Global well-posedness and symmetries for dissipative active scalar equations with positive-order couplings

2013/05/14 by Lucas C. F. Ferreira, Ferreira, Lucas C. F., Lidiane S. M. Lima +1
Mathematics · #35A01 #35B06 #35B40 #35Q35 #35R11 #76D03 #86A10 #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP #msc:35A01 #msc:35B06 #msc:35B40 #msc:35Q35 #msc:35R11 #msc:76D03 #msc:86A10

paper · pdf · doi:10.48550/arxiv.1305.2987

arxiv created 2014/01/02 · arxiv updated 2014/01/03

Abstract

We consider a family of dissipative active scalar equations outside the L²-space. This was introduced in [D. Chae, P. Constantin, J. Wu, to appear in IUMJ (2014)] and its velocity fields are coupled with the active scalar via a class of multiplier operators which morally behave as derivatives of positive order. We prove global well-posedness and time decay of solutions, without smallness assumptions, for initial data belonging to the critical Lebesgue space L^(n)/(γ-β)(ℝⁿ) which is a class larger than that of the above reference. Symmetry properties of solutions are investigated depending on the symmetry of initial data and coupling operators.

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