Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
low regularity; global well-posedness; Klein-Gordon-Schrödinger equation; higher order coupling
low regularity; global well-posedness; Klein-Gordon-Schrödinger equation; higher order coupling
Summary:
Global well-posedness for the Klein-Gordon-Schrödinger system with generalized higher order coupling, which is a system of PDEs in two variables arising from quantum physics, is proven. It is shown that the system is globally well-posed in $(u,n)\in L^2\times L^2$ under some conditions on the nonlinearity (the coupling term), by using the $L^2$ conservation law for $u$ and controlling the growth of $n$ via the estimates in the local theory. In particular, this extends the well-posedness results for such a system in Miao, Xu (2007) for some exponents to other dimensions and in lower regularity spaces.
References:
[1] Bachelot, A.: Problème de Cauchy pour des systèmes hyperboliques semi-linéaires. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 1 (1984), 453-478 French. DOI 10.1016/S0294-1449(16)30414-0 | MR 0778979 | Zbl 0566.35068
[2] Bekiranov, D., Ogawa, T., Ponce, G.: Interaction equations for short and long dispersive waves. J. Funct. Anal. 158 (1998), 357-388. DOI 10.1006/jfan.1998.3257 | MR 1648479 | Zbl 0909.35123
[3] Biler, P.: Attractors for the system of Schrödinger and Klein-Gordon equations with Yukawa coupling. SIAM J. Math. Anal. 21 (1990), 1190-1212. DOI 10.1137/0521065 | MR 1062399 | Zbl 0725.35020
[4] Cazenave, T.: Semilinear Schrödinger Equations. Courant Lecture Notes in Mathematics 10. AMS, Providence (2003). DOI 10.1090/cln/010 | MR 2002047 | Zbl 1055.35003
[5] Colliander, J.: Wellposedness for Zakharov systems with generalized nonlinearity. J. Differ. Equations 148 (1998), 351-363. DOI 10.1006/jdeq.1998.3445 | MR 1643187 | Zbl 0921.35162
[6] Colliander, J., Holmer, J., Tzirakis, N.: Low regularity global well-posedness for the Zakharov and Klein-Gordon-Schrödinger systems. Trans. Am. Math. Soc. 360 (2008), 4619-4638. DOI 10.1090/S0002-9947-08-04295-5 | MR 2403699 | Zbl 1158.35085
[7] Fukuda, I., Tsutsumi, M.: On the Yukawa-coupled Klein-Gordon-Schrödinger equations in three space dimensions. Proc. Japan Acad. 51 (1975), 402-405. DOI 10.3792/pja/1195518563 | MR 0380160 | Zbl 0313.35065
[8] Fukuda, I., Tsutsumi, M.: On coupled Klein-Gordon-Schrödinger equations. III: Higher order interaction, decay and blow-up. Math. Jap. 24 (1979), 307-321. MR 0550215 | Zbl 0415.35071
[9] Ginibre, J., Tsutsumi, Y., Velo, G.: On the Cauchy problem for the Zakharov system. J. Funct. Anal. 151 (1997), 384-436. DOI 10.1006/jfan.1997.3148 | MR 1491547 | Zbl 0894.35108
[10] Miao, C., Xu, G.: Low regularity global well-posedness for the Klein-Gordon-Schrödinger system with the higher-order Yukawa coupling. Differ. Integral Equ. 20 (2007), 643-656. MR 2319459 | Zbl 1212.35454
[11] Pecher, H.: Some new well-posedness results for the Klein-Gordon-Schrödinger system. Differ. Integral Equ. 25 (2012), 117-142. MR 2906550 | Zbl 1249.35309
[12] Tao, T.: Nonlinear Dispersive Equations: Local and Global Analysis. CBMS Regional Conference Series in Mathematics 106. AMS, Providence (2006). DOI 10.1090/cbms/106 | MR 2233925 | Zbl 1106.35001
[13] Tzirakis, N.: The Cauchy problem for the Klein-Gordon-Schrödinger system in low dimensions below the energy space. Commun. Partial Differ. Equations 30 (2005), 605-641. DOI 10.1081/PDE-200059260 | MR 2153510 | Zbl 1075.35082
Search
Partner of
