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

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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

Abstract
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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 and 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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[12]	J. Slattery, Advanced Transport Phenomena, Cambridge University Press, Cambridge, 1999.
[13]	W. Strauss, Partial Differential Equations. An Introduction, John Wiley and Sons, Inc., New York, 1992.
[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.
[15]	Z. Warhaft, Passive scalars in turbulent flows, Annu. Rev. Fluid Mech., 32 (2000), 203-240. doi: 10.1146/annurev.fluid.32.1.203.
[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.

show all references

References:

[1]	J. Beale, Large-time regularity of viscous surface waves,, Arch. Rational Mech. Anal., 84 (): 307. doi: 10.1007/BF00250586.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[12]	J. Slattery, Advanced Transport Phenomena, Cambridge University Press, Cambridge, 1999.
[13]	W. Strauss, Partial Differential Equations. An Introduction, John Wiley and Sons, Inc., New York, 1992.
[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.
[15]	Z. Warhaft, Passive scalars in turbulent flows, Annu. Rev. Fluid Mech., 32 (2000), 203-240. doi: 10.1146/annurev.fluid.32.1.203.
[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.

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