Inhomogeneities in 3 dimensional oscillatory media

Gabriela Jaramillo

doi:10.3934/nhm.2015.10.387

Article Contents

2015, Volume 10, Issue 2: 387-399. Doi: 10.3934/nhm.2015.10.387

This issue Previous Article Stability of conductivities in an inverse problem in the reaction-diffusion system in electrocardiology Next Article Self-similar control systems and applications to zygodactyl bird's foot

Inhomogeneities in 3 dimensional oscillatory media

Gabriela Jaramillo^1,

1.
University of Minnesota, School of Mathematics, 127 Vincent Hall, 206 Church St SE, Minneapolis, MN 55455

Received: January 2014

Revised: December 2014

Published: June 2015

Abstract Related Papers Cited by

Abstract

We consider localized perturbations to spatially homogeneous oscillations in dimension 3 using the complex Ginzburg-Landau equation as a prototype. In particular, we will focus on inhomogeneities that locally change the phase of the oscillations. In the usual translation invariant spaces and at $ \epsilon=0$ the linearization about these spatially homogeneous solutions result in an operator with zero eigenvalue embedded in the essential spectrum. In contrast, we show that when considered as an operator between Kondratiev spaces, the linearization is a Fredholm operator. These spaces consist of functions with algebraical localization that increases with each derivative. We use this result to construct solutions close to the equilibrium via the Implicit Function Theorem and derive asymptotics for wavenumbers in the far field.

Keywords:

Mathematics Subject Classification: Primary: 35B36, 35Q56; Secondary: 46E35.

Citation:

Related Papers

Cited by

References

[1]	R. A. Adams and J. J. F. Fournier, Sobolev Spaces, 2nd edition, Pure and Applied Mathematics, 140, Elsevier/Academic Press, Amsterdam, 2003.
[2]	I. S. Aranson and L. Kramer, The world of the complex Ginzburg-Landau equation, Reviews of Modern Physics, 74 (2002), 99-143.doi: 10.1103/RevModPhys.74.99.
[3]	G. Jaramillo and A. Scheel, Deformation of striped patterns by inhomogeneities, Mathematical Methods in the Applied Sciences, 38 (2015), 51-65.doi: 10.1002/mma.3049.
[4]	A.-K. Kassam, Solving reaction-diffusion equations 10 times faster, 2003.
[5]	A.-K. Kassam and L. N. Trefethen, Fourth-order time-stepping for stiff pdes, SIAM Journal on Scientific Computing, 26 (2005), 1214-1233.doi: 10.1137/S1064827502410633.
[6]	R. Kollár and A. Scheel, Coherent structures generated by inhomogeneities in oscillatory media, SIAM J. Appl. Dyn. Syst., 6 (2007), 236-262.doi: 10.1137/060666950.
[7]	V. A. Kondrat'ev, Boundary value problems for elliptic equations in domains with conical or angular points, Trudy Moskov Mat. Obšč., 16 (1967), 209-292.
[8]	R. B. Lockhart, Fredholm properties of a class of elliptic operators on noncompact manifolds, Duke Math. J., 48 (1981), 289-312.doi: 10.1215/S0012-7094-81-04817-1.
[9]	R. B. Lockhart and R. C. McOwen, On elliptic systems in $\mathbbR^n$, Acta Math., 150 (1983), 125-135.doi: 10.1007/BF02392969.
[10]	R. B. Lockhart and R. C. McOwen, Elliptic differential operators on noncompact manifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 12 (1985), 409-447.
[11]	R. C. McOwen, The behavior of the laplacian on weighted Sobolev spaces, Communications on Pure and Applied Mathematics, 32 (1979), 783-795.doi: 10.1002/cpa.3160320604.
[12]	A. Melcher, G. Schneider and H. Uecker, A hopf-bifurcation theorem for the vorticity formulation of the Navier-Stokes equations in $\mathbbR^3$, Communications in Partial Differential Equations, 33 (2008), 772-783.doi: 10.1080/03605300802038536.
[13]	V. Milisic and U. Razafison, Weighted Sobolev spaces for the Laplace equation in periodic infinite strips, preprint, arXiv:1302.4253.
[14]	L. Nirenberg and H. F. Walker, The null spaces of elliptic partial differential operators in $\mathbbR^n$, J. Math. Anal. Appl., 42 (1973), 271-301.doi: 10.1016/0022-247X(73)90138-8.
[15]	M. Specovius-Neugebauer and W. Wendland, Exterior stokes problems and decay at infinity, Mathematical Methods in the Applied Sciences, 8 (1986), 351-367.doi: 10.1002/mma.1670080124.
[16]	M. Stich and A. S. Mikhailov, Target patterns in two-dimensional heterogeneous oscillatory reaction-diffusion systems, Physica D: Nonlinear Phenomena, 215 (2006), 38-45.doi: 10.1016/j.physd.2006.01.011.