May  2019, 18(3): 1359-1374. doi: 10.3934/cpaa.2019066

A remark on local well-posedness for nonlinear Schrödinger equations with power nonlinearity-an alternative approach

Department of Mathematics, Shimane University, Matsue 690-8504, Japan

Received  June 2018 Revised  August 2018 Published  November 2018

Fund Project: Supported in part by JSPS, Grant-in-Aid for Scientific Research (C) #25400176.

We study the nonlinear Schrödinger equation (NLS)
$\partial_t u +i \Delta u = i\lambda |u|^{p-1} u$
in
$\mathit{\boldsymbol{R}}^{1+n}$
, where
$n\ge 3$
,
$p>1$
, and
$\lambda \in \mathit{\boldsymbol{C}}$
. We prove that (NLS) is locally well-posed in
$H^s$
if
$1<s<\min\{4;n/2\}$
and
$\max\{1;s/2\}< p< 1+4/(n-2s)$
. To obtain a good lower bound for
$p$
, we use fractional order Besov spaces for the time variable. The use of such spaces together with time cut-off makes it difficult to derive positive powers of time length from nonlinear estimates, so that it is difficult to apply the contraction mapping principle. For the proof we improve Pecher's inequality (1997), which is a modification of the Strichartz estimate, and apply this inequality to the nonlinear problem together with paraproduct formula.
Citation: Takeshi Wada. A remark on local well-posedness for nonlinear Schrödinger equations with power nonlinearity-an alternative approach. Communications on Pure and Applied Analysis, 2019, 18 (3) : 1359-1374. doi: 10.3934/cpaa.2019066
References:
[1]

H. Amann, Operator-valued Fourier multipliers, vector-valued Besov spaces, and applications, Math. Nachr., 186 (1997), 5-56.  doi: 10.1002/mana.3211860102.

[2]

J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Springer-Verlag, Berlin-New York, 1976.

[3]

T. CazenaveD. Fang and Z. Han, Local well-posedness for the $H^2$-critical nonlinear Schrödinger equation, Trans. Amer. Math. Soc., 368 (2016), 7911-7934.  doi: 10.1090/tran6683.

[4]

T. Cazenave and F. B. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in $H^s$, Nonlinear Anal., 14 (1990), 807-836.  doi: 10.1016/0362-546X(90)90023-A.

[5]

D. Fang and Z. Han, On the well-posedness for NLS in $H^s$, J. Funct. Anal., 264 (2013), 1438-1455.  doi: 10.1016/j.jfa.2013.01.005.

[6]

J. GinibreT. Ozawa and G. Velo, On the existence of the wave operators for a class of nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor., 60 (1994), 211-239. 

[7]

J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations. Ⅰ. The Cauchy problem, general case, J. Funct. Anal., 32 (1979), 1-32.  doi: 10.1016/0022-1236(79)90076-4.

[8]

J. Ginibre and G. Velo, The global Cauchy problem for the nonlinear Schrödinger equation revisited, Ann. Inst. H. Poincaré Anal. Non Linéaire, 2 (1985), 309-327. 

[9]

J. Ginibre and G. Velo, Scattering theory in the energy space for a class of nonlinear wave equations, Comm. Math. Phys., 123 (1989), 535-573. 

[10]

J. Ginibre and G. Velo, Generalized Strichartz inequalities for the wave equation, J. Funct. Anal., 133 (1995), 50-68.  doi: 10.1006/jfan.1995.1119.

[11]

T. Kato, On nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor., 46 (1987), 113-129. 

[12]

T. Kato, Nonlinear Schrödinger equations, in Schrödinger Operators, Lecture Notes in Phys., 345, Springer, Berlin (1989), 218–263. doi: 10.1007/3-540-51783-9_22.

[13]

T. Kato, On nonlinear Schrödinger equations. Ⅱ. $H^s$-solutions and unconditional well-posedness, J. Anal. Math., 67 (1995), 281-306.  doi: 10.1007/BF02787794.

[14]

M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980. 

[15]

M. Nakamura and T. Ozawa, Low energy scattering for nonlinear Schrödinger equations in fractional order Sobolev spaces, Rev. Math. Phys., 9 (1997), 397-410.  doi: 10.1142/S0129055X97000154.

[16]

M. Nakamura and T. Wada, Modified Strichartz estimates with an application to the critical nonlinear Schrödinger equation, Nonlinear Anal., 130 (2016), 138-156.  doi: 10.1016/j.na.2015.09.023.

[17]

H. Pecher, Solutions of semilinear Schrödinger equations in $H^s$, Ann. Inst. H. Poincaré Phys. Théor., 67 (1997), 259-296. 

[18]

H. Y. Schmeisser, Vector-valued Sobolev and Besov spaces, in Seminar Analysis of the KarlWeierstraß-Institute of Mathematics 1985/86 (Berlin, 1985/86), Teubner-Texte Math. 96, Teubner, Leipzig (1987), 4–44.

[19]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amsterdam-New York-Oxford, 1978.

[20]

Y. Tsutsumi, Global strong solutions for nonlinear Schrödinger equations, Nonlinear Anal., 11 (1987), 1143-1154.  doi: 10.1016/0362-546X(87)90003-4.

[21]

Y. Tsutsumi, $L^2$-solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcial. Ekvac., 30 (1987), 115-125. 

[22]

H. Uchizono and T. Wada, Continuous dependence for nonlinear Schrödinger equation in $H^s$, J. Math. Sci. Univ. Tokyo, 19 (2012), 57-68. 

[23]

H. Uchizono and T. Wada, On well-posedness for nonlinear Schrödinger equations with power nonlinearity in fractional order Sobolev spaces, J. Math. Anal. Appl., 395 (2012), 56-62.  doi: 10.1016/j.jmaa.2012.04.079.

show all references

References:
[1]

H. Amann, Operator-valued Fourier multipliers, vector-valued Besov spaces, and applications, Math. Nachr., 186 (1997), 5-56.  doi: 10.1002/mana.3211860102.

[2]

J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Springer-Verlag, Berlin-New York, 1976.

[3]

T. CazenaveD. Fang and Z. Han, Local well-posedness for the $H^2$-critical nonlinear Schrödinger equation, Trans. Amer. Math. Soc., 368 (2016), 7911-7934.  doi: 10.1090/tran6683.

[4]

T. Cazenave and F. B. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in $H^s$, Nonlinear Anal., 14 (1990), 807-836.  doi: 10.1016/0362-546X(90)90023-A.

[5]

D. Fang and Z. Han, On the well-posedness for NLS in $H^s$, J. Funct. Anal., 264 (2013), 1438-1455.  doi: 10.1016/j.jfa.2013.01.005.

[6]

J. GinibreT. Ozawa and G. Velo, On the existence of the wave operators for a class of nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor., 60 (1994), 211-239. 

[7]

J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations. Ⅰ. The Cauchy problem, general case, J. Funct. Anal., 32 (1979), 1-32.  doi: 10.1016/0022-1236(79)90076-4.

[8]

J. Ginibre and G. Velo, The global Cauchy problem for the nonlinear Schrödinger equation revisited, Ann. Inst. H. Poincaré Anal. Non Linéaire, 2 (1985), 309-327. 

[9]

J. Ginibre and G. Velo, Scattering theory in the energy space for a class of nonlinear wave equations, Comm. Math. Phys., 123 (1989), 535-573. 

[10]

J. Ginibre and G. Velo, Generalized Strichartz inequalities for the wave equation, J. Funct. Anal., 133 (1995), 50-68.  doi: 10.1006/jfan.1995.1119.

[11]

T. Kato, On nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor., 46 (1987), 113-129. 

[12]

T. Kato, Nonlinear Schrödinger equations, in Schrödinger Operators, Lecture Notes in Phys., 345, Springer, Berlin (1989), 218–263. doi: 10.1007/3-540-51783-9_22.

[13]

T. Kato, On nonlinear Schrödinger equations. Ⅱ. $H^s$-solutions and unconditional well-posedness, J. Anal. Math., 67 (1995), 281-306.  doi: 10.1007/BF02787794.

[14]

M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980. 

[15]

M. Nakamura and T. Ozawa, Low energy scattering for nonlinear Schrödinger equations in fractional order Sobolev spaces, Rev. Math. Phys., 9 (1997), 397-410.  doi: 10.1142/S0129055X97000154.

[16]

M. Nakamura and T. Wada, Modified Strichartz estimates with an application to the critical nonlinear Schrödinger equation, Nonlinear Anal., 130 (2016), 138-156.  doi: 10.1016/j.na.2015.09.023.

[17]

H. Pecher, Solutions of semilinear Schrödinger equations in $H^s$, Ann. Inst. H. Poincaré Phys. Théor., 67 (1997), 259-296. 

[18]

H. Y. Schmeisser, Vector-valued Sobolev and Besov spaces, in Seminar Analysis of the KarlWeierstraß-Institute of Mathematics 1985/86 (Berlin, 1985/86), Teubner-Texte Math. 96, Teubner, Leipzig (1987), 4–44.

[19]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amsterdam-New York-Oxford, 1978.

[20]

Y. Tsutsumi, Global strong solutions for nonlinear Schrödinger equations, Nonlinear Anal., 11 (1987), 1143-1154.  doi: 10.1016/0362-546X(87)90003-4.

[21]

Y. Tsutsumi, $L^2$-solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcial. Ekvac., 30 (1987), 115-125. 

[22]

H. Uchizono and T. Wada, Continuous dependence for nonlinear Schrödinger equation in $H^s$, J. Math. Sci. Univ. Tokyo, 19 (2012), 57-68. 

[23]

H. Uchizono and T. Wada, On well-posedness for nonlinear Schrödinger equations with power nonlinearity in fractional order Sobolev spaces, J. Math. Anal. Appl., 395 (2012), 56-62.  doi: 10.1016/j.jmaa.2012.04.079.

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