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December  2020, 10(4): 699-714. doi: 10.3934/mcrf.2020016

The Kato smoothing effect for the nonlinear regularized Schrödinger equation on compact manifolds

Department of Mathematics, Faculty of Sciences of Tunis, University of Tunis El Manar, 2092 El Manar II, Tunisia, LR LAMMDA, (LR16ES13), ESSTHS, Tunisia

* Corresponding author: Imen El Khal El Taief

Received  June 2019 Revised  October 2019 Published  December 2019

We establish Strichartz estimates for the regularized Schrödinger equation on a two dimensional compact Riemannian manifold without boundary. As a consequence we deduce global existence and uniqueness results for the Cauchy problem for the nonlinear regularized Schrödinger equation and we prove under the geometric control condition the Kato smoothing effect for solutions of this equation in this particular geometries.

Citation: Lassaad Aloui, Imen El Khal El Taief. The Kato smoothing effect for the nonlinear regularized Schrödinger equation on compact manifolds. Mathematical Control & Related Fields, 2020, 10 (4) : 699-714. doi: 10.3934/mcrf.2020016
References:
[1]

L. Aloui, Smoothing effect for regularized Schrödinger equation on compact manifolds, Collect. Math., 59 (2008), 53-62.  doi: 10.1007/BF03191181.  Google Scholar

[2]

L. Aloui, Smoothing effect for regularized Schrödinger equation on bounded domains, Asymptotic Analysis, 59 (2008), 179-193.  doi: 10.3233/ASY-2008-0892.  Google Scholar

[3]

L. Aloui, M. Khenissi and L. Robbiano, The Kato smoothing effect for regularized Schrödinger equations in exterior domains, Ann. Inst. H. Poincaré Anal. Non Linéaire, 34 (2017), 1759–1792, arXiv: 1204.1904v1. doi: 10.1016/j.anihpc.2016.12.006.  Google Scholar

[4]

C. BardosG. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary, SIAM J. Control Optim., 30 (1992), 1024-1065.  doi: 10.1137/0330055.  Google Scholar

[5]

M. Ben-Artzi and A. Devinatz, Regularity and decay of solutions to the Stark evolution equation, J. Funct. Anal., 154 (1998), 501-512.  doi: 10.1006/jfan.1997.3211.  Google Scholar

[6]

M. Ben-Artzi and S. Klainerman, Decay and regularity for the Schrödinger equation, J. Anal. Math., 58 (1992), 25-37.  doi: 10.1007/BF02790356.  Google Scholar

[7]

N. Burq, Smoothing effect for Schrödinger boundary value problems, Duke Math. J., 123 (2004), 403-427.  doi: 10.1215/S0012-7094-04-12326-7.  Google Scholar

[8]

N. BurqP. Gérard and N. Tzvetkov, On nonlinear Schrödinger equations in exterior domain, Ann. Inst. H. Poincaré Anal. Non. Linéaire, 21 (2004), 295-318.  doi: 10.1016/j.anihpc.2003.03.002.  Google Scholar

[9]

N. BurqP. Gérard and N. Tzvetkov, Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds, Amer. J. Math., 126 (2004), 569-605.  doi: 10.1353/ajm.2004.0016.  Google Scholar

[10]

P. Constantin and J.-C. Saut, Local smoothing properties of dispersive equations, J. Amer. Math. Soc., 1 (1988), 413-439.  doi: 10.1090/S0894-0347-1988-0928265-0.  Google Scholar

[11]

P. Constantin and J.-C. Saut, Local smoothing properties of Schrödinger equations, Indiana Univ. Math. J., 38 (1989), 791-810.  doi: 10.1512/iumj.1989.38.38037.  Google Scholar

[12]

E. B. Davies, The functional calculus, J. London Math. Soc. (2), 52 (1995), 166-176.  doi: 10.1112/jlms/52.1.166.  Google Scholar

[13]

B. DehmanP. Gérard and G. Lebeau, Stabilization and control for the nonlinear Schrödinger equation on a compact surface, Math. Z., 254 (2006), 729-749.  doi: 10.1007/s00209-006-0005-3.  Google Scholar

[14] M. Dimassi and J. Sjöstrand, Spectral Asymptotics in the Semi-Classical Limit, London Mathematical Society Lecture Note Series, 268. Cambridge University Press, Cambridge, 1999.  doi: 10.1017/CBO9780511662195.  Google Scholar
[15]

S. Doï, Remarks on the Cauchy problem for Schrödinger type equations, Communications in Partial Differential Equations, 21 (1996), 163-178.  doi: 10.1080/03605309608821178.  Google Scholar

[16]

S. Doï, Smoothing effects for Schrödinger evolution equation and global behaviour of geodesic flow, Math. Ann., 318 (2000), 355-389.  doi: 10.1007/s002080000128.  Google Scholar

[17]

S. Doï, Smoothing effects of Schrödinger evolution groups on Riemannian manifolds, Duke Math. J., 82 (1996), 679-706.  doi: 10.1215/S0012-7094-96-08228-9.  Google Scholar

[18]

O. Ivanovici and F. Planchon, Square function and heat flow estimates on domains, Comm. Partial Differential Equations, 42 (2017), 1447–1466, arXiv: 0812.2733v2. doi: 10.1080/03605302.2017.1365267.  Google Scholar

[19]

R. B. Melrose, Singularities and energy decay in acoustical scattering, Duke Math. J., 46 (1979), 43-59.  doi: 10.1215/S0012-7094-79-04604-0.  Google Scholar

[20]

F. Nier, A variational formulation of Schrödinger-Poisson systems in dimension $d\leq3$, Comm. Partial Differential Equations, 18 (1993), 1125-1147.  doi: 10.1080/03605309308820966.  Google Scholar

[21]

J. V. Ralston, Solutions of the wave equation with localized energy, Comm. Pure Appl. Math., 22 (1969), 807-823.  doi: 10.1002/cpa.3160220605.  Google Scholar

[22]

J. Rauch and M. Taylor, Exponential decay of solutions of hyperbolic equations in bounded domains, Indiana Univ. Math. J., 24 (1974), 79-86.  doi: 10.1512/iumj.1975.24.24004.  Google Scholar

[23]

L. Robbiano and C. Zuily, icrolocal analytic smoothing effect for the Schrödinger equation, Duke Mathematical Journal, 100 (1999), 93-129.  doi: 10.1215/S0012-7094-99-10003-2.  Google Scholar

[24]

P. Sjölin, Regularity of solutions to the Schrödinger equation, Duke Math. J., 55 (1987), 699-715.  doi: 10.1215/S0012-7094-87-05535-9.  Google Scholar

[25]

J. Szeftel, Microlocal dispersive smoothing for the nonlinear Schrödinger equation, SIAM Journal on Mathematical Analysis, 37 (2005), 549-597.  doi: 10.1137/S0036141003432109.  Google Scholar

[26]

L. Vega, Schrödinger equations: Pointwise convergence to the initial data, Proc. Amer. Math. Soc., 102 (1988), 874-878.  doi: 10.2307/2047326.  Google Scholar

[27]

X. P. Wang, Time-decay of scattering solutions and classical trajectories, Ann. I.H.P. Phys. Théor., 47 (1987), 25-37.  doi: 10.1119/1.14979.  Google Scholar

show all references

References:
[1]

L. Aloui, Smoothing effect for regularized Schrödinger equation on compact manifolds, Collect. Math., 59 (2008), 53-62.  doi: 10.1007/BF03191181.  Google Scholar

[2]

L. Aloui, Smoothing effect for regularized Schrödinger equation on bounded domains, Asymptotic Analysis, 59 (2008), 179-193.  doi: 10.3233/ASY-2008-0892.  Google Scholar

[3]

L. Aloui, M. Khenissi and L. Robbiano, The Kato smoothing effect for regularized Schrödinger equations in exterior domains, Ann. Inst. H. Poincaré Anal. Non Linéaire, 34 (2017), 1759–1792, arXiv: 1204.1904v1. doi: 10.1016/j.anihpc.2016.12.006.  Google Scholar

[4]

C. BardosG. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary, SIAM J. Control Optim., 30 (1992), 1024-1065.  doi: 10.1137/0330055.  Google Scholar

[5]

M. Ben-Artzi and A. Devinatz, Regularity and decay of solutions to the Stark evolution equation, J. Funct. Anal., 154 (1998), 501-512.  doi: 10.1006/jfan.1997.3211.  Google Scholar

[6]

M. Ben-Artzi and S. Klainerman, Decay and regularity for the Schrödinger equation, J. Anal. Math., 58 (1992), 25-37.  doi: 10.1007/BF02790356.  Google Scholar

[7]

N. Burq, Smoothing effect for Schrödinger boundary value problems, Duke Math. J., 123 (2004), 403-427.  doi: 10.1215/S0012-7094-04-12326-7.  Google Scholar

[8]

N. BurqP. Gérard and N. Tzvetkov, On nonlinear Schrödinger equations in exterior domain, Ann. Inst. H. Poincaré Anal. Non. Linéaire, 21 (2004), 295-318.  doi: 10.1016/j.anihpc.2003.03.002.  Google Scholar

[9]

N. BurqP. Gérard and N. Tzvetkov, Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds, Amer. J. Math., 126 (2004), 569-605.  doi: 10.1353/ajm.2004.0016.  Google Scholar

[10]

P. Constantin and J.-C. Saut, Local smoothing properties of dispersive equations, J. Amer. Math. Soc., 1 (1988), 413-439.  doi: 10.1090/S0894-0347-1988-0928265-0.  Google Scholar

[11]

P. Constantin and J.-C. Saut, Local smoothing properties of Schrödinger equations, Indiana Univ. Math. J., 38 (1989), 791-810.  doi: 10.1512/iumj.1989.38.38037.  Google Scholar

[12]

E. B. Davies, The functional calculus, J. London Math. Soc. (2), 52 (1995), 166-176.  doi: 10.1112/jlms/52.1.166.  Google Scholar

[13]

B. DehmanP. Gérard and G. Lebeau, Stabilization and control for the nonlinear Schrödinger equation on a compact surface, Math. Z., 254 (2006), 729-749.  doi: 10.1007/s00209-006-0005-3.  Google Scholar

[14] M. Dimassi and J. Sjöstrand, Spectral Asymptotics in the Semi-Classical Limit, London Mathematical Society Lecture Note Series, 268. Cambridge University Press, Cambridge, 1999.  doi: 10.1017/CBO9780511662195.  Google Scholar
[15]

S. Doï, Remarks on the Cauchy problem for Schrödinger type equations, Communications in Partial Differential Equations, 21 (1996), 163-178.  doi: 10.1080/03605309608821178.  Google Scholar

[16]

S. Doï, Smoothing effects for Schrödinger evolution equation and global behaviour of geodesic flow, Math. Ann., 318 (2000), 355-389.  doi: 10.1007/s002080000128.  Google Scholar

[17]

S. Doï, Smoothing effects of Schrödinger evolution groups on Riemannian manifolds, Duke Math. J., 82 (1996), 679-706.  doi: 10.1215/S0012-7094-96-08228-9.  Google Scholar

[18]

O. Ivanovici and F. Planchon, Square function and heat flow estimates on domains, Comm. Partial Differential Equations, 42 (2017), 1447–1466, arXiv: 0812.2733v2. doi: 10.1080/03605302.2017.1365267.  Google Scholar

[19]

R. B. Melrose, Singularities and energy decay in acoustical scattering, Duke Math. J., 46 (1979), 43-59.  doi: 10.1215/S0012-7094-79-04604-0.  Google Scholar

[20]

F. Nier, A variational formulation of Schrödinger-Poisson systems in dimension $d\leq3$, Comm. Partial Differential Equations, 18 (1993), 1125-1147.  doi: 10.1080/03605309308820966.  Google Scholar

[21]

J. V. Ralston, Solutions of the wave equation with localized energy, Comm. Pure Appl. Math., 22 (1969), 807-823.  doi: 10.1002/cpa.3160220605.  Google Scholar

[22]

J. Rauch and M. Taylor, Exponential decay of solutions of hyperbolic equations in bounded domains, Indiana Univ. Math. J., 24 (1974), 79-86.  doi: 10.1512/iumj.1975.24.24004.  Google Scholar

[23]

L. Robbiano and C. Zuily, icrolocal analytic smoothing effect for the Schrödinger equation, Duke Mathematical Journal, 100 (1999), 93-129.  doi: 10.1215/S0012-7094-99-10003-2.  Google Scholar

[24]

P. Sjölin, Regularity of solutions to the Schrödinger equation, Duke Math. J., 55 (1987), 699-715.  doi: 10.1215/S0012-7094-87-05535-9.  Google Scholar

[25]

J. Szeftel, Microlocal dispersive smoothing for the nonlinear Schrödinger equation, SIAM Journal on Mathematical Analysis, 37 (2005), 549-597.  doi: 10.1137/S0036141003432109.  Google Scholar

[26]

L. Vega, Schrödinger equations: Pointwise convergence to the initial data, Proc. Amer. Math. Soc., 102 (1988), 874-878.  doi: 10.2307/2047326.  Google Scholar

[27]

X. P. Wang, Time-decay of scattering solutions and classical trajectories, Ann. I.H.P. Phys. Théor., 47 (1987), 25-37.  doi: 10.1119/1.14979.  Google Scholar

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