A mean field type flow. I: compactness of solutions to a perturbed mean field type equation. (English) Zbl 1326.35031
Author’s abstract: We initiate the study of an evolution problem associated to a mean field type equation. In the present paper, we give a compactness result for solutions of a perturbed mean field type equation. Our method is based on blow-up analysis relying on integral estimates. This result will be used in our work [Pac. J. Math. 276, No. 2, 321-345 (2015; Zbl 1331.53097)] to prove the convergence of a flow associated to a mean field type equation.
Reviewer: Patrick Winkert (Berlin)
MSC:
| 35B33 | Critical exponents in context of PDEs |
| 35J20 | Variational methods for second-order elliptic equations |
| 53C44 | Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010) |
| 58E20 | Harmonic maps, etc. |
| 35B44 | Blow-up in context of PDEs |
Citations:
Zbl 1331.53097References:
| [1] | Aubin, T.: Some Nonlinear Problems in Riemannian Geometry. Springer Monographs in Mathematics. Springer, Berlin (1998) · Zbl 0896.53003 · doi:10.1007/978-3-662-13006-3 |
| [2] | Brezis, H., Merle, F.: Uniform estimates and blow-up behavior for solutions of \[-\Delta u=V(x)e^u\]-Δu=V(x)eu in two dimensions. Comm. Partial Differ. Equ. 16(8-9), 1223-1253 (1991) · Zbl 0746.35006 · doi:10.1080/03605309108820797 |
| [3] | Caffarelli, L.A., Yang, Y.S.: Vortex condensation in the Chern-Simons Higgs model: an existence theorem. Comm. Math. Phys. 168(2), 321-336 (1995) · Zbl 0846.58063 · doi:10.1007/BF02101552 |
| [4] | Caglioti, E., Lions, P.-L., Marchioro, C., Pulvirenti, M.: A special class of stationary flows for two-dimensional Euler equations: a statistical mechanics description. Comm. Math. Phys. 143(3), 501-525 (1992) · Zbl 0745.76001 · doi:10.1007/BF02099262 |
| [5] | Casteras, J.-B. A Mean Field Type Flow Part \[ii\] ii: Existence and Convergence. Preprint (2013) · Zbl 1282.35217 |
| [6] | Chen, C.-C., Lin, C.-S.: Topological degree for a mean field equation on Riemann surfaces. Comm. Pure Appl. Math. 56(12), 1667-1727 (2003) · Zbl 1032.58010 · doi:10.1002/cpa.10107 |
| [7] | Chen, W.X., Li, C.: Classification of solutions of some nonlinear elliptic equations. Duke Math. J. 63(3), 615-622 (1991) · Zbl 0768.35025 · doi:10.1215/S0012-7094-91-06325-8 |
| [8] | Ding, W., Jost, J., Li, J., Wang, G.: Existence results for mean field equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 16(5), 653-666 (1999) · Zbl 0937.35055 |
| [9] | Djadli, Z.: Existence result for the mean field problem on Riemann surfaces of all genuses. Commun. Contemp. Math. 10(2), 205-220 (2008) · Zbl 1151.53035 · doi:10.1142/S0219199708002776 |
| [10] | Druet, O., Robert, F.: Bubbling phenomena for fourth-order four-dimensional PDEs with exponential growth. Proc. Am. Math. Soc. 134(3), 897-908 (2006) (electronic) · Zbl 1083.58018 |
| [11] | Han, J.: Asymptotic limit for condensate solutions in the abelian Chern-Simons Higgs model. Proc. Am. Math. Soc. 131(6), 1839-1845 (2003) (electronic) · Zbl 1036.35034 |
| [12] | Li, Y.Y.: Harnack type inequality: the method of moving planes. Comm. Math. Phys. 200(2), 421-444 (1999) · Zbl 0928.35057 · doi:10.1007/s002200050536 |
| [13] | Li, Y.Y., Shafrir, I.: Blow-up analysis for solutions of \[-\Delta u=Ve^u\]-Δu=Veu in dimension two. Indiana Univ. Math. J. 43(4), 1255-1270 (1994) · Zbl 0842.35011 · doi:10.1512/iumj.1994.43.43054 |
| [14] | Malchiodi, A.: Compactness of solutions to some geometric fourth-order equations. J. Reine Angew. Math. 594, 137-174 (2006) · Zbl 1098.53032 |
| [15] | Malchiodi, A.: Morse theory and a scalar field equation on compact surfaces. Adv. Differ. Equ. 13(11-12), 1109-1129 (2008) · Zbl 1175.53052 |
| [16] | Struwe, M., Tarantello, G. On multivortex solutions in Chern-Simons gauge theory. Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 1(1), 109-121 (1998) · Zbl 0912.58046 |
| [17] | Tarantello, G.: Multiple condensate solutions for the Chern-Simons-Higgs theory. J. Math. Phys. 37(8), 3769-3796 (1996) · Zbl 0863.58081 · doi:10.1063/1.531601 |
| [18] | Yang, Y.: Solitons in Field Theory and Nonlinear Analysis. Springer Monographs in Mathematics. Springer, New York (2001) · Zbl 0982.35003 · doi:10.1007/978-1-4757-6548-9 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
