June  2015, 10(2): 387-399. doi: 10.3934/nhm.2015.10.387

Inhomogeneities in 3 dimensional oscillatory media

1. 

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

Received  January 2014 Revised  December 2014 Published  April 2015

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.
Citation: Gabriela Jaramillo. Inhomogeneities in 3 dimensional oscillatory media. Networks & Heterogeneous Media, 2015, 10 (2) : 387-399. doi: 10.3934/nhm.2015.10.387
References:
[1]

R. A. Adams and J. J. F. Fournier, Sobolev Spaces,, 2nd edition, (2003). Google Scholar

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I. S. Aranson and L. Kramer, The world of the complex Ginzburg-Landau equation,, Reviews of Modern Physics, 74 (2002), 99. doi: 10.1103/RevModPhys.74.99. Google Scholar

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G. Jaramillo and A. Scheel, Deformation of striped patterns by inhomogeneities,, Mathematical Methods in the Applied Sciences, 38 (2015), 51. doi: 10.1002/mma.3049. Google Scholar

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A.-K. Kassam, Solving reaction-diffusion equations 10 times faster,, 2003., (). Google Scholar

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A.-K. Kassam and L. N. Trefethen, Fourth-order time-stepping for stiff pdes,, SIAM Journal on Scientific Computing, 26 (2005), 1214. doi: 10.1137/S1064827502410633. Google Scholar

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R. Kollár and A. Scheel, Coherent structures generated by inhomogeneities in oscillatory media,, SIAM J. Appl. Dyn. Syst., 6 (2007), 236. doi: 10.1137/060666950. Google Scholar

[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. Google Scholar

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R. B. Lockhart, Fredholm properties of a class of elliptic operators on noncompact manifolds,, Duke Math. J., 48 (1981), 289. doi: 10.1215/S0012-7094-81-04817-1. Google Scholar

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R. B. Lockhart and R. C. McOwen, On elliptic systems in $\mathbbR^n$,, Acta Math., 150 (1983), 125. doi: 10.1007/BF02392969. Google Scholar

[10]

R. B. Lockhart and R. C. McOwen, Elliptic differential operators on noncompact manifolds,, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 12 (1985), 409. Google Scholar

[11]

R. C. McOwen, The behavior of the laplacian on weighted Sobolev spaces,, Communications on Pure and Applied Mathematics, 32 (1979), 783. doi: 10.1002/cpa.3160320604. Google Scholar

[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. doi: 10.1080/03605300802038536. Google Scholar

[13]

V. Milisic and U. Razafison, Weighted Sobolev spaces for the Laplace equation in periodic infinite strips,, preprint, (). Google Scholar

[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. doi: 10.1016/0022-247X(73)90138-8. Google Scholar

[15]

M. Specovius-Neugebauer and W. Wendland, Exterior stokes problems and decay at infinity,, Mathematical Methods in the Applied Sciences, 8 (1986), 351. doi: 10.1002/mma.1670080124. Google Scholar

[16]

M. Stich and A. S. Mikhailov, Target patterns in two-dimensional heterogeneous oscillatory reaction-diffusion systems,, Physica D: Nonlinear Phenomena, 215 (2006), 38. doi: 10.1016/j.physd.2006.01.011. Google Scholar

show all references

References:
[1]

R. A. Adams and J. J. F. Fournier, Sobolev Spaces,, 2nd edition, (2003). Google Scholar

[2]

I. S. Aranson and L. Kramer, The world of the complex Ginzburg-Landau equation,, Reviews of Modern Physics, 74 (2002), 99. doi: 10.1103/RevModPhys.74.99. Google Scholar

[3]

G. Jaramillo and A. Scheel, Deformation of striped patterns by inhomogeneities,, Mathematical Methods in the Applied Sciences, 38 (2015), 51. doi: 10.1002/mma.3049. Google Scholar

[4]

A.-K. Kassam, Solving reaction-diffusion equations 10 times faster,, 2003., (). Google Scholar

[5]

A.-K. Kassam and L. N. Trefethen, Fourth-order time-stepping for stiff pdes,, SIAM Journal on Scientific Computing, 26 (2005), 1214. doi: 10.1137/S1064827502410633. Google Scholar

[6]

R. Kollár and A. Scheel, Coherent structures generated by inhomogeneities in oscillatory media,, SIAM J. Appl. Dyn. Syst., 6 (2007), 236. doi: 10.1137/060666950. Google Scholar

[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. Google Scholar

[8]

R. B. Lockhart, Fredholm properties of a class of elliptic operators on noncompact manifolds,, Duke Math. J., 48 (1981), 289. doi: 10.1215/S0012-7094-81-04817-1. Google Scholar

[9]

R. B. Lockhart and R. C. McOwen, On elliptic systems in $\mathbbR^n$,, Acta Math., 150 (1983), 125. doi: 10.1007/BF02392969. Google Scholar

[10]

R. B. Lockhart and R. C. McOwen, Elliptic differential operators on noncompact manifolds,, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 12 (1985), 409. Google Scholar

[11]

R. C. McOwen, The behavior of the laplacian on weighted Sobolev spaces,, Communications on Pure and Applied Mathematics, 32 (1979), 783. doi: 10.1002/cpa.3160320604. Google Scholar

[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. doi: 10.1080/03605300802038536. Google Scholar

[13]

V. Milisic and U. Razafison, Weighted Sobolev spaces for the Laplace equation in periodic infinite strips,, preprint, (). Google Scholar

[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. doi: 10.1016/0022-247X(73)90138-8. Google Scholar

[15]

M. Specovius-Neugebauer and W. Wendland, Exterior stokes problems and decay at infinity,, Mathematical Methods in the Applied Sciences, 8 (1986), 351. doi: 10.1002/mma.1670080124. Google Scholar

[16]

M. Stich and A. S. Mikhailov, Target patterns in two-dimensional heterogeneous oscillatory reaction-diffusion systems,, Physica D: Nonlinear Phenomena, 215 (2006), 38. doi: 10.1016/j.physd.2006.01.011. Google Scholar

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