Anal. PDE, 2 (2009), 361-366. doi: 10.2140/apde.2009.2.361. [11] J. Wu, Global regularity for a class of generalized magnetohydrodynamic equations, J. Math. Fluid Mech., 13 (2011), 295-305. doi: 10.1007/s00021-009-0017-y.

show all references

##### References:
 [1] R. Askey, Radial Characteristic Functions, University of Wisconsin-Madison, Mathematics Research Center, 1262, 1973. [2] A. Córdoba and D. Córdoba, A maximum principle applied to quasi-geostrophic equations, Comm. Math. Phys., 249 (2004), 511-528. doi: 10.1007/s00220-004-1055-1. [3] C. H. Chan, M. Czubak and L. Silvestre, Eventual regularization of the slightly supercritical fractional Burgers equation, Discrete Contin. Dyn. Syst., 27 (2010), 847-861. doi: 10.3934/dcds.2010.27.847. [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), 807-821. doi: 10.1512/iumj.2009.58.3505. [5] M. Dabkowski, A. Kiselev, L. Silvestre and V. Vicol, Global well-posedness of slightly supercritical active scalar equations,, Analysis and PDE, (). [6] M. Dabkowski, A. Kiselev and V. Vicol, Global well-posedness for a slightly supercritical surface quasi-geostrophic equation, Nonlinearity, 25 (2012), 1525-1535. doi: 10.1088/0951-7715/25/5/1525. [7] T. Hmidi, On a maximum principle and its application to the logarithmically critical Boussinesq system, Anal. PDE, 4 (2011), 247-284. doi: 10.2140/apde.2011.4.247. [8] N. Ju, The maximum principle and the global attractor for the dissipative 2D quasi-geostrophic equations, Comm. Math. Phys., 255 (2005), 161-181. doi: 10.1007/s00220-004-1256-7. [9] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30 Princeton University Press, Princeton, N.J. 1970. [10] T. Tao, Global regularity for a logarithmically supercritical hyperdissipative Navier-Stokes equation, Anal. PDE, 2 (2009), 361-366. doi: 10.2140/apde.2009.2.361. [11] J. Wu, Global regularity for a class of generalized magnetohydrodynamic equations, J. Math. Fluid Mech., 13 (2011), 295-305. doi: 10.1007/s00021-009-0017-y.
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