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January  2020, 19(1): 425-453. doi: 10.3934/cpaa.2020022

On the Schrödinger-Debye system in compact Riemannian manifolds

Department of Mathematics, State University of Campinas, Campinas-SP, 13083-859, Brazil

* Corresponding author

Received  December 2018 Revised  May 2019 Published  July 2019

Fund Project: The first author is supported by CAPES and CNPq, Brazil. The second author is partially supported by FAPESP (2016/25864-6) Brazil and CNPq (308131/2017-7) Brazil.

We consider the initial value problem (IVP) associated with the Schrödinger-Debye system posed on a $d$-dimensional compact Riemannian manifold $M $ and prove the local well-posedness result for given data $ (u_0, v_0)\in H^s(M)\times (H^s(M)\cap L^{\infty}(M))$ whenever $s>\frac{d}2-\frac12 $, $d\geq 2 $. For $d=2 $, we apply a sharp version of the Gagliardo-Nirenberg inequality in compact manifold to derive an a priori estimate for the $H^1 $-solution and use it to prove the global well-posedness result in this space.

Citation: Marcelo Nogueira, Mahendra Panthee. On the Schrödinger-Debye system in compact Riemannian manifolds. Communications on Pure and Applied Analysis, 2020, 19 (1) : 425-453. doi: 10.3934/cpaa.2020022
References:
[1]

N. Anantharaman and G. Revière, Dispersion and controllability for the Schrödinger equation on negative curved manifolds, Analysis and PDE, 5 (2012), 313–337. doi: 10.2140/apde.2012.5.313.

[2]

R. Anton, Cubic nonlinear Schrödinger equation on three dimensional balls with radial data, Comm. in PDE, 33 (2008), 1862–1889. doi: 10.1080/03605300802402591.

[3]

R. Anton, Strichartz inequalities for lipschitz metrics on manifolds and nonlinear Schrödinger equation on domains, Bull. Soc. math. France, 136 (2008), 27–65. doi: 10.24033/bsmf.2548.

[4]

A. Arbieto and C. Matheus, On the periodic Schrödinger-Debye equation, Comm. Pure and Applied Anal., 7 (2008), 699–713. doi: 10.3934/cpaa.2008.7.699.

[5]

S. Alinhac and P. Gérard, Pseudo-differential operators and the Nash-Moser theorem, Graduate Studies in Mathematics, 82 (2007). doi: 10.1090/gsm/082.

[6]

B. Bidégaray, On the Cauchy problem for systems occurring in nonlinear optics, Adv. Diff. Equat., 3 (1998), 473–496.

[7]

B. Bidégaray, The Cauchy problem for Schrödinger-Debye equations, Math. Models Methods Appl. Sci., 10 (2000), 307–315. doi: 10.1142/S0218202500000185.

[8]

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and application to nonlinear evolution equations I. Schrödinger equations, Geom. and Funct. Anal., 3 (1993), 107–156. doi: 10.1007/BF01896020.

[9]

J. Bourgain, Exponential sums and nonlinear Schrödinger equations, Geom. and Funct. Anal., 3 (1993), 157–178. doi: 10.1007/BF01896021.

[10]

M. D. Blair, H. F. Smith and C. D. Sogge, On Strichartz estimates for Schrödinger operators in compact manifolds with boundary, Proc. of the Amer. Math. Soc., 136 (2008), 247–256. doi: 10.1090/S0002-9939-07-09114-9.

[11]

M. D. Blair, H. F. Smith and C. D. Sogge, Strichartz estimates and the nonlinear Schrödinger equation on manifolds with boundary, Math. Ann., 354 (2012), 1397–1430. doi: 10.1007/s00208-011-0772-y.

[12]

N. Burq, P. Gérard and N. Tzvetkov, Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds, Amer. J. Math., 126 (2004), 569–605.

[13]

X. Carvajal and P. Gamboa, Global well-posedness for the critical Schrödinger-Debye system, Dynamics of PDE., 11 (2014), 251–268. doi: 10.4310/DPDE.2014.v11.n3.a3.

[14]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, New York University, 2003. doi: 10.1090/cln/010.

[15]

J. Ceccon and M. Montenegro, Optimal $L^{p}$- Riemannian Gagliardo-Nirenberg inequalities, Mathematische Zeitschrift, 258 (2008), 851–873. doi: 10.1007/s00209-007-0202-8.

[16]

A. J. Corcho, F. Oliveira and J. D. Silva, Local and global well-posedness for the critical Schrödinger-Debye system, Proc. of the Amer. Math. Soc., 144 (2013), 3485–3499. doi: 10.1090/S0002-9939-2013-11612-6.

[17]

A. J. Corcho and F. Linares, Well-posedness for the Schrödinger-Debye equation, Contemporary Mathematics, 362 (2004), 113–131. doi: 10.1090/conm/362/06608.

[18]

P. Gérard, Nonlinear Schrödinger equations in compact manifolds, European Congress of Mathematics, Stockholm 2004, Editor Ary Laptov, 121–139.

[19]

J. Ginibre, Y. Tsutsumi and G. Velo, On the Cauchy problem for the Zakharov system, J. Funct. Anal., 151 (1997), 384–436. doi: 10.1006/jfan.1997.3148.

[20]

J. Ginibre and G. Velo, The global Cauchy problem for the nonlinear Schrödinger equation, Ann. Inst. H. Poincaré-AN, 3 (1985), 309–327.

[21]

J. Ginibre, Le problème de Cauchy pour des EDP semi-linéaires périodiques en variables d'espace (d'après Bourgain), Séminaire Bourbaki, Exp. 796, Astérisque, 237 (1996), 163–187.

[22]

M. Grillakis, On nonlinear Schrödinger equations, Comm. Partial Differential Equations, 25 (2000), 1827–1844. doi: 10.1080/03605300008821569.

[23]

A. Hassell, T. Tao and J. Wunsch, Sharp Strichartz estimates on nontrapping asymptotically conic manifolds, Amer. Journal of Math., 128 (2006), 963–1024.

[24]

Z. Hani, A bilinear oscillatory integral estimate and bilinear refinements to Strichartz estimate on closed manifolds, Analysis and PDE, 5 (2012), 339–363. doi: 10.2140/apde.2012.5.339.

[25]

O. Ivanovici, On the Schrödinger equation outside strictly convex obstacles, Analysis and PDE., 3 (2010), 261–292. doi: 10.2140/apde.2010.3.261.

[26]

J. C. Jiang, Bilinear Strichartz estimates for Schrödinger operators in two-dimensional compact manifolds with boundary and cubic NLS, Diff. and Integral Equations, 24 (2012), 83–108.

[27]

T. Kato, On nonlinear Schrödinger equations, Ann. Inst. H. Poincaré, Physique théorique, 46 (1987), 113–129.

[28]

M. Kell and T. Tao, End point Strichartz estimates, Amer. J. Math., 120 (1998), 955-980.

[29]

F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equations, Second edition, Universitext, Springer, New York, 2015. doi: 10.1007/978-1-4939-2181-2.

[30]

H. Mizutani and N. Tzvetkov, Strichartz estimates for non-elliptic Schrödinger equations on compact manifolds, Communications in Partial Differential Equations, 40 (2015), 1182–1195. doi: 10.1080/03605302.2015.1010211.

[31]

A. C. Newell and J. V. Moloney, Nonlinear Optics, Addison-Wesley, 1992.

[32]

I. Pesenson, An approach to spectral problems on Riemannian manifolds, Pacific Journal of Mathematics, 215 (2004), 183–199. doi: 10.2140/pjm.2004.215.183.

[33]

I. Rodnianski and T. Tao, Longtime decay estimates for the Schrödinger equation on manifolds, Mathematical Aspects of Nonlinear Dispersive Equations, J. Bourgain, Carlos E. Kenig and S. Klainerman eds., Annals of Mathematics Studies, 163 (2007), 223–253.

show all references

References:
[1]

N. Anantharaman and G. Revière, Dispersion and controllability for the Schrödinger equation on negative curved manifolds, Analysis and PDE, 5 (2012), 313–337. doi: 10.2140/apde.2012.5.313.

[2]

R. Anton, Cubic nonlinear Schrödinger equation on three dimensional balls with radial data, Comm. in PDE, 33 (2008), 1862–1889. doi: 10.1080/03605300802402591.

[3]

R. Anton, Strichartz inequalities for lipschitz metrics on manifolds and nonlinear Schrödinger equation on domains, Bull. Soc. math. France, 136 (2008), 27–65. doi: 10.24033/bsmf.2548.

[4]

A. Arbieto and C. Matheus, On the periodic Schrödinger-Debye equation, Comm. Pure and Applied Anal., 7 (2008), 699–713. doi: 10.3934/cpaa.2008.7.699.

[5]

S. Alinhac and P. Gérard, Pseudo-differential operators and the Nash-Moser theorem, Graduate Studies in Mathematics, 82 (2007). doi: 10.1090/gsm/082.

[6]

B. Bidégaray, On the Cauchy problem for systems occurring in nonlinear optics, Adv. Diff. Equat., 3 (1998), 473–496.

[7]

B. Bidégaray, The Cauchy problem for Schrödinger-Debye equations, Math. Models Methods Appl. Sci., 10 (2000), 307–315. doi: 10.1142/S0218202500000185.

[8]

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and application to nonlinear evolution equations I. Schrödinger equations, Geom. and Funct. Anal., 3 (1993), 107–156. doi: 10.1007/BF01896020.

[9]

J. Bourgain, Exponential sums and nonlinear Schrödinger equations, Geom. and Funct. Anal., 3 (1993), 157–178. doi: 10.1007/BF01896021.

[10]

M. D. Blair, H. F. Smith and C. D. Sogge, On Strichartz estimates for Schrödinger operators in compact manifolds with boundary, Proc. of the Amer. Math. Soc., 136 (2008), 247–256. doi: 10.1090/S0002-9939-07-09114-9.

[11]

M. D. Blair, H. F. Smith and C. D. Sogge, Strichartz estimates and the nonlinear Schrödinger equation on manifolds with boundary, Math. Ann., 354 (2012), 1397–1430. doi: 10.1007/s00208-011-0772-y.

[12]

N. Burq, P. Gérard and N. Tzvetkov, Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds, Amer. J. Math., 126 (2004), 569–605.

[13]

X. Carvajal and P. Gamboa, Global well-posedness for the critical Schrödinger-Debye system, Dynamics of PDE., 11 (2014), 251–268. doi: 10.4310/DPDE.2014.v11.n3.a3.

[14]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, New York University, 2003. doi: 10.1090/cln/010.

[15]

J. Ceccon and M. Montenegro, Optimal $L^{p}$- Riemannian Gagliardo-Nirenberg inequalities, Mathematische Zeitschrift, 258 (2008), 851–873. doi: 10.1007/s00209-007-0202-8.

[16]

A. J. Corcho, F. Oliveira and J. D. Silva, Local and global well-posedness for the critical Schrödinger-Debye system, Proc. of the Amer. Math. Soc., 144 (2013), 3485–3499. doi: 10.1090/S0002-9939-2013-11612-6.

[17]

A. J. Corcho and F. Linares, Well-posedness for the Schrödinger-Debye equation, Contemporary Mathematics, 362 (2004), 113–131. doi: 10.1090/conm/362/06608.

[18]

P. Gérard, Nonlinear Schrödinger equations in compact manifolds, European Congress of Mathematics, Stockholm 2004, Editor Ary Laptov, 121–139.

[19]

J. Ginibre, Y. Tsutsumi and G. Velo, On the Cauchy problem for the Zakharov system, J. Funct. Anal., 151 (1997), 384–436. doi: 10.1006/jfan.1997.3148.

[20]

J. Ginibre and G. Velo, The global Cauchy problem for the nonlinear Schrödinger equation, Ann. Inst. H. Poincaré-AN, 3 (1985), 309–327.

[21]

J. Ginibre, Le problème de Cauchy pour des EDP semi-linéaires périodiques en variables d'espace (d'après Bourgain), Séminaire Bourbaki, Exp. 796, Astérisque, 237 (1996), 163–187.

[22]

M. Grillakis, On nonlinear Schrödinger equations, Comm. Partial Differential Equations, 25 (2000), 1827–1844. doi: 10.1080/03605300008821569.

[23]

A. Hassell, T. Tao and J. Wunsch, Sharp Strichartz estimates on nontrapping asymptotically conic manifolds, Amer. Journal of Math., 128 (2006), 963–1024.

[24]

Z. Hani, A bilinear oscillatory integral estimate and bilinear refinements to Strichartz estimate on closed manifolds, Analysis and PDE, 5 (2012), 339–363. doi: 10.2140/apde.2012.5.339.

[25]

O. Ivanovici, On the Schrödinger equation outside strictly convex obstacles, Analysis and PDE., 3 (2010), 261–292. doi: 10.2140/apde.2010.3.261.

[26]

J. C. Jiang, Bilinear Strichartz estimates for Schrödinger operators in two-dimensional compact manifolds with boundary and cubic NLS, Diff. and Integral Equations, 24 (2012), 83–108.

[27]

T. Kato, On nonlinear Schrödinger equations, Ann. Inst. H. Poincaré, Physique théorique, 46 (1987), 113–129.

[28]

M. Kell and T. Tao, End point Strichartz estimates, Amer. J. Math., 120 (1998), 955-980.

[29]

F. Linares and G. Ponce, Introduction to Nonlinear Dispersive Equations, Second edition, Universitext, Springer, New York, 2015. doi: 10.1007/978-1-4939-2181-2.

[30]

H. Mizutani and N. Tzvetkov, Strichartz estimates for non-elliptic Schrödinger equations on compact manifolds, Communications in Partial Differential Equations, 40 (2015), 1182–1195. doi: 10.1080/03605302.2015.1010211.

[31]

A. C. Newell and J. V. Moloney, Nonlinear Optics, Addison-Wesley, 1992.

[32]

I. Pesenson, An approach to spectral problems on Riemannian manifolds, Pacific Journal of Mathematics, 215 (2004), 183–199. doi: 10.2140/pjm.2004.215.183.

[33]

I. Rodnianski and T. Tao, Longtime decay estimates for the Schrödinger equation on manifolds, Mathematical Aspects of Nonlinear Dispersive Equations, J. Bourgain, Carlos E. Kenig and S. Klainerman eds., Annals of Mathematics Studies, 163 (2007), 223–253.

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