Abstract
In this paper, we consider the cauchy problem for the 2D Groma–Balogh model (Acta Mater 47:3647–3654, 1999). From the works Cannone et al. (Arch Ration Mech Anal 196:71–96, 2010) and El Hajj (Ann Inst Henri Poincaré Anal Nonlinéaire 27:21–35, 2010), one can see global well-posedness for this model is an open question. However, we can prove longtime well-posedness. In particular, we show that this model admits a unique solution with the lifespan \(T^\star \) satisfying \(T^\star \log ^2(1+T^\star )\gtrsim \epsilon ^{-2}\) if the initial data is of size \(\epsilon \). To achieve this, we first establish some new decay estimates concerning the operator \(e^{-{\mathcal {R}}_{12}^2t}\). Then, we prove the longtime well-posedness by utilizing the weak dissipation to deal with the nonlinear terms.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alvarez, O., Hoch, P., Le Bouar, Y., Monneau, R.: Dislocation dynamics: short-time existence and uniqueness of the solution. Arch. Ration. Mech. Anal. 181, 449–504 (2006)
Barles, G., Cardaliaguet, P., Ley, O., Monneau, R.: General existence results and uniqueness for dislocation equations. SIAM J. Math. Anal. 40, 44–69 (2008)
Cannone, M., El Hajj, A., Monneau, R., Ribaud, F.: Global existence for a system of non-linear and non-local transport equations describing the dynamics of dislocation densities. Arch. Ration. Mech. Anal. 196, 71–96 (2010)
El Hajj, A.: Short time existence and uniqueness in Hölder spaces for the 2D dynamics of dislocation densities. Ann. Inst. Henri. Poincaré Anal. Non Linéaire 27, 21–35 (2010)
Grafakos, L.: Modern Fourier Analysis. 2nd Edn, Grad. Text in Math., 250. Springer, New York (2008)
Groma, I.: Link between the microscopic and mesoscopic length-scale description of the collective behavior of dislocations. Phys. Rev. B 56, 5807 (1997)
Groma, I., Balogh, P.: Investigation of dislocation pattern formation in two-dimensional self-consistent field approximation. Acta Mater. 47, 3647–3654 (1999)
Hirth, J., Lothe, J.: Theory of Dislocations, vol. 1982, 2nd edn, pp. 2233–2247. Wiley, New York (2008)
Kato, T., Ponce, G.: Commutator estimates and the Euler and Navier–Stokes equations. Comm. Pure Appl. Math. 41, 891–907 (1988)
Kenig, C., Ponce, G., Vega, L.: Well-posedness of the initial value problem for the Korteweg-de Vries equation. J. Am. Math. Soc. 4, 323–347 (1991)
Kiselev, A., Nazarov, F., Volberg, A.: Global well-posedness for the critical 2D dissipative quasi-geostrophic equation. Invent. Math. 167, 445–453 (2007)
Li, D., Miao, C., Xue, L.: On the well-posedness of a 2D nonlinear and nonlocal system arising from the dislocation dynamics. Commun. Contemp. Math. 16, 577–596 (2014)
Majda, A.J., Bertozzi, A.L.: Vorticity and Incompressible Flow. Cambridge University Press, Cambridge (2001)
Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton (1970)
Acknowledgments
Wan is partially supported by NSF of China under Grants 11571306. Chen is partially supported by NSF of China under Grants 11271330.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Paul Newton.
Rights and permissions
About this article
Cite this article
Wan, R., Chen, J. Longtime Well-posedness for the 2D Groma–Balogh Model. J Nonlinear Sci 26, 1817–1831 (2016). https://doi.org/10.1007/s00332-016-9320-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00332-016-9320-y