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March  2011, 10(2): 653-665. doi: 10.3934/cpaa.2011.10.653

Global wellposedness for a transport equation with super-critial dissipation

Xumin Gu 1,

1. 

School of Mathematical Sciences, Fudan University, Shanghai 200433, China

Received  May 2010 Revised  October 2010 Published  December 2010

We study a one-dimensional transport equation with non-local velocity and supercritial dissipation. Using the methods of modulus of continuity introduced in [1] and fractional Laplacian representaiton introduced in [2], we prove its global well-posedness for small periodic initial data in Holder spaces.
Keywords: global regularity., Super-critical dissipation, modulus of continuity, fractional Laplacian.
Mathematics Subject Classification: Primary: 35Q35; Secondary: 76D0.
Citation: Xumin Gu. Global wellposedness for a transport equation with super-critial dissipation. Communications on Pure and Applied Analysis, 2011, 10 (2) : 653-665. doi: 10.3934/cpaa.2011.10.653
References:
[1]

A. Kiselev, F. nazarov and A. Volberg, Global well-posedness for the critical 2D dissipative quasi-geostrophic equation, Invent. Mathe., 167 (2007), 445-453. doi: doi:10.1007/s00222-006-0020-3.

[2]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian,, preprint, (). 

[3]

P. Constantin, P. D. Lax and A. Majda, A simple one-dimensional model for the three-dimensional vorticity equation, Comm. Pure Appl. Math., 38 (1985), 715-724. doi: doi:10.1002/cpa.3160380605.

[4]

P. Constantin, A. Majda and E. Tabak, Formation of strong fronts in the 2-D quasigeostrophic thermal active scalar, Nonlinearity, 7 (1994), 1495-1533. doi: doi:10.1088/0951-7715/7/6/001.

[5]

D. Córdoba, Nonexistence of simple hyperbolic blow-up for the quasi-geostrophic equation, Ann. of Math., 148 (1998), 1135-1152. doi: doi:10.2307/121037.

[6]

K. Y. Wang, Global well-posedness for a transport equation with non-local velocity and critical diffusion, Commun. Pure Appl. Anal., 7 (2008), 1203-1210. doi: doi:10.3934/cpaa.2008.7.1203.

[7]

A. Córdoba and D. Córdoba, A maximum principle applied to quasi-geostrophic equations, Comm. Math. Phys., 249 (2004), 511-528.

[8]

H. Okamoto, T. Sakajo and M. Wunsch, On a generalization of the Constantin-Lax-Majda equation, Nonlinearity, 21 (2008), 2447-2461. doi: doi:10.1088/0951-7715/21/10/013.

[9]

X. W. Yu, Remarks on the global regularity for the super-critical 2D dissipative quasi-geostrophic equation, J. Math. Anal. Appl., 339 (2008), 359-371. doi: doi:10.1016/j.jmaa.2007.06.064.

[10]

G. R. Baker, X. Li and A. C. Morlet, Analytic structure of two 1D-transport equations with nonlocal fluxes, Physica D: Nonlinear Phenomena, 91 (1996), 349-375. doi: doi:10.1016/0167-2789(95)00271-5.

[11]

A. Córdoba, D. Córdoba and M. A. Fontelos, Integral inequalities for the Hilbert transform applied to a nonlocal transport equation, J. Math. Pures Appl., 86 (2006), 529-540. doi: doi:10.1016/j.matpur.2006.08.002.

[12]

A. Córdoba, D. Córdoba and M. A. Fontelos, Formation of singularities for a transport equation with nonlocal velocity, Ann. of Math., 162 (2005), 1377-1389. doi: doi:10.4007/annals.2005.162.1377.

show all references

References:
[1]

A. Kiselev, F. nazarov and A. Volberg, Global well-posedness for the critical 2D dissipative quasi-geostrophic equation, Invent. Mathe., 167 (2007), 445-453. doi: doi:10.1007/s00222-006-0020-3.

[2]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian,, preprint, (). 

[3]

P. Constantin, P. D. Lax and A. Majda, A simple one-dimensional model for the three-dimensional vorticity equation, Comm. Pure Appl. Math., 38 (1985), 715-724. doi: doi:10.1002/cpa.3160380605.

[4]

P. Constantin, A. Majda and E. Tabak, Formation of strong fronts in the 2-D quasigeostrophic thermal active scalar, Nonlinearity, 7 (1994), 1495-1533. doi: doi:10.1088/0951-7715/7/6/001.

[5]

D. Córdoba, Nonexistence of simple hyperbolic blow-up for the quasi-geostrophic equation, Ann. of Math., 148 (1998), 1135-1152. doi: doi:10.2307/121037.

[6]

K. Y. Wang, Global well-posedness for a transport equation with non-local velocity and critical diffusion, Commun. Pure Appl. Anal., 7 (2008), 1203-1210. doi: doi:10.3934/cpaa.2008.7.1203.

[7]

A. Córdoba and D. Córdoba, A maximum principle applied to quasi-geostrophic equations, Comm. Math. Phys., 249 (2004), 511-528.

[8]

H. Okamoto, T. Sakajo and M. Wunsch, On a generalization of the Constantin-Lax-Majda equation, Nonlinearity, 21 (2008), 2447-2461. doi: doi:10.1088/0951-7715/21/10/013.

[9]

X. W. Yu, Remarks on the global regularity for the super-critical 2D dissipative quasi-geostrophic equation, J. Math. Anal. Appl., 339 (2008), 359-371. doi: doi:10.1016/j.jmaa.2007.06.064.

[10]

G. R. Baker, X. Li and A. C. Morlet, Analytic structure of two 1D-transport equations with nonlocal fluxes, Physica D: Nonlinear Phenomena, 91 (1996), 349-375. doi: doi:10.1016/0167-2789(95)00271-5.

[11]

A. Córdoba, D. Córdoba and M. A. Fontelos, Integral inequalities for the Hilbert transform applied to a nonlocal transport equation, J. Math. Pures Appl., 86 (2006), 529-540. doi: doi:10.1016/j.matpur.2006.08.002.

[12]

A. Córdoba, D. Córdoba and M. A. Fontelos, Formation of singularities for a transport equation with nonlocal velocity, Ann. of Math., 162 (2005), 1377-1389. doi: doi:10.4007/annals.2005.162.1377.

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