Validity of Whitham’s modulation equations for dissipative systems with a conservation law: phase dynamics in a generalized Ginzburg-Landau system. (English) Zbl 1525.35007
This paper focuses on Whitham modulation equations for dissipative systems coupled with a conservation law, proving a first result towards a rigorous approximation result, in the setting of the generalised Ginzburg-Landau system (i.e. a complex Ginzburg-Landau equation coupled with a conservation law).
The main result shows that the error between the approximation by means of Whitham modulation equations and solution of the system can be rigorously bounded. The strategy is as follows: at first approximate solutions are constructed, then the error estimate is proven.
The tools used to carry out the proof are Cauchy-Kovalevskaya theory in Gevrey spaces, as well as a Fourier decomposition that allows for separation of stable and center modes, with a semiderivative, corresponding to a vanishing Fourier multiplier, controlling the nonlinear terms for the center modes. One of the difficulties in the arguments lies in the estimates used to prove the main theorem, that require some careful tuning for the time scale to be the correct one.
The main result shows that the error between the approximation by means of Whitham modulation equations and solution of the system can be rigorously bounded. The strategy is as follows: at first approximate solutions are constructed, then the error estimate is proven.
The tools used to carry out the proof are Cauchy-Kovalevskaya theory in Gevrey spaces, as well as a Fourier decomposition that allows for separation of stable and center modes, with a semiderivative, corresponding to a vanishing Fourier multiplier, controlling the nonlinear terms for the center modes. One of the difficulties in the arguments lies in the estimates used to prove the main theorem, that require some careful tuning for the time scale to be the correct one.
Reviewer: Luigi Amedeo Bianchi (Povo)
MSC:
| 35A10 | Cauchy-Kovalevskaya theorems |
| 35B10 | Periodic solutions to PDEs |
| 35Q56 | Ginzburg-Landau equations |
Keywords:
Whitham modulation equations; validity; wave trains; generalized Ginzburg-Landau system; Cauchy-Kovalevskaya theory; energy estimates in Gevrey spaces; local decomposition in Fourier spaceReferences:
| [1] | A. S. ALNAHDI, J. NIESEN, AND A. M. RUCKLIDGE, Localized patterns in periodically forced systems, SIAM J. Appl. Dyn. Syst. 13 (2014), no. 3, 1311-1327. https://dx.doi. org/10.1137/130948495. MR3264560. · Zbl 1310.35036 · doi:10.1137/130948495.MR3264560 |
| [2] | T. J. BRIDGES, A. KOSTIANKO, AND G. SCHNEIDER, A proof of validity for multiphase Whitham modulation theory, Proc. A. 476 (2020), no. 2243, 20200203, 19 pp. http:// dx.doi.org/10.1098/rspa.2020.0203. MR4187951. · Zbl 1472.35332 · doi:10.1098/rspa.2020.0203.MR4187951 |
| [3] | T. J. BRIDGES, A. KOSTIANKO, AND S. ZELIK, Validity of the hyperbolic Whitham modulation equations in Sobolev spaces, J. Differential Equations 274 (2021), 971-995. https://dx.doi.org/10.1016/j.jde.2020.11.019. MR4188999. · Zbl 1455.35233 · doi:10.1137/130948495 |
| [4] | T.J. BRIDGES, Symmetry, Phase Modulation and Nonlinear Waves, Cambridge Mono-graphs on Applied and Computational Mathematics, vol. 31, Cambridge University Press, Cambridge, 2017. https://dx.doi.org/10.1017/9781316986769. MR3702013. · doi:10.1098/rspa.2020.0203 |
| [5] | H.-C. CHANG AND E. A. DEMEKHIN, Complex Wave Dynamics on Thin Films, Studies in Interface Science, vol. 14, Elsevier Science B.V., Amsterdam, 2002. MR2362445. |
| [6] | P. COLLET AND J.-P. ECKMANN, The time dependent amplitude equation for the Swift-Hohenberg problem, Comm. Math. Phys. 132 (1990), no. 1, 139-153. http://dx.doi. org/10.1007/BF02278004. MR1069205. · Zbl 0756.35096 · doi:10.1016/j.jde.2020.11.019 |
| [7] | W. CRAIG, An existence theory for water waves and the Boussinesq and Korteweg-de Vries scal-ing limits, Comm. Partial Differential Equations 10 (1985), no. 8, 787-1003. https:// dx.doi.org/10.1080/03605308508820396. MR795808. · Zbl 0577.76030 · doi:10.1017/9781316986769 |
| [8] | W.-P. DÜLL AND G. SCHNEIDER, Validity of Whitham’s equations for the modulation of periodic traveling waves in the NLS equation, J. Nonlinear Sci. 19 (2009), no. 5, 453-466. https://dx.doi.org/10.1007/s00332-009-9043-4. MR2540165. · Zbl 1179.35305 · doi:10.1007/s00332-009-9043-4.MR2540165 |
| [9] | A. DOELMAN, B. SANDSTEDE, A. SCHEEL, AND G. SCHNEIDER, The dynamics of modulated wave trains, Mem. Amer. Math. Soc. 199 (2009), no. 934, viii+105. https:// dx.doi.org/10.1090/memo/0934. MR2507940. · Zbl 1179.35005 · doi:10.1090/memo/0934.MR2507940 |
| [10] | T. HAAS, Amplitude equations for Boussinesq and Ginzburg-Landau-like models, PhD thesis (2019), University of Stuttgart. |
| [11] | T. HAAS AND G. SCHNEIDER, Failure of the N-wave interaction approximation without imposing periodic boundary conditions, ZAMM Z. Angew. Math. Mech. 100 (2020), no. 6, e201900230, 16 pp. https://dx.doi.org/10.1002/zamm.201900230. MR4135777. · Zbl 1179.35305 · doi:10.1007/s00332-009-9043-4 |
| [12] | T. HÄCKER, G. SCHNEIDER, AND D. ZIMMERMANN, Justification of the Ginzburg-Landau approximation in case of marginally stable long waves, J. Nonlinear Sci. 21 (2011), no. 1, 93-113. https://dx.doi.org/10.1007/s00332-010-9077-7. MR2775604. [JNR + 19] M. A. JOHNSON, P. NOBLE, L. M. RODRIGUES, Z. YANG, AND K. ZUMBRUN, Spectral stability of inviscid roll waves, Comm. Math. Phys. 367 (2019), no. 1, 265-316. https://dx.doi.org/10.1007/s00220-018-3277-7. MR3933411. · Zbl 07806611 · doi:10.1002/zamm.201900230 |
| [13] | M. A. JOHNSON, P. NOBLE, L. M. RODRIGUES, AND K. ZUMBRUN, Nonlo-calized modulation of periodic reaction diffusion waves: The Whitham equation, Arch. Ration. Mech. Anal. 207 (2013), no. 2, 669-692. https://dx.doi.org/10.1007/ s00205-012-0572-x. MR3005326. · Zbl 1270.35106 · doi:10.1007/s00332-010-9077-7 |
| [14] | , Behavior of periodic solutions of viscous conservation laws under localized and nonlo-calized perturbations, Invent. Math. 197 (2014), no. 1, 115-213. https://dx.doi.org/ 10.1007/s00222-013-0481-0. MR3219516. · doi:10.1007/s00220-018-3277-7 |
| [15] | , Spectral stability of periodic wave trains of the Korteweg-de Vries/Kuramoto-Sivashinsky equation in the Korteweg-de Vries limit, Trans. Amer. Math. Soc. 367 (2015), no. 3, 2159-2212. https://dx.doi.org/10.1090/S0002-9947-2014-06274-0. MR3286511. · doi:10.1007/s00205-012-0572-x |
| [16] | M. A. JOHNSON AND K. ZUMBRUN, Rigorous justification of the Whitham modulation equations for the generalized Korteweg-de Vries equation, Stud. Appl. Math. 125 (2010), no. 1, 69-89. https://dx.doi.org/10.1111/j.1467-9590.2010.00482.x. MR2676781. · doi:10.1007/s00222-013-0481-0 |
| [17] | M. A. JOHNSON, K. ZUMBRUN, AND J. C. BRONSKI, On the modulation equations and stability of periodic generalized Korteweg-de Vries waves via Bloch decompositions, Phys. D 239 (2010), no. 23-24, 2057-2065. https://dx.doi.org/10.1016/j.physd.2010. 07.012. MR2733113. · Zbl 1211.37087 · doi:10.1090/S0002-9947-2014-06274-0 |
| [18] | M. A. JOHNSON, K. ZUMBRUN, AND P. NOBLE, Nonlinear stability of viscous roll waves, SIAM J. Math. Anal. 43 (2011), no. 2, 577-611. https://dx.doi.org/10.1137/ 100785454. MR2784868. · Zbl 1240.35037 · doi:10.1111/j.1467-9590.2010.00482.x |
| [19] | L. A. KALYAKIN, Asymptotic decay of a one-dimensional wave packet in a nonlinear disper-sive medium, Mat. Sb. (N.S.) 132(174) (1987), no. 4, 470-495, 592 (Russian); English transl., Math. USSR-Sb. 60 (1988), no. 2, 457-483. https://dx.doi.org/10.1070/ SM1988v060n02ABEH003181. MR886641. · Zbl 0699.35135 · doi:10.1016/j.physd.2010.07.012 |
| [20] | T. KATO, Perturbation Theory for Linear Operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the 1980 edition. MR1335452. · doi:10.1137/100785454 |
| [21] | I. MELBOURNE AND G. SCHNEIDER, Phase dynamics in the complex Ginzburg-Landau equation, J. Differential Equations 199 (2004), no. 1, 22-46. https://dx.doi.org/10. 1016/j.jde.2003.11.004. MR2041510. · Zbl 1059.35139 · doi:10.1016/j.jde.2003.11.004.MR2041510 |
| [22] | , Phase dynamics in the real Ginzburg-Landau equation, Math. Nachr. 263/264 (2004), 171-180. https://dx.doi.org/10.1002/mana.200310129. MR2029750. · Zbl 1057.35064 · doi:10.1070/SM1988v060n02ABEH003181 |
| [23] | M. REED AND B. SIMON, Methods of Modern Mathematical Physics. II: Fourier Anal-ysis, Self-adjointness, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR0493420. · Zbl 0308.47002 |
| [24] | M. V. SAFONOV, The abstract Cauchy-Kovalevskaya theorem in a weighted Banach space, Comm. Pure Appl. Math. 48 (1995), no. 6, 629-637. https://dx.doi.org/10.1002/ cpa.3160480604. MR1338472. · Zbl 0836.35004 · doi:10.1002/cpa.3160480604.MR1338472 |
| [25] | G. SCHNEIDER, Validity and limitation of the Newell-Whitehead equation, Math. Nachr. 176 (1995), 249-263. https://dx.doi.org/10.1002/mana.19951760118. MR1361138. · Zbl 0844.35120 · doi:10.1002/mana.19951760118.MR1361138 |
| [26] | D. SERRE, Spectral stability of periodic solutions of viscous conservation laws: Large wave-length analysis, Comm. Partial Differential Equations 30 (2005), no. 1-3, 259-282. https://dx.doi.org/10.1081/PDE-200044492. MR2131054. · Zbl 1131.35046 · doi:10.1002/cpa.3160480604 |
| [27] | G. SCHNEIDER, D. A. SUNNY, AND D. ZIMMERMANN, The NLS approximation makes wrong predictions for the water wave problem in case of small surface tension and spatially periodic boundary conditions, J. Dynam. Differential Equations 27 (2015), no. 3-4, 1077-1099. https://dx.doi.org/10.1007/s10884-014-9350-9. MR3435146. · Zbl 1344.35138 · doi:10.1002/mana.19951760118 |
| [28] | G. SCHNEIDER AND H. UECKER, Nonlinear PDEs: A Dynamical Systems Approach, Graduate Studies in Mathematics, vol. 182, American Mathematical Society, Providence, RI, 2017. https://dx.doi.org/10.1090/gsm/182. MR3702025. · doi:10.1081/PDE-200044492 |
| [29] | M. TAKASHIMA, Surface tension driven instability in a horizontal liquid layer with a de-formable free surface. i: Stationary convection, Journal of the Physical Society of Japan 50 (1981), no. 8, 2745-2750. https://dx.doi.org/10.1143/JPSJ.50.2751. · Zbl 1344.35138 · doi:10.1007/s10884-014-9350-9 |
| [30] | A. VAN HARTEN, On the validity of the Ginzburg-Landau equation, J. Nonlinear Sci. 1 (1991), no. 4, 397-422. https://dx.doi.org/10.1007/BF02429847. MR1143976. · Zbl 0795.35112 · doi:10.1007/BF02429847.MR1143976 |
| [31] | G. B. WHITHAM, A general approach to linear and non-linear dispersive waves using a Lagrangian, J. Fluid Mech. 22 (1965), 273-283. https://dx.doi.org/10.1017/ S0022112065000745. MR182236. · Zbl 1467.35244 · doi:10.1090/gsm/182 |
| [32] | , Non-linear dispersive waves, Proc. Roy. Soc. London Ser. A 283 (1965), 238-261. https://dx.doi.org/10.1098/rspa.1965.0019. MR176724. · Zbl 0125.44202 · doi:10.1143/JPSJ.50.2751 |
| [33] | , Linear and Nonlinear Waves, Pure and Applied Mathematics; A Wiley-Interscience Seires of Texts, Monographs and Tracts, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. MR0483954. · Zbl 0795.35112 · doi:10.1007/BF02429847 |
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