August  2020, 13(4): 837-867. doi: 10.3934/krm.2020029

Local well-posedness of the Boltzmann equation with polynomially decaying initial data

1. 

Department of Mathematics, University of Arizona, Tucson, AZ 85721, USA

2. 

Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901, USA

3. 

Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, USA

Received  October 2019 Revised  February 2020 Published  May 2020

Fund Project: The first author was partially supported by NSF grant DMS-2003110. The second author was partially supported by a Ralph E. Powe Award from ORAU. The third author was partially supported by NSF grant DMS-2012333

We consider the Cauchy problem for the spatially inhomogeneous non-cutoff Boltzmann equation with polynomially decaying initial data in the velocity variable. We establish short-time existence for any initial data with this decay in a fifth order Sobolev space by working in a mixed $ L^2 $ and $ L^\infty $ space that allows to compensate for potential moment generation and obtaining new estimates on the collision operator that are well-adapted to this space. Our results improve the range of parameters for which the Boltzmann equation is well-posed in this decay regime, as well as relax the restrictions on the initial regularity. As an application, we can combine our existence result with the recent conditional regularity estimates of Imbert-Silvestre (arXiv:1909.12729 [math.AP]) to conclude solutions can be continued for as long as the mass, energy, and entropy densities remain under control. This continuation criterion was previously only available in the restricted range of parameters of previous well-posedness results for polynomially decaying initial data.

Citation: Christopher Henderson, Stanley Snelson, Andrei Tarfulea. Local well-posedness of the Boltzmann equation with polynomially decaying initial data. Kinetic and Related Models, 2020, 13 (4) : 837-867. doi: 10.3934/krm.2020029
References:
[1]

R. Alexandre, Around 3D Boltzmann non linear operator without angular cutoff, a new formulation, M2AN Math. Model. Numer. Anal., 34 (2000), 575-590.  doi: 10.1051/m2an:2000157.

[2]

R. Alexandre, Some solutions of the Boltzmann equation without angular cutoff, J. Statist. Phys., 104 (2001), 327-358.  doi: 10.1023/A:1010317913642.

[3]

R. AlexandreL. DesvillettesC. Villani and B. Wennberg, Entropy dissipation and long-range interactions, Arch. Ration. Mech. Anal., 152 (2000), 327-355.  doi: 10.1007/s002050000083.

[4]

R. AlexandreY. MorimotoS. UkaiC.-J. Xu and T. Yang, Regularizing effect and local existence for the non-cutoff Boltzmann equation, Arch. Ration. Mech. Anal., 198 (2010), 39-123.  doi: 10.1007/s00205-010-0290-1.

[5]

R. AlexandreY. MorimotoS. UkaiC.-J. Xu and T. Yang, The Boltzmann equation without angular cutoff in the whole space: II, Global existence for hard potential, Anal. Appl. (Singap.), 9 (2011), 113-134.  doi: 10.1142/S0219530511001777.

[6]

R. AlexandreY. MorimotoS. UkaiC.-J. Xu and T. Yang, The Boltzmann equation without angular cutoff in the whole space: Qualitative properties of solutions, Arch. Ration. Mech. Anal., 202 (2011), 599-661.  doi: 10.1007/s00205-011-0432-0.

[7]

R. AlexandreY. MorimotoS. UkaiC.-J. Xu and T. Yang, Bounded solutions of the Boltzmann equation in the whole space, Kinet. Relat. Models, 41 (2011), 17-40.  doi: 10.3934/krm.2011.4.17.

[8]

R. AlexandreY. MorimotoS. UkaiC.-J. Xu and T. Yang, Global existence and full regularity of the Boltzmann equation without angular cutoff, Comm. Math. Phys., 304 (2011), 513-581.  doi: 10.1007/s00220-011-1242-9.

[9]

R. AlexandreY. MorimotoS. UkaiC.-J. Xu and T. Yang, Uniqueness of solutions for the non-cutoff Boltzmann equation with soft potential, Kinet. Relat. Models, 4 (2011), 919-934.  doi: 10.3934/krm.2011.4.919.

[10]

R. AlexandreY. MorimotoS. UkaiC.-J. Xu and T. Yang, The Boltzmann equation without angular cutoff in the whole space: I, Global existence for soft potential, J. Funct. Anal., 262 (2012), 915-1010.  doi: 10.1016/j.jfa.2011.10.007.

[11]

R. AlexandreY. MorimotoS. UkaiC.-J. Xu and T. Yang, Local existence with mild regularity for the Boltzmann equation, Kinet. Relat. Models, 6 (2013), 1011-1041.  doi: 10.3934/krm.2013.6.1011.

[12]

R. Alexandre and C. Villani, On the Boltzmann equation for long-range interactions, Comm. Pure Appl. Math., 55 (2002), 30-70.  doi: 10.1002/cpa.10012.

[13]

R. Alexandre and C. Villani, On the Landau approximation in plasma physics, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 21 (2004), 61-95.  doi: 10.1016/j.anihpc.2002.12.001.

[14]

R. Alonso, Y. Morimoto, W. Sun and T. Yang, Non-cutoff Boltzmann equation with polynomial decay perturbation, preprint, 2019, arXiv: 1812.05299.

[15]

T. Carleman, Sur la théorie de l'équation intégrodifférentielle de Boltzmann, Acta Math., 60 (1933), 91-146.  doi: 10.1007/BF02398270.

[16]

S. Chaturvedi, Local existence for the Landau equation with hard potentials, preprint, 2019, arXiv: 1910.11866.

[17]

S. Chaturvedi, Stability of vacuum for the Landau equation with hard potentials, preprint, 2020, arXiv: 2001.07208.

[18]

Y. Chen and L.-B. He, Smoothing estimates for Boltzmann equation with full-range interactions: Spatially homogeneous case, Arch. Ration. Mech. Anal., 201 (2011), 501-548.  doi: 10.1007/s00205-010-0393-8.

[19]

L. Desvillettes and C. Mouhot, About $L^p$ estimates for the spatially homogeneous Boltzmann equation, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 22 (2005), 127-142.  doi: 10.1016/j.anihpc.2004.03.002.

[20]

L. Desvillettes and C. Mouhot, Stability and uniqueness for the spatially homogeneous Boltzmann equation with long-range interactions, Arch. Ration. Mech. Anal., 193 (2009), 227-253.  doi: 10.1007/s00205-009-0233-x.

[21]

L. Desvillettes and B. Wennberg, Smoothness of the solution of the spatially homogeneous Boltzmann equation without cutoff, Comm. Part. Diff. Eq., 29 (2004), 133-155.  doi: 10.1081/PDE-120028847.

[22]

R. J. DiPerna and P. L. Lions, On the Cauchy Problem for Boltzmann equations: Global existence and weak stability, Ann. Math., 130 (1989), 321-366.  doi: 10.2307/1971423.

[23]

A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964.

[24]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2001. doi: 10.1007/978-3-642-61798-0.

[25]

T. Goudon, On Boltzmann equations and Fokker-Planck asymptotics: Influence of grazing collisions, J. Statist. Phys., 89 (1997), 751-776.  doi: 10.1007/BF02765543.

[26]

P. T. Gressman and R. M. Strain, Global classical solutions of the Boltzmann equation without angular cut-off, J. Amer. Math. Soc., 24 (2011), 771-847.  doi: 10.1090/S0894-0347-2011-00697-8.

[27]

L.-B. He and X. Yang, Well-posedness and asymptotics of grazing collisions limit of Boltzmann equation with Coulomb interaction, SIAM J. Math. Anal., 46 (2014), 4104-4165.  doi: 10.1137/140965983.

[28]

L.-B. He and J.-C. Jiang, On the global dynamics of the inhomogeneous Boltzmann equations without angular cutoff: Hard potentials and Maxwellian molecules, preprint, 2017, arXiv: 1710.00315.

[29]

C. HendersonS. Snelson and A. Tarfulea, Local existence, lower mass bounds, and a new continuation criterion for the Landau equation, J. Differential Equations, 266 (2019), 1536-1577.  doi: 10.1016/j.jde.2018.08.005.

[30]

C. Henderson, S. Snelson, and A. Tarfulea, Local solutions of the Landau equation with rough, slowly decaying initial data, preprint, 2019, arXiv: 1909.05914.

[31]

F. Hérau, D. Tonon, and I. Tristani, Regularization estimates and Cauchy theory for inhomogeneous Boltzmann equation for hard potentials without cut-off, arXiv: 1710.01098.

[32]

C. Imbert, C. Mouhot, and L. Silvestre, Decay estimates for large velocities in the Boltzmann equation without cut-off, preprint, 2019, arXiv: 1804.06135.

[33]

C. Imbert and L. Silvestre, The Schauder estimate for kinetic integral equations, preprint, 2019, arXiv: 1812.11870.

[34]

C. Imbert and L. Silvestre, Global regularity estimates for the Boltzmann equation without cut-off, preprint, 2019, arXiv: 1909.12729.

[35]

C. Imbert and L. Silvestre, Weak Harnack inequality for the Boltzmann equation without cut-off, J. Eur. Math. Soc., 22 (2020), 507-592.  doi: 10.4171/JEMS/928.

[36]

X. Lu and C. Mouhot, On measure solutions of the Boltzmann equation, part I: Moment production and stability estimates, J. Differential Equations, 252 (2012), 3305-3363.  doi: 10.1016/j.jde.2011.10.021.

[37]

J. Luk, Stability of Vacuum for the Landau Equation with Moderately Soft Potentials, Ann. PDE, 5 (2019), 101 pp. doi: 10.1007/s40818-019-0067-2.

[38]

Y. MorimotoS. Wang and T. Yang, Measure valued solutions to the spatially homogeneous Boltzmann equation without angular cutoff, J. Stat. Phys., 165 (2016), 866-906.  doi: 10.1007/s10955-016-1655-0.

[39]

Y. Morimoto and T. Yang, Local existence of polynomial decay solutions to the Boltzmann equation for soft potentials, Anal. Appl. (Singap.), 13 (2015), 663-683.  doi: 10.1142/S0219530514500079.

[40]

L. Silvestre, A new regularization mechanism for the Boltzmann equation without cut-off, Comm. Math. Phys., 348 (2016), 69-100.  doi: 10.1007/s00220-016-2757-x.

[41]

C. Villani, On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations, Arch. Ration. Mech. Anal., 143 (1998), 273-307.  doi: 10.1007/s002050050106.

[42]

C. Villani, A review of mathematical topics in collisional kinetic theory, in Handbook of Mathematical Fluid Dynamics, Vol. I, North-Holland, Amsterdam, (2002), 71–305. doi: 10.1016/S1874-5792(02)80004-0.

show all references

References:
[1]

R. Alexandre, Around 3D Boltzmann non linear operator without angular cutoff, a new formulation, M2AN Math. Model. Numer. Anal., 34 (2000), 575-590.  doi: 10.1051/m2an:2000157.

[2]

R. Alexandre, Some solutions of the Boltzmann equation without angular cutoff, J. Statist. Phys., 104 (2001), 327-358.  doi: 10.1023/A:1010317913642.

[3]

R. AlexandreL. DesvillettesC. Villani and B. Wennberg, Entropy dissipation and long-range interactions, Arch. Ration. Mech. Anal., 152 (2000), 327-355.  doi: 10.1007/s002050000083.

[4]

R. AlexandreY. MorimotoS. UkaiC.-J. Xu and T. Yang, Regularizing effect and local existence for the non-cutoff Boltzmann equation, Arch. Ration. Mech. Anal., 198 (2010), 39-123.  doi: 10.1007/s00205-010-0290-1.

[5]

R. AlexandreY. MorimotoS. UkaiC.-J. Xu and T. Yang, The Boltzmann equation without angular cutoff in the whole space: II, Global existence for hard potential, Anal. Appl. (Singap.), 9 (2011), 113-134.  doi: 10.1142/S0219530511001777.

[6]

R. AlexandreY. MorimotoS. UkaiC.-J. Xu and T. Yang, The Boltzmann equation without angular cutoff in the whole space: Qualitative properties of solutions, Arch. Ration. Mech. Anal., 202 (2011), 599-661.  doi: 10.1007/s00205-011-0432-0.

[7]

R. AlexandreY. MorimotoS. UkaiC.-J. Xu and T. Yang, Bounded solutions of the Boltzmann equation in the whole space, Kinet. Relat. Models, 41 (2011), 17-40.  doi: 10.3934/krm.2011.4.17.

[8]

R. AlexandreY. MorimotoS. UkaiC.-J. Xu and T. Yang, Global existence and full regularity of the Boltzmann equation without angular cutoff, Comm. Math. Phys., 304 (2011), 513-581.  doi: 10.1007/s00220-011-1242-9.

[9]

R. AlexandreY. MorimotoS. UkaiC.-J. Xu and T. Yang, Uniqueness of solutions for the non-cutoff Boltzmann equation with soft potential, Kinet. Relat. Models, 4 (2011), 919-934.  doi: 10.3934/krm.2011.4.919.

[10]

R. AlexandreY. MorimotoS. UkaiC.-J. Xu and T. Yang, The Boltzmann equation without angular cutoff in the whole space: I, Global existence for soft potential, J. Funct. Anal., 262 (2012), 915-1010.  doi: 10.1016/j.jfa.2011.10.007.

[11]

R. AlexandreY. MorimotoS. UkaiC.-J. Xu and T. Yang, Local existence with mild regularity for the Boltzmann equation, Kinet. Relat. Models, 6 (2013), 1011-1041.  doi: 10.3934/krm.2013.6.1011.

[12]

R. Alexandre and C. Villani, On the Boltzmann equation for long-range interactions, Comm. Pure Appl. Math., 55 (2002), 30-70.  doi: 10.1002/cpa.10012.

[13]

R. Alexandre and C. Villani, On the Landau approximation in plasma physics, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 21 (2004), 61-95.  doi: 10.1016/j.anihpc.2002.12.001.

[14]

R. Alonso, Y. Morimoto, W. Sun and T. Yang, Non-cutoff Boltzmann equation with polynomial decay perturbation, preprint, 2019, arXiv: 1812.05299.

[15]

T. Carleman, Sur la théorie de l'équation intégrodifférentielle de Boltzmann, Acta Math., 60 (1933), 91-146.  doi: 10.1007/BF02398270.

[16]

S. Chaturvedi, Local existence for the Landau equation with hard potentials, preprint, 2019, arXiv: 1910.11866.

[17]

S. Chaturvedi, Stability of vacuum for the Landau equation with hard potentials, preprint, 2020, arXiv: 2001.07208.

[18]

Y. Chen and L.-B. He, Smoothing estimates for Boltzmann equation with full-range interactions: Spatially homogeneous case, Arch. Ration. Mech. Anal., 201 (2011), 501-548.  doi: 10.1007/s00205-010-0393-8.

[19]

L. Desvillettes and C. Mouhot, About $L^p$ estimates for the spatially homogeneous Boltzmann equation, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 22 (2005), 127-142.  doi: 10.1016/j.anihpc.2004.03.002.

[20]

L. Desvillettes and C. Mouhot, Stability and uniqueness for the spatially homogeneous Boltzmann equation with long-range interactions, Arch. Ration. Mech. Anal., 193 (2009), 227-253.  doi: 10.1007/s00205-009-0233-x.

[21]

L. Desvillettes and B. Wennberg, Smoothness of the solution of the spatially homogeneous Boltzmann equation without cutoff, Comm. Part. Diff. Eq., 29 (2004), 133-155.  doi: 10.1081/PDE-120028847.

[22]

R. J. DiPerna and P. L. Lions, On the Cauchy Problem for Boltzmann equations: Global existence and weak stability, Ann. Math., 130 (1989), 321-366.  doi: 10.2307/1971423.

[23]

A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964.

[24]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2001. doi: 10.1007/978-3-642-61798-0.

[25]

T. Goudon, On Boltzmann equations and Fokker-Planck asymptotics: Influence of grazing collisions, J. Statist. Phys., 89 (1997), 751-776.  doi: 10.1007/BF02765543.

[26]

P. T. Gressman and R. M. Strain, Global classical solutions of the Boltzmann equation without angular cut-off, J. Amer. Math. Soc., 24 (2011), 771-847.  doi: 10.1090/S0894-0347-2011-00697-8.

[27]

L.-B. He and X. Yang, Well-posedness and asymptotics of grazing collisions limit of Boltzmann equation with Coulomb interaction, SIAM J. Math. Anal., 46 (2014), 4104-4165.  doi: 10.1137/140965983.

[28]

L.-B. He and J.-C. Jiang, On the global dynamics of the inhomogeneous Boltzmann equations without angular cutoff: Hard potentials and Maxwellian molecules, preprint, 2017, arXiv: 1710.00315.

[29]

C. HendersonS. Snelson and A. Tarfulea, Local existence, lower mass bounds, and a new continuation criterion for the Landau equation, J. Differential Equations, 266 (2019), 1536-1577.  doi: 10.1016/j.jde.2018.08.005.

[30]

C. Henderson, S. Snelson, and A. Tarfulea, Local solutions of the Landau equation with rough, slowly decaying initial data, preprint, 2019, arXiv: 1909.05914.

[31]

F. Hérau, D. Tonon, and I. Tristani, Regularization estimates and Cauchy theory for inhomogeneous Boltzmann equation for hard potentials without cut-off, arXiv: 1710.01098.

[32]

C. Imbert, C. Mouhot, and L. Silvestre, Decay estimates for large velocities in the Boltzmann equation without cut-off, preprint, 2019, arXiv: 1804.06135.

[33]

C. Imbert and L. Silvestre, The Schauder estimate for kinetic integral equations, preprint, 2019, arXiv: 1812.11870.

[34]

C. Imbert and L. Silvestre, Global regularity estimates for the Boltzmann equation without cut-off, preprint, 2019, arXiv: 1909.12729.

[35]

C. Imbert and L. Silvestre, Weak Harnack inequality for the Boltzmann equation without cut-off, J. Eur. Math. Soc., 22 (2020), 507-592.  doi: 10.4171/JEMS/928.

[36]

X. Lu and C. Mouhot, On measure solutions of the Boltzmann equation, part I: Moment production and stability estimates, J. Differential Equations, 252 (2012), 3305-3363.  doi: 10.1016/j.jde.2011.10.021.

[37]

J. Luk, Stability of Vacuum for the Landau Equation with Moderately Soft Potentials, Ann. PDE, 5 (2019), 101 pp. doi: 10.1007/s40818-019-0067-2.

[38]

Y. MorimotoS. Wang and T. Yang, Measure valued solutions to the spatially homogeneous Boltzmann equation without angular cutoff, J. Stat. Phys., 165 (2016), 866-906.  doi: 10.1007/s10955-016-1655-0.

[39]

Y. Morimoto and T. Yang, Local existence of polynomial decay solutions to the Boltzmann equation for soft potentials, Anal. Appl. (Singap.), 13 (2015), 663-683.  doi: 10.1142/S0219530514500079.

[40]

L. Silvestre, A new regularization mechanism for the Boltzmann equation without cut-off, Comm. Math. Phys., 348 (2016), 69-100.  doi: 10.1007/s00220-016-2757-x.

[41]

C. Villani, On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations, Arch. Ration. Mech. Anal., 143 (1998), 273-307.  doi: 10.1007/s002050050106.

[42]

C. Villani, A review of mathematical topics in collisional kinetic theory, in Handbook of Mathematical Fluid Dynamics, Vol. I, North-Holland, Amsterdam, (2002), 71–305. doi: 10.1016/S1874-5792(02)80004-0.

[1]

Lassaad Aloui, Slim Tayachi. Local well-posedness for the inhomogeneous nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems, 2021, 41 (11) : 5409-5437. doi: 10.3934/dcds.2021082

[2]

Boris Kolev. Local well-posedness of the EPDiff equation: A survey. Journal of Geometric Mechanics, 2017, 9 (2) : 167-189. doi: 10.3934/jgm.2017007

[3]

Hartmut Pecher. Local well-posedness for the nonlinear Dirac equation in two space dimensions. Communications on Pure and Applied Analysis, 2014, 13 (2) : 673-685. doi: 10.3934/cpaa.2014.13.673

[4]

Jae Min Lee, Stephen C. Preston. Local well-posedness of the Camassa-Holm equation on the real line. Discrete and Continuous Dynamical Systems, 2017, 37 (6) : 3285-3299. doi: 10.3934/dcds.2017139

[5]

Yongye Zhao, Yongsheng Li, Wei Yan. Local Well-posedness and Persistence Property for the Generalized Novikov Equation. Discrete and Continuous Dynamical Systems, 2014, 34 (2) : 803-820. doi: 10.3934/dcds.2014.34.803

[6]

Andrea Giorgini. On the Swift-Hohenberg equation with slow and fast dynamics: well-posedness and long-time behavior. Communications on Pure and Applied Analysis, 2016, 15 (1) : 219-241. doi: 10.3934/cpaa.2016.15.219

[7]

Chao Yang. Sharp condition of global well-posedness for inhomogeneous nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : 4631-4642. doi: 10.3934/dcdss.2021136

[8]

Tong Li. Well-posedness theory of an inhomogeneous traffic flow model. Discrete and Continuous Dynamical Systems - B, 2002, 2 (3) : 401-414. doi: 10.3934/dcdsb.2002.2.401

[9]

Daoyuan Fang, Ruizhao Zi. On the well-posedness of inhomogeneous hyperdissipative Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3517-3541. doi: 10.3934/dcds.2013.33.3517

[10]

Hongjie Dong, Dapeng Du. Global well-posedness and a decay estimate for the critical dissipative quasi-geostrophic equation in the whole space. Discrete and Continuous Dynamical Systems, 2008, 21 (4) : 1095-1101. doi: 10.3934/dcds.2008.21.1095

[11]

Benjamin Dodson. Global well-posedness and scattering for the defocusing, cubic nonlinear Schrödinger equation when $n = 3$ via a linear-nonlinear decomposition. Discrete and Continuous Dynamical Systems, 2013, 33 (5) : 1905-1926. doi: 10.3934/dcds.2013.33.1905

[12]

Tristan Roy. Adapted linear-nonlinear decomposition and global well-posedness for solutions to the defocusing cubic wave equation on $\mathbb{R}^{3}$. Discrete and Continuous Dynamical Systems, 2009, 24 (4) : 1307-1323. doi: 10.3934/dcds.2009.24.1307

[13]

Kenji Nakanishi, Hideo Takaoka, Yoshio Tsutsumi. Local well-posedness in low regularity of the MKDV equation with periodic boundary condition. Discrete and Continuous Dynamical Systems, 2010, 28 (4) : 1635-1654. doi: 10.3934/dcds.2010.28.1635

[14]

Sergey Zelik, Jon Pennant. Global well-posedness in uniformly local spaces for the Cahn-Hilliard equation in $\mathbb{R}^3$. Communications on Pure and Applied Analysis, 2013, 12 (1) : 461-480. doi: 10.3934/cpaa.2013.12.461

[15]

Luiz Gustavo Farah. Local solutions in Sobolev spaces and unconditional well-posedness for the generalized Boussinesq equation. Communications on Pure and Applied Analysis, 2009, 8 (5) : 1521-1539. doi: 10.3934/cpaa.2009.8.1521

[16]

Luigi Forcella, Kazumasa Fujiwara, Vladimir Georgiev, Tohru Ozawa. Local well-posedness and blow-up for the half Ginzburg-Landau-Kuramoto equation with rough coefficients and potential. Discrete and Continuous Dynamical Systems, 2019, 39 (5) : 2661-2678. doi: 10.3934/dcds.2019111

[17]

Nikolaos Bournaveas. Local well-posedness for a nonlinear dirac equation in spaces of almost critical dimension. Discrete and Continuous Dynamical Systems, 2008, 20 (3) : 605-616. doi: 10.3934/dcds.2008.20.605

[18]

Seckin Demirbas. Local well-posedness for 2-D Schrödinger equation on irrational tori and bounds on Sobolev norms. Communications on Pure and Applied Analysis, 2017, 16 (5) : 1517-1530. doi: 10.3934/cpaa.2017072

[19]

Nobu Kishimoto. Local well-posedness for the Cauchy problem of the quadratic Schrödinger equation with nonlinearity $\bar u^2$. Communications on Pure and Applied Analysis, 2008, 7 (5) : 1123-1143. doi: 10.3934/cpaa.2008.7.1123

[20]

Qifan Li. Local well-posedness for the periodic Korteweg-de Vries equation in analytic Gevrey classes. Communications on Pure and Applied Analysis, 2012, 11 (3) : 1097-1109. doi: 10.3934/cpaa.2012.11.1097

2021 Impact Factor: 1.398

Metrics

  • PDF downloads (256)
  • HTML views (121)
  • Cited by (2)

[Back to Top]