Construction of a solitary wave solution of the nonlinear focusing schrödinger equation outside a strictly convex obstacle in the <inline-formula><tex-math id="M1">\begin{document} L^2 \end{document}</tex-math></inline-formula>-supercritical case

Previous Article
Large time behavior of exchange-driven growth
DCDS Home
This Issue
Next Article
Asymptotic behavior of minimal solutions of $ -\Delta u = \lambda f(u) $ as $ \lambda\to-\infty $

February 2021, 41(2): 701-746. doi: 10.3934/dcds.2020298

Construction of a solitary wave solution of the nonlinear focusing schrödinger equation outside a strictly convex obstacle in the $ L^2 $-supercritical case

Oussama Landoulsi

Univ. Sorbonne Paris Nord, Institut Galilée, LAGA, UMR 7539, 99 Avenue Jean Baptiste Clément, 93430 Villetaneuse, France

Received March 2019 Revised June 2020 Published February 2021 Early access August 2020

We consider the focusing $ L^2 $-supercritical Schrödinger equation in the exterior of a smooth, compact, strictly convex obstacle $ \Theta \subset \mathbb{R}^3 $. We construct a solution behaving asymptotically as a solitary wave on $ \mathbb{R}^3, $ for large times. When the velocity of the solitary wave is high, the existence of such a solution can be proved by a classical fixed point argument. To construct solutions with arbitrary nonzero velocity, we use a compactness argument similar to the one that was introduced by F.Merle in 1990 to construct solutions of the NLS equation blowing up at several points together with a topological argument using Brouwer's theorem to control the unstable direction of the linearized operator at the soliton. These solutions are arbitrarily close to the scattering threshold given by a previous work of R. Killip, M. Visan and X. Zhang, which is the same as the one on the whole Euclidean space given by S. Roundenko and J. Holmer in the radial case and by the previous authors with T. Duyckaerts in the non-radial case.

Keywords: Focusing NLS equation outside obstacle, compactness, existence for positive time, large-time behavior of solutions outside obstacle.

Mathematics Subject Classification: Primary: 35Q55, 35C08; Secondary: 35B40.

Citation: Oussama Landoulsi. Construction of a solitary wave solution of the nonlinear focusing schrödinger equation outside a strictly convex obstacle in the $ L^2 $-supercritical case. Discrete and Continuous Dynamical Systems, 2021, 41 (2) : 701-746. doi: 10.3934/dcds.2020298

References:

[1]	H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. II. Existence of infinitely many solutions, Arch. Rational Mech. Anal., 82 (1983), 347-375. doi: 10.1007/BF00250556.
[2]	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.
[3]	N. Burq, P. Gérard and N. Tzvetkov, Two singular dynamics of the nonlinear Schrödinger equation on a plane domain, Geom. Funct. Anal., 13 (2003), 1-19. doi: 10.1007/s000390300000.
[4]	N. Burq, P. Gérard and N. Tzvetkov, On nonlinear Schrödinger equations in exterior domains, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 295-318. doi: 10.1016/j.anihpc.2003.03.002.
[5]	T. Cazenave, Semilinear Schrödinger Equations, vol. 10 of Courant Lecture Notes in Mathematics, New York University Courant Institute of Mathematical Sciences, New York, 2003. doi: 10.1090/cln/010.
[6]	T. Cazenave and P.-L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys., 85 (1982), 549-561. doi: 10.1007/BF01403504.
[7]	V. Combet, Multi-existence of multi-solitons for the supercritical nonlinear Schrödinger equation in one dimension, Discrete Contin. Dyn. Syst., 34 (2014), 1961-1993. doi: 10.3934/dcds.2014.34.1961.
[8]	R. Côte, Y. Martel and F. Merle, Construction of multi-soliton solutions for the $L^2$-supercritical gKdV and NLS equations, Rev. Mat. Iberoam., 27 (2011), 273-302. doi: 10.4171/RMI/636.
[9]	T. Duyckaerts, J. Holmer and S. Roudenko, Scattering for the non-radial 3D cubic nonlinear Schrödinger equation, Math. Res. Lett., 15 (2008), 1233-1250. doi: 10.4310/MRL.2008.v15.n6.a13.
[10]	T. Duyckaerts and F. Merle, Dynamic of threshold solutions for energy-critical NLS, Geom. Funct. Anal., 18 (2009), 1787-1840. doi: 10.1007/s00039-009-0707-x.
[11]	T. Duyckaerts and S. Roudenko, Threshold solutions for the focusing 3d cubic Schrödinger equation., Rev. Mat. Iberoam., 26 (2010), 1-56. doi: 10.4171/RMI/592.
[12]	D. Fang, J. Xie and T. Cazenave, Scattering for the focusing energy-subcritical nonlinear Schrödinger equation, Sci. China Math., 54 (2011), 2037-2062. doi: 10.1007/s11425-011-4283-9.
[13]	N. Godet, Blow-up in several points for the nonlinear Schrödinger equation on a bounded domain, Differential Integral Equations, 24 (2011), 505-517.
[14]	M. Grillakis, Analysis of the linearization around a critical point of an infinite-dimensional Hamiltonian system, Comm. Pure Appl. Math., 43 (1990), 299-333. doi: 10.1002/cpa.3160430302.
[15]	M. Grillakis, Analysis of the linearization around a critical point of an infinite-dimensional Hamiltonian system, Comm. Pure Appl. Math., 43 (1990), 299-333. doi: 10.1002/cpa.3160430302.
[16]	M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry. I, J. Funct. Anal., 74 (1987), 160-197. doi: 10.1016/0022-1236(87)90044-9.
[17]	C. D. Guevara, Global behavior of finite energy solutions to the $d$-dimensional focusing nonlinear Schrödinger equation, Appl. Math. Res. Express. AMRX, 2014 (2014), 177-243.
[18]	J. Holmer and S. Roudenko, A sharp condition for scattering of the radial 3D cubic nonlinear Schrödinger equation, Comm. Math. Phys., 282 (2008), 435-467. doi: 10.1007/s00220-008-0529-y.
[19]	O. Ivanovici, On the Schrödinger equation outside strictly convex obstacles, Analysis and PDE, 3 (2010), 261-293. doi: 10.2140/apde.2010.3.261.
[20]	O. Ivanovici and G. Lebeau, Dispersion for the wave and the Schrödinger equations outside strictly convex obstacles and counterexamples, C. R. Math. Acad. Sci. Paris, 355 (2017), 774-779. doi: 10.1016/j.crma.2017.05.011.
[21]	O. Ivanovici and F. Planchon, On the energy critical Schrödinger equation in $3D$ non-trapping domains, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 1153-1177. doi: 10.1016/j.anihpc.2010.04.001.
[22]	C. E. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case, Invent. Math., 166 (2006), 645-675. doi: 10.1007/s00222-006-0011-4.
[23]	C. E. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation., Acta Math., 201 (2008), 147-212. doi: 10.1007/s11511-008-0031-6.
[24]	R. Killip, M. Visan and X. Zhang, Riesz transforms outside a convex obstacle, International Mathematics Research Notices, 2016 (2016), 5875-5921. doi: 10.1093/imrn/rnv338.
[25]	R. Killip, M. Visan and X. Zhang, The focusing cubic NLS on exterior domains in three dimensions, Appl. Math. Res. Express. AMRX, 2016 (2016), 146-180. doi: 10.1093/amrx/abv012.
[26]	M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p = 0$ in ${\bf{R}}^n$, Arch. Rational Mech. Anal., 105 (1989), 243-266. doi: 10.1007/BF00251502.
[27]	D. Lafontaine, Strichartz estimates without loss outside two strictly convex obstacles, arXiv preprint, arXiv: 1709.03836, (2017).
[28]	D. Lafontaine, Strichartz estimates without loss outside many strictly convex obstacles, arXiv preprint, arXiv: 1811.12357, (2018).
[29]	D. Li, H. Smith and X. Zhang, Global well-posedness and scattering for defocusing energy-critical NLS in the exterior of balls with radial data, Math. Res. Lett., 19 (2012), 213-232. doi: 10.4310/MRL.2012.v19.n1.a17.
[30]	Y. Martel and F. Merle, Multi solitary waves for nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 849-864. doi: 10.1016/j.anihpc.2006.01.001.
[31]	F. Merle, Construction of solutions with exactly $k$ blow-up points for the Schrödinger equation with critical nonlinearity, Comm. Math. Phys., 129 (1990), 223-240. doi: 10.1007/BF02096981.
[32]	F. Planchon and L. Vega, Bilinear virial identities and applications, Ann. Sci. Éc. Norm. Supér. (4), 42 (2009), 261–290. doi: 10.24033/asens.2096.
[33]	F. Planchon and L. Vega, Scattering for solutions of NLS in the exterior of a 2D star-shaped obstacle, Math. Res. Lett., 19 (2012), 887-897. doi: 10.4310/MRL.2012.v19.n4.a12.
[34]	W. Schlag, Spectral theory and nonlinear partial differential equations: A survey, Discrete Contin. Dyn. Syst., 15 (2006), 703-723. doi: 10.3934/dcds.2006.15.703.
[35]	F. A. Shakra, On 2D nonlinear Schrödinger equation on non-trapping exterior domains, Rev. Mat. Iberoam., 31 (2015), 657-680.
[36]	T. Tao, Nonlinear Dispersive Equations, vol. 106 of CBMS Regional Conference Series in Mathematics, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 2006. Local and global analysis. doi: 10.1090/cbms/106.
[37]	M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87 (1982/83), 567-576.
[38]	M. I. Weinstein, Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal., 16 (1985), 472-491. doi: 10.1137/0516034.
[39]	M. I. Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolution equations, Comm. Pure Appl. Math., 39 (1986), 51-67. doi: 10.1002/cpa.3160390103.
[40]	K. Yang, The focusing NLS on exterior domains in three dimensions, Commun. Pure Appl. Anal., 16 (2017), 2269-2297. doi: 10.3934/cpaa.2017112.

show all references

References:

[1]	H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. II. Existence of infinitely many solutions, Arch. Rational Mech. Anal., 82 (1983), 347-375. doi: 10.1007/BF00250556.
[2]	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.
[3]	N. Burq, P. Gérard and N. Tzvetkov, Two singular dynamics of the nonlinear Schrödinger equation on a plane domain, Geom. Funct. Anal., 13 (2003), 1-19. doi: 10.1007/s000390300000.
[4]	N. Burq, P. Gérard and N. Tzvetkov, On nonlinear Schrödinger equations in exterior domains, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 295-318. doi: 10.1016/j.anihpc.2003.03.002.
[5]	T. Cazenave, Semilinear Schrödinger Equations, vol. 10 of Courant Lecture Notes in Mathematics, New York University Courant Institute of Mathematical Sciences, New York, 2003. doi: 10.1090/cln/010.
[6]	T. Cazenave and P.-L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys., 85 (1982), 549-561. doi: 10.1007/BF01403504.
[7]	V. Combet, Multi-existence of multi-solitons for the supercritical nonlinear Schrödinger equation in one dimension, Discrete Contin. Dyn. Syst., 34 (2014), 1961-1993. doi: 10.3934/dcds.2014.34.1961.
[8]	R. Côte, Y. Martel and F. Merle, Construction of multi-soliton solutions for the $L^2$-supercritical gKdV and NLS equations, Rev. Mat. Iberoam., 27 (2011), 273-302. doi: 10.4171/RMI/636.
[9]	T. Duyckaerts, J. Holmer and S. Roudenko, Scattering for the non-radial 3D cubic nonlinear Schrödinger equation, Math. Res. Lett., 15 (2008), 1233-1250. doi: 10.4310/MRL.2008.v15.n6.a13.
[10]	T. Duyckaerts and F. Merle, Dynamic of threshold solutions for energy-critical NLS, Geom. Funct. Anal., 18 (2009), 1787-1840. doi: 10.1007/s00039-009-0707-x.
[11]	T. Duyckaerts and S. Roudenko, Threshold solutions for the focusing 3d cubic Schrödinger equation., Rev. Mat. Iberoam., 26 (2010), 1-56. doi: 10.4171/RMI/592.
[12]	D. Fang, J. Xie and T. Cazenave, Scattering for the focusing energy-subcritical nonlinear Schrödinger equation, Sci. China Math., 54 (2011), 2037-2062. doi: 10.1007/s11425-011-4283-9.
[13]	N. Godet, Blow-up in several points for the nonlinear Schrödinger equation on a bounded domain, Differential Integral Equations, 24 (2011), 505-517.
[14]	M. Grillakis, Analysis of the linearization around a critical point of an infinite-dimensional Hamiltonian system, Comm. Pure Appl. Math., 43 (1990), 299-333. doi: 10.1002/cpa.3160430302.
[15]	M. Grillakis, Analysis of the linearization around a critical point of an infinite-dimensional Hamiltonian system, Comm. Pure Appl. Math., 43 (1990), 299-333. doi: 10.1002/cpa.3160430302.
[16]	M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry. I, J. Funct. Anal., 74 (1987), 160-197. doi: 10.1016/0022-1236(87)90044-9.
[17]	C. D. Guevara, Global behavior of finite energy solutions to the $d$-dimensional focusing nonlinear Schrödinger equation, Appl. Math. Res. Express. AMRX, 2014 (2014), 177-243.
[18]	J. Holmer and S. Roudenko, A sharp condition for scattering of the radial 3D cubic nonlinear Schrödinger equation, Comm. Math. Phys., 282 (2008), 435-467. doi: 10.1007/s00220-008-0529-y.
[19]	O. Ivanovici, On the Schrödinger equation outside strictly convex obstacles, Analysis and PDE, 3 (2010), 261-293. doi: 10.2140/apde.2010.3.261.
[20]	O. Ivanovici and G. Lebeau, Dispersion for the wave and the Schrödinger equations outside strictly convex obstacles and counterexamples, C. R. Math. Acad. Sci. Paris, 355 (2017), 774-779. doi: 10.1016/j.crma.2017.05.011.
[21]	O. Ivanovici and F. Planchon, On the energy critical Schrödinger equation in $3D$ non-trapping domains, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 1153-1177. doi: 10.1016/j.anihpc.2010.04.001.
[22]	C. E. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case, Invent. Math., 166 (2006), 645-675. doi: 10.1007/s00222-006-0011-4.
[23]	C. E. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation., Acta Math., 201 (2008), 147-212. doi: 10.1007/s11511-008-0031-6.
[24]	R. Killip, M. Visan and X. Zhang, Riesz transforms outside a convex obstacle, International Mathematics Research Notices, 2016 (2016), 5875-5921. doi: 10.1093/imrn/rnv338.
[25]	R. Killip, M. Visan and X. Zhang, The focusing cubic NLS on exterior domains in three dimensions, Appl. Math. Res. Express. AMRX, 2016 (2016), 146-180. doi: 10.1093/amrx/abv012.
[26]	M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p = 0$ in ${\bf{R}}^n$, Arch. Rational Mech. Anal., 105 (1989), 243-266. doi: 10.1007/BF00251502.
[27]	D. Lafontaine, Strichartz estimates without loss outside two strictly convex obstacles, arXiv preprint, arXiv: 1709.03836, (2017).
[28]	D. Lafontaine, Strichartz estimates without loss outside many strictly convex obstacles, arXiv preprint, arXiv: 1811.12357, (2018).
[29]	D. Li, H. Smith and X. Zhang, Global well-posedness and scattering for defocusing energy-critical NLS in the exterior of balls with radial data, Math. Res. Lett., 19 (2012), 213-232. doi: 10.4310/MRL.2012.v19.n1.a17.
[30]	Y. Martel and F. Merle, Multi solitary waves for nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 849-864. doi: 10.1016/j.anihpc.2006.01.001.
[31]	F. Merle, Construction of solutions with exactly $k$ blow-up points for the Schrödinger equation with critical nonlinearity, Comm. Math. Phys., 129 (1990), 223-240. doi: 10.1007/BF02096981.
[32]	F. Planchon and L. Vega, Bilinear virial identities and applications, Ann. Sci. Éc. Norm. Supér. (4), 42 (2009), 261–290. doi: 10.24033/asens.2096.
[33]	F. Planchon and L. Vega, Scattering for solutions of NLS in the exterior of a 2D star-shaped obstacle, Math. Res. Lett., 19 (2012), 887-897. doi: 10.4310/MRL.2012.v19.n4.a12.
[34]	W. Schlag, Spectral theory and nonlinear partial differential equations: A survey, Discrete Contin. Dyn. Syst., 15 (2006), 703-723. doi: 10.3934/dcds.2006.15.703.
[35]	F. A. Shakra, On 2D nonlinear Schrödinger equation on non-trapping exterior domains, Rev. Mat. Iberoam., 31 (2015), 657-680.
[36]	T. Tao, Nonlinear Dispersive Equations, vol. 106 of CBMS Regional Conference Series in Mathematics, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 2006. Local and global analysis. doi: 10.1090/cbms/106.
[37]	M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87 (1982/83), 567-576.
[38]	M. I. Weinstein, Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal., 16 (1985), 472-491. doi: 10.1137/0516034.
[39]	M. I. Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolution equations, Comm. Pure Appl. Math., 39 (1986), 51-67. doi: 10.1002/cpa.3160390103.
[40]	K. Yang, The focusing NLS on exterior domains in three dimensions, Commun. Pure Appl. Anal., 16 (2017), 2269-2297. doi: 10.3934/cpaa.2017112.

[1]	Walter Allegretto, Yanping Lin, Shuqing Ma. Existence and long time behaviour of solutions to obstacle thermistor equations. Discrete and Continuous Dynamical Systems, 2002, 8 (3) : 757-780. doi: 10.3934/dcds.2002.8.757
[2]	Qiwei Wu, Liping Luan. Large-time behavior of solutions to unipolar Euler-Poisson equations with time-dependent damping. Communications on Pure and Applied Analysis, 2021, 20 (3) : 995-1023. doi: 10.3934/cpaa.2021003
[3]	Shifeng Geng, Lina Zhang. Large-time behavior of solutions for the system of compressible adiabatic flow through porous media with nonlinear damping. Communications on Pure and Applied Analysis, 2014, 13 (6) : 2211-2228. doi: 10.3934/cpaa.2014.13.2211
[4]	Linlin Li, Bedreddine Ainseba. Large-time behavior of matured population in an age-structured model. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2561-2580. doi: 10.3934/dcdsb.2020195
[5]	Marco Di Francesco, Yahya Jaafra. Multiple large-time behavior of nonlocal interaction equations with quadratic diffusion. Kinetic and Related Models, 2019, 12 (2) : 303-322. doi: 10.3934/krm.2019013
[6]	Qiwei Wu. Large-time behavior of solutions to the bipolar quantum Euler-Poisson system with critical time-dependent over-damping. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2022008
[7]	Wen Tan, Bo-Qing Dong, Zhi-Min Chen. Large-time regular solutions to the modified quasi-geostrophic equation in Besov spaces. Discrete and Continuous Dynamical Systems, 2019, 39 (7) : 3749-3765. doi: 10.3934/dcds.2019152
[8]	Goro Akagi, Kei Matsuura. Well-posedness and large-time behaviors of solutions for a parabolic equation involving $p(x)$-Laplacian. Conference Publications, 2011, 2011 (Special) : 22-31. doi: 10.3934/proc.2011.2011.22
[9]	Jerry L. Bona, Laihan Luo. Large-time asymptotics of the generalized Benjamin-Ono-Burgers equation. Discrete and Continuous Dynamical Systems - S, 2011, 4 (1) : 15-50. doi: 10.3934/dcdss.2011.4.15
[10]	Dongfen Bian, Boling Guo. Global existence and large time behavior of solutions to the electric-magnetohydrodynamic equations. Kinetic and Related Models, 2013, 6 (3) : 481-503. doi: 10.3934/krm.2013.6.481
[11]	Laurent Bourgeois, Dmitry Ponomarev, Jérémi Dardé. An inverse obstacle problem for the wave equation in a finite time domain. Inverse Problems and Imaging, 2019, 13 (2) : 377-400. doi: 10.3934/ipi.2019019
[12]	Leonardo Kosloff, Tomas Schonbek. Existence and decay of solutions of the 2D QG equation in the presence of an obstacle. Discrete and Continuous Dynamical Systems - S, 2014, 7 (5) : 1025-1043. doi: 10.3934/dcdss.2014.7.1025
[13]	Kin Ming Hui, Soojung Kim. Asymptotic large time behavior of singular solutions of the fast diffusion equation. Discrete and Continuous Dynamical Systems, 2017, 37 (11) : 5943-5977. doi: 10.3934/dcds.2017258
[14]	Kazuhiro Ishige, Asato Mukai. Large time behavior of solutions of the heat equation with inverse square potential. Discrete and Continuous Dynamical Systems, 2018, 38 (8) : 4041-4069. doi: 10.3934/dcds.2018176
[15]	Hiroshi Takeda. Large time behavior of solutions for a nonlinear damped wave equation. Communications on Pure and Applied Analysis, 2016, 15 (1) : 41-55. doi: 10.3934/cpaa.2016.15.41
[16]	Nakao Hayashi, Elena I. Kaikina, Pavel I. Naumkin. Large time behavior of solutions to the generalized derivative nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems, 1999, 5 (1) : 93-106. doi: 10.3934/dcds.1999.5.93
[17]	Zhong Tan, Yong Wang, Fanhui Xu. Large-time behavior of the full compressible Euler-Poisson system without the temperature damping. Discrete and Continuous Dynamical Systems, 2016, 36 (3) : 1583-1601. doi: 10.3934/dcds.2016.36.1583
[18]	Youshan Tao, Lihe Wang, Zhi-An Wang. Large-time behavior of a parabolic-parabolic chemotaxis model with logarithmic sensitivity in one dimension. Discrete and Continuous Dynamical Systems - B, 2013, 18 (3) : 821-845. doi: 10.3934/dcdsb.2013.18.821
[19]	Ken Shirakawa, Hiroshi Watanabe. Large-time behavior for a PDE model of isothermal grain boundary motion with a constraint. Conference Publications, 2015, 2015 (special) : 1009-1018. doi: 10.3934/proc.2015.1009
[20]	Jishan Fan, Fei Jiang. Large-time behavior of liquid crystal flows with a trigonometric condition in two dimensions. Communications on Pure and Applied Analysis, 2016, 15 (1) : 73-90. doi: 10.3934/cpaa.2016.15.73

2020 Impact Factor: 1.392

Tools

Metrics

Other articles
by authors

on AIMS
Oussama Landoulsi
on Google Scholar
Oussama Landoulsi

2020 Impact Factor: 1.392

Tools

Article outline

2020 Impact Factor: 1.392

Tools

Metrics

Other articles
by authors

on AIMS
Oussama Landoulsi
on Google Scholar
Oussama Landoulsi

[Back to Top]