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March  2016, 36(3): 1383-1413. doi: 10.3934/dcds.2016.36.1383

Passive scalars, moving boundaries, and Newton's law of cooling

Juhi Jang 1, and  Ian Tice 2,

1. 

Department of Mathematics, University of California, Riverside, Riverside, CA 92521
2. 

Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, United States

Received  January 2015 Revised  June 2015 Published  August 2015

We study the evolution of passive scalars in both rigid and moving slab-like domains, in both horizontally periodic and infinite contexts. The scalar is required to satisfy Robin-type boundary conditions corresponding to Newton's law of cooling, which lead to nontrivial equilibrium configurations. We study the equilibration rate of the passive scalar in terms of the parameters in the boundary condition and the equilibration rates of the background velocity field and moving domain.
Keywords: equilibration rates, energy estimates, calculus of variations., Passive scalars, Newton's law of cooling, free boundary problems.
Mathematics Subject Classification: Primary: 35Q35, 35B40, 35A23; Secondary: 35A23, 35Q30, 58J6.
Citation: Juhi Jang, Ian Tice. Passive scalars, moving boundaries, and Newton's law of cooling. Discrete & Continuous Dynamical Systems, 2016, 36 (3) : 1383-1413. doi: 10.3934/dcds.2016.36.1383
References:
[1]

J. Beale, Large-time regularity of viscous surface waves,, Arch. Rational Mech. Anal., 84 (): 307.  doi: 10.1007/BF00250586.  Google Scholar

[2]

J. Beale and T. Nishida, Large-time behavior of viscous surface waves, Recent topics in nonlinear PDE, II (Sendai, 1984), 1-14, North-Holland Math. Stud., 128, North-Holland, Amsterdam, 1985. doi: 10.1016/S0304-0208(08)72355-7.  Google Scholar

[3]

P. Constantin, A. Kiselev, L. Ryzhik and A. Zlatoš, Diffusion and mixing in fluid flow, Ann. of Math. (2), 168 (2008), 643-674. doi: 10.4007/annals.2008.168.643.  Google Scholar

[4]

A. Kiselev, R. Shterenberg and A. Zlatoš, Relaxation enhancement by time-periodic flows, Indiana Univ. Math. J., 57 (2008), 2137-2152. doi: 10.1512/iumj.2008.57.3349.  Google Scholar

[5]

Y. Guo and I. Tice, Almost exponential decay of periodic viscous surface waves without surface tension, Arch. Rational Mech. Anal., 207 (2013), 459-531. doi: 10.1007/s00205-012-0570-z.  Google Scholar

[6]

Y. Guo and I. Tice, Decay of viscous surface waves without surface tension in horizontally infinite domains, Anal. PDE, 6 (2013), 1429-1533. doi: 10.2140/apde.2013.6.1429.  Google Scholar

[7]

Y. Guo and I. Tice, Local well-posedness of the viscous surface wave problem without surface tension, Anal. PDE, 6 (2013), 287-369. doi: 10.2140/apde.2013.6.287.  Google Scholar

[8]

Y. Guo and Y. Wang, Decay of dissipative equations and negative Sobolev spaces, Comm. Partial Differential Equations, 37 (2012), 2165-2208. doi: 10.1080/03605302.2012.696296.  Google Scholar

[9]

Z. Lin, J.-L. Thiffeault and C. Doering, Optimal stirring strategies for passive scalar mixing, J. Fluid Mech., 675 (2011), 465-476. doi: 10.1017/S0022112011000292.  Google Scholar

[10]

A. Majda and P. Kramer, Simplified models for turbulent diffusion: Theory, numerical modelling, and physical phenomena, Phys. Rep., 314 (1999), 237-574. doi: 10.1016/S0370-1573(98)00083-0.  Google Scholar

[11]

T. Nishida, Y. Teramoto and H. Yoshihara, Global in time behavior of viscous surface waves: Horizontally periodic motion, J. Math. Kyoto Univ., 44 (2004), 271-323.  Google Scholar

[12]

J. Slattery, Advanced Transport Phenomena, Cambridge University Press, Cambridge, 1999. Google Scholar

[13]

W. Strauss, Partial Differential Equations. An Introduction, John Wiley and Sons, Inc., New York, 1992.  Google Scholar

[14]

J.-L. Thiffeault, Using multiscale norms to quantify mixing and transport, Nonlinearity, 25 (2012), R1-R44. doi: 10.1088/0951-7715/25/2/R1.  Google Scholar

[15]

Z. Warhaft, Passive scalars in turbulent flows, Annu. Rev. Fluid Mech., 32 (2000), 203-240. doi: 10.1146/annurev.fluid.32.1.203.  Google Scholar

[16]

A. Zlatoš, Diffusion in fluid flow: Dissipation enhancement by flows in 2D, Comm. Partial Differential Equations, 35 (2010), 496-534. doi: 10.1080/03605300903362546.  Google Scholar

show all references

References:
[1]

J. Beale, Large-time regularity of viscous surface waves,, Arch. Rational Mech. Anal., 84 (): 307.  doi: 10.1007/BF00250586.  Google Scholar

[2]

J. Beale and T. Nishida, Large-time behavior of viscous surface waves, Recent topics in nonlinear PDE, II (Sendai, 1984), 1-14, North-Holland Math. Stud., 128, North-Holland, Amsterdam, 1985. doi: 10.1016/S0304-0208(08)72355-7.  Google Scholar

[3]

P. Constantin, A. Kiselev, L. Ryzhik and A. Zlatoš, Diffusion and mixing in fluid flow, Ann. of Math. (2), 168 (2008), 643-674. doi: 10.4007/annals.2008.168.643.  Google Scholar

[4]

A. Kiselev, R. Shterenberg and A. Zlatoš, Relaxation enhancement by time-periodic flows, Indiana Univ. Math. J., 57 (2008), 2137-2152. doi: 10.1512/iumj.2008.57.3349.  Google Scholar

[5]

Y. Guo and I. Tice, Almost exponential decay of periodic viscous surface waves without surface tension, Arch. Rational Mech. Anal., 207 (2013), 459-531. doi: 10.1007/s00205-012-0570-z.  Google Scholar

[6]

Y. Guo and I. Tice, Decay of viscous surface waves without surface tension in horizontally infinite domains, Anal. PDE, 6 (2013), 1429-1533. doi: 10.2140/apde.2013.6.1429.  Google Scholar

[7]

Y. Guo and I. Tice, Local well-posedness of the viscous surface wave problem without surface tension, Anal. PDE, 6 (2013), 287-369. doi: 10.2140/apde.2013.6.287.  Google Scholar

[8]

Y. Guo and Y. Wang, Decay of dissipative equations and negative Sobolev spaces, Comm. Partial Differential Equations, 37 (2012), 2165-2208. doi: 10.1080/03605302.2012.696296.  Google Scholar

[9]

Z. Lin, J.-L. Thiffeault and C. Doering, Optimal stirring strategies for passive scalar mixing, J. Fluid Mech., 675 (2011), 465-476. doi: 10.1017/S0022112011000292.  Google Scholar

[10]

A. Majda and P. Kramer, Simplified models for turbulent diffusion: Theory, numerical modelling, and physical phenomena, Phys. Rep., 314 (1999), 237-574. doi: 10.1016/S0370-1573(98)00083-0.  Google Scholar

[11]

T. Nishida, Y. Teramoto and H. Yoshihara, Global in time behavior of viscous surface waves: Horizontally periodic motion, J. Math. Kyoto Univ., 44 (2004), 271-323.  Google Scholar

[12]

J. Slattery, Advanced Transport Phenomena, Cambridge University Press, Cambridge, 1999. Google Scholar

[13]

W. Strauss, Partial Differential Equations. An Introduction, John Wiley and Sons, Inc., New York, 1992.  Google Scholar

[14]

J.-L. Thiffeault, Using multiscale norms to quantify mixing and transport, Nonlinearity, 25 (2012), R1-R44. doi: 10.1088/0951-7715/25/2/R1.  Google Scholar

[15]

Z. Warhaft, Passive scalars in turbulent flows, Annu. Rev. Fluid Mech., 32 (2000), 203-240. doi: 10.1146/annurev.fluid.32.1.203.  Google Scholar

[16]

A. Zlatoš, Diffusion in fluid flow: Dissipation enhancement by flows in 2D, Comm. Partial Differential Equations, 35 (2010), 496-534. doi: 10.1080/03605300903362546.  Google Scholar

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