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doi: 10.3934/eect.2022002
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Local well-posedness of the coupled KdV-KdV systems on $ \mathbb{R} $

Xin Yang 1,, and  Bing-Yu Zhang 2,

1. 

Department of Mathematics, University of California, Riverside, Riverside, CA 92521, USA
2. 

Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH 45221, USA

*Corresponding author: Xin Yang

Received  August 2021 Revised  November 2021 Early access January 2022

Inspired by the recent successful completion of the study of the well-posedness theory for the Cauchy problem of the Korteweg-de Vries (KdV) equation
$ u_t +uu_x +u_{xxx} = 0, \quad \left. u \right |_{t = 0} = u_{0} $
in the space
$ H^{s} (\mathbb{R}) $
 (or
$ H^{s} (\mathbb{T}) $
), we study the well-posedness of the Cauchy problem for a class of coupled KdV-KdV (cKdV) systems
$ \left\{\begin{array}{rcl} u_t+a_{1}u_{xxx} & = & c_{11}uu_x+c_{12}vv_x+d_{11}u_{x}v+d_{12}uv_{x}, \\ v_t+a_{2}v_{xxx}& = & c_{21}uu_x+c_{22}vv_x +d_{21}u_{x}v+d_{22}uv_{x}, \\ \left. (u, v)\right |_{t = 0} & = & (u_{0}, v_{0}) \end{array}\right. $
in the space
$ \mathcal{H}^s (\mathbb{R}) : = H^s (\mathbb{R})\times H^s (\mathbb{R}) $
. Typical examples include the Gear-Grimshaw system, the Hirota-Satsuma system and the Majda-Biello system, to name a few.
In this paper we look for those values of
$ s\in \mathbb{R} $
 for which the cKdV systems are well-posed in
$ \mathcal{H}^s ( \mathbb {R}) $
. The key ingredients in the proofs are the bilinear estimates in both divergence and non-divergence forms under the Fourier restriction space norms. Sharp results are established for all four types of the bilinear estimates that are associated to the cKdV systems. In contrast to the lone critical index
$ -\frac{3}{4} $
 for the single KdV equation, the critical indexes for the cKdV systems are
$ -\frac{13}{12} $
,
$ -\frac{3}{4} $
,
$ 0 $
 and
$ \frac{3}{4} $
.
As a result, the cKdV systems are classified into four classes, each of which corresponds to a unique index
$ s^{*}\in\{-\frac{13}{12}, \, -\frac{3}{4}, \, 0, \, \frac{3}{4}\} $
 such that any system in this class is locally analytically well-posed if
$ s>s^{*} $
 while the bilinear estimate fails if
$ s<s^{*} $
.
Keywords: KdV-KdV systems, Gear-Grimshaw system, Hirota-Satsuma system, Majda-Biello system, Local well-posedness, Fourier restriction space, Bilinear estimates.
Mathematics Subject Classification: Primary: 35Q53; 35E15; 35G55; 35L56; 35D30.
Citation: Xin Yang, Bing-Yu Zhang. Local well-posedness of the coupled KdV-KdV systems on $ \mathbb{R} $. Evolution Equations and Control Theory, doi: 10.3934/eect.2022002
References:
[1]

B. Alvarez and X. Carvajal, On the local well-posedness for some systems of coupled KdV equations, Nonlinear Anal., 69 (2008), 692-715.  doi: 10.1016/j.na.2007.06.009.

[2]

J. M. Ash, J. Cohen and G. Wang, On strongly interacting internal solitary waves, J. Fourier Anal. Appl., 2 (1996), 507-517.  doi: 10.1007/s00041-001-4041-4.

[3]

D. Bekiranov, T. Ogawa and G. Ponce, Weak solvability and well-posedness of a coupled Schrödinger-Korteweg de Vries equation for capillary-gravity wave interactions, Proc. Amer. Math. Soc., 125 (1997), 2907-2919.  doi: 10.1090/S0002-9939-97-03941-5.

[4]

J. L. Bona, G. Ponce, J.-C. Saut and M. M. Tom, A model system for strong interaction between internal solitary waves, Comm. Math. Phys., 143 (1992), 287-313. 

[5]

J. L. Bona and R. Scott, Solutions of the Korteweg-de Vries equation in fractional order Sobolev spaces, Duke Math. J., 43 (1976), 87-99. 

[6]

J. L. Bona and R. Smith, The initial-value problem for the Korteweg-de Vries equation, Philos. Trans. Roy. Soc. London Ser. A, 278 (1975), 555-601.  doi: 10.1098/rsta.1975.0035.

[7]

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

[8]

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. II. The KdV-equation, Geom. Funct. Anal., 3 (1993), 209-262.  doi: 10.1007/BF01895688.

[9]

E. Cerpa and E. Crépeau, Boundary controllability for the nonlinear Korteweg-de Vries equation on any critical domain, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 457-475.  doi: 10.1016/j.anihpc.2007.11.003.

[10]

M. Christ, J. Colliander and T. Tao, Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations, Amer. J. Math., 125 (2003), 1235-1293.  doi: 10.1353/ajm.2003.0040.

[11]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified KdV on $\Bbb R$ and $\Bbb T$, J. Amer. Math. Soc., 16 (2003), 705-749.  doi: 10.1090/S0894-0347-03-00421-1.

[12]

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.

[13]

J.-M. Coron and E. Crépeau, Exact boundary controllability of a nonlinear KdV equation with a critical length, J. Eur. Math. Soc., 6 (2004), 367-398. 

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X. Feng, Global well-posedness of the initial value problem for the Hirota-Satsuma system, Manuscripta Math., 84 (1994), 361-378.  doi: 10.1007/BF02567462.

[15]

J. A. Gear and R. Grimshaw, Weak and strong interactions between internal solitary waves, Stud. Appl. Math., 70 (1984), 235-258.  doi: 10.1002/sapm1984703235.

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Z. Guo, Global well-posedness of Korteweg-de Vries equation in $H^{-3/4}(\Bbb R)$, J. Math. Pures Appl., 91 (2009), 583-597.  doi: 10.1016/j.matpur.2009.01.012.

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R. Hirota and J. Satsuma, Soliton solutions of a coupled Korteweg-de Vries equation, Phys. Lett. A, 85 (1981), 407-408.  doi: 10.1016/0375-9601(81)90423-0.

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[19]

T. Kappeler and P. Topalov, Global wellposedness of KdV in $H^{-1}(\Bbb T, \Bbb R)$, Duke Math. J., 135 (2006), 327-360.  doi: 10.1215/S0012-7094-06-13524-X.

[20]

T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations, Lecture Notes in Math., 448 (1975), 25-70. 

[21]

T. Kato, On the Korteweg-de\thinspace Vries equation, Manuscripta Math., 28 (1979), 89-99.  doi: 10.1007/BF01647967.

[22]

T. Kato, The Cauchy problem for the Korteweg-de Vries equation, In Nonlinear Partial Differential Equations and Their Applications, Res. Notes in Math., 53 (1981), 293–307.

[23]

T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation, Studies in applied mathematics, 8 (1983), 93-128. 

[24]

C. E. Kenig, G. Ponce and L. Vega, On the (generalized) Korteweg-de Vries equation, Duke Math. J., 59 (1989), 585-610.  doi: 10.1215/S0012-7094-89-05927-9.

[25]

C. E. Kenig, G. Ponce and L. Vega, Oscillatory integrals and regularity of dispersive equations, Indiana Univ. Math. J., 40 (1991), 33-69.  doi: 10.1512/iumj.1991.40.40003.

[26]

C. E. Kenig, G. Ponce and L. Vega, Well-posedness of the initial value problem for the Korteweg-de Vries equation, J. Amer. Math. Soc., 4 (1991), 323-347.  doi: 10.1090/S0894-0347-1991-1086966-0.

[27]

C. E. Kenig, G. Ponce and L. Vega, The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices, Duke Math. J., 71 (1993), 1-21.  doi: 10.1215/S0012-7094-93-07101-3.

[28]

C. E. Kenig, G. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math., 46 (1993), 527-620.  doi: 10.1002/cpa.3160460405.

[29]

C. E. Kenig, G. Ponce and L. Vega, A bilinear estimate with applications to the KdV equation, J. Amer. Math. Soc., 9 (1996), 573-603.  doi: 10.1090/S0894-0347-96-00200-7.

[30]

R. Killip and M. Vişan, KdV is well-posed in $H^{-1}$, Ann. of Math., 190 (2019), 249-305.  doi: 10.4007/annals.2019.190.1.4.

[31]

N. Kishimoto, Well-posedness of the Cauchy problem for the Korteweg-de Vries equation at the critical regularity, Differential Integral Equations, 22 (2009), 447-464. 

[32]

C. Laurent, L. Rosier and B.-Y. Zhang, Control and stabilization of the Korteweg-de Vries equation on a periodic Domain, Comm. Partial Differential Equations, 35 (2010), 707-744.  doi: 10.1080/03605300903585336.

[33]

F. Linares and M. Panthee, On the Cauchy problem for a coupled system of KdV equations, Commun. Pure Appl. Anal., 3 (2004), 417-431.  doi: 10.3934/cpaa.2004.3.417.

[34]

A. J. Majda and J. A. Biello, The nonlinear interaction of barotropic and equatorial baroclinic Rossby waves, J. Atmospheric Sci., 60 (2003), 1809-1821.  doi: 10.1175/1520-0469(2003)060<1809:TNIOBA>2.0.CO;2.

[35]

L. Molinet, A note on ill posedness for the KdV equation, Differential Integral Equations, 24 (2011), 759-765. 

[36]

L. Molinet, Sharp ill-posedness results for the KdV and mKdV equations on the torus, Adv. Math., 230 (2012), 1895-1930.  doi: 10.1016/j.aim.2012.03.026.

[37]

T. Oh, Diophantine conditions in global well-posedness for coupled KdV-type systems, Electron. J. Differential Equations, (2009), 48 pp.

[38]

T. Oh, Diophantine conditions in well-posedness theory of coupled KdV-type systems: Local theory, Int. Math. Res. Not. IMRN, 18 (2009), 3516-3556.  doi: 10.1093/imrn/rnp063.

[39]

L. Rosier, Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain, ESAIM Control Optim. Cal. Var., 2 (1997), 33-55.  doi: 10.1051/cocv:1997102.

[40]

L. Rosier and B.-Y. Zhang, Control and stabilization of the korteweg-de Vries equation: Recent progress, J. Syst. Sci. Complex, 22 (2009), 647-682.  doi: 10.1007/s11424-009-9194-2.

[41]

D.-L. Russell and B.-Y. Zhang, Exact controllability and stabilizability of the Korteweg-de Vries equation, Trans. Amer. Math. Soc., 348 (1996), 3643-3672.  doi: 10.1090/S0002-9947-96-01672-8.

[42]

J. C. Saut and R. Temam, Remarks on the Korteweg-de Vries equation, Israel J. Math., 24 (1976), 78-87.  doi: 10.1007/BF02761431.

[43]

J.-C. Saut and N. Tzvetkov, On a model system for the oblique interaction of internal gravity waves, M2AN Math. Model. Numer. Anal., Special issue for R. Temam's 60th birthday 34 (2000), 501–523. doi: 10.1051/m2an:2000153.

[44]

A. Sjöberg, On the Korteweg-de Vries equation: Existence and uniqueness, Department of Computer Sciences, Uppsala University, Uppsala, Sweden, 1967.

[45]

A. Sjöberg, On the Korteweg-de Vries equation: Existence and uniqueness, J. Math. Anal. Appl., 29 (1970), 569-579.  doi: 10.1016/0022-247X(70)90068-5.

[46]

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

[47]

T. Tao, Multilinear weighted convolution of $L^2$-functions, and applications to nonlinear dispersive equations, Amer. J. Math., 123 (2001), 839-908.  doi: 10.1353/ajm.2001.0035.

[48]

L. Tartar, Interpolation non linéaire et régularité, J. Functional Analysis, 9 (1972), 469-489.  doi: 10.1016/0022-1236(72)90022-5.

[49]

R. Temam, Sur un problème non linéaire, J. Math. Pures Appl., 48 (1969), 159-172. 

[50]

M. Tsutsumi and T. Mukasa, Parabolic regularizations for the generalized Korteweg-de Vries equation, Funkcial. Ekvac., 14 (1971), 89-110. 

[51]

M. Tsutsumi, T. Mukasa and R. Iino, On the generalized Korteweg-de Vries equation, Proc. Japan Acad., 46 (1970), 921-925. 

[52]

B.-Y. Zhang, Analyticity of solutions of the generalized Kortweg-de Vries equation with respect to their initial values, SIAM J. Math. Anal., 26 (1995), 1488-1513.  doi: 10.1137/S0036141093242600.

[53]

B.-Y. Zhang, A remark on the Cauchy problem for the Korteweg-de Vries equation on a periodic domain, Differential Integral Equations, 8 (1995), 1191-1204. 

[54]

B.-Y. Zhang, Taylor series expansion for solutions of the Korteweg-de Vries equation with respect to their initial values, J. Funct. Anal., 129 (1995), 293-324.  doi: 10.1006/jfan.1995.1052.

[55]

B.-Y. Zhang, Exact boundary controllability of the Korteweg-de Vries equation, SIAM J. Cont. Optim., 37 (1999), 543-565.  doi: 10.1137/S0363012997327501.

[56]

B.-Y. Zhang, Well-posedness and control of the Korteweg-de Vries equation on a bounded domain, Fifth International Congress of Chinese Mathematicians, AMS/IP Stud. Adv. Math. AMS, Providence, RI, 51 (2012), 931–956.

show all references

References:
[1]

B. Alvarez and X. Carvajal, On the local well-posedness for some systems of coupled KdV equations, Nonlinear Anal., 69 (2008), 692-715.  doi: 10.1016/j.na.2007.06.009.

[2]

J. M. Ash, J. Cohen and G. Wang, On strongly interacting internal solitary waves, J. Fourier Anal. Appl., 2 (1996), 507-517.  doi: 10.1007/s00041-001-4041-4.

[3]

D. Bekiranov, T. Ogawa and G. Ponce, Weak solvability and well-posedness of a coupled Schrödinger-Korteweg de Vries equation for capillary-gravity wave interactions, Proc. Amer. Math. Soc., 125 (1997), 2907-2919.  doi: 10.1090/S0002-9939-97-03941-5.

[4]

J. L. Bona, G. Ponce, J.-C. Saut and M. M. Tom, A model system for strong interaction between internal solitary waves, Comm. Math. Phys., 143 (1992), 287-313. 

[5]

J. L. Bona and R. Scott, Solutions of the Korteweg-de Vries equation in fractional order Sobolev spaces, Duke Math. J., 43 (1976), 87-99. 

[6]

J. L. Bona and R. Smith, The initial-value problem for the Korteweg-de Vries equation, Philos. Trans. Roy. Soc. London Ser. A, 278 (1975), 555-601.  doi: 10.1098/rsta.1975.0035.

[7]

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

[8]

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. II. The KdV-equation, Geom. Funct. Anal., 3 (1993), 209-262.  doi: 10.1007/BF01895688.

[9]

E. Cerpa and E. Crépeau, Boundary controllability for the nonlinear Korteweg-de Vries equation on any critical domain, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 457-475.  doi: 10.1016/j.anihpc.2007.11.003.

[10]

M. Christ, J. Colliander and T. Tao, Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations, Amer. J. Math., 125 (2003), 1235-1293.  doi: 10.1353/ajm.2003.0040.

[11]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified KdV on $\Bbb R$ and $\Bbb T$, J. Amer. Math. Soc., 16 (2003), 705-749.  doi: 10.1090/S0894-0347-03-00421-1.

[12]

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.

[13]

J.-M. Coron and E. Crépeau, Exact boundary controllability of a nonlinear KdV equation with a critical length, J. Eur. Math. Soc., 6 (2004), 367-398. 

[14]

X. Feng, Global well-posedness of the initial value problem for the Hirota-Satsuma system, Manuscripta Math., 84 (1994), 361-378.  doi: 10.1007/BF02567462.

[15]

J. A. Gear and R. Grimshaw, Weak and strong interactions between internal solitary waves, Stud. Appl. Math., 70 (1984), 235-258.  doi: 10.1002/sapm1984703235.

[16]

Z. Guo, Global well-posedness of Korteweg-de Vries equation in $H^{-3/4}(\Bbb R)$, J. Math. Pures Appl., 91 (2009), 583-597.  doi: 10.1016/j.matpur.2009.01.012.

[17]

R. Hirota and J. Satsuma, Soliton solutions of a coupled Korteweg-de Vries equation, Phys. Lett. A, 85 (1981), 407-408.  doi: 10.1016/0375-9601(81)90423-0.

[18]

Y. Kametaka, Korteweg -de vries equation, i, ii, iii, iv, Proc. Japan Acad., 45 (1969), 661-665. 

[19]

T. Kappeler and P. Topalov, Global wellposedness of KdV in $H^{-1}(\Bbb T, \Bbb R)$, Duke Math. J., 135 (2006), 327-360.  doi: 10.1215/S0012-7094-06-13524-X.

[20]

T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations, Lecture Notes in Math., 448 (1975), 25-70. 

[21]

T. Kato, On the Korteweg-de\thinspace Vries equation, Manuscripta Math., 28 (1979), 89-99.  doi: 10.1007/BF01647967.

[22]

T. Kato, The Cauchy problem for the Korteweg-de Vries equation, In Nonlinear Partial Differential Equations and Their Applications, Res. Notes in Math., 53 (1981), 293–307.

[23]

T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation, Studies in applied mathematics, 8 (1983), 93-128. 

[24]

C. E. Kenig, G. Ponce and L. Vega, On the (generalized) Korteweg-de Vries equation, Duke Math. J., 59 (1989), 585-610.  doi: 10.1215/S0012-7094-89-05927-9.

[25]

C. E. Kenig, G. Ponce and L. Vega, Oscillatory integrals and regularity of dispersive equations, Indiana Univ. Math. J., 40 (1991), 33-69.  doi: 10.1512/iumj.1991.40.40003.

[26]

C. E. Kenig, G. Ponce and L. Vega, Well-posedness of the initial value problem for the Korteweg-de Vries equation, J. Amer. Math. Soc., 4 (1991), 323-347.  doi: 10.1090/S0894-0347-1991-1086966-0.

[27]

C. E. Kenig, G. Ponce and L. Vega, The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices, Duke Math. J., 71 (1993), 1-21.  doi: 10.1215/S0012-7094-93-07101-3.

[28]

C. E. Kenig, G. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math., 46 (1993), 527-620.  doi: 10.1002/cpa.3160460405.

[29]

C. E. Kenig, G. Ponce and L. Vega, A bilinear estimate with applications to the KdV equation, J. Amer. Math. Soc., 9 (1996), 573-603.  doi: 10.1090/S0894-0347-96-00200-7.

[30]

R. Killip and M. Vişan, KdV is well-posed in $H^{-1}$, Ann. of Math., 190 (2019), 249-305.  doi: 10.4007/annals.2019.190.1.4.

[31]

N. Kishimoto, Well-posedness of the Cauchy problem for the Korteweg-de Vries equation at the critical regularity, Differential Integral Equations, 22 (2009), 447-464. 

[32]

C. Laurent, L. Rosier and B.-Y. Zhang, Control and stabilization of the Korteweg-de Vries equation on a periodic Domain, Comm. Partial Differential Equations, 35 (2010), 707-744.  doi: 10.1080/03605300903585336.

[33]

F. Linares and M. Panthee, On the Cauchy problem for a coupled system of KdV equations, Commun. Pure Appl. Anal., 3 (2004), 417-431.  doi: 10.3934/cpaa.2004.3.417.

[34]

A. J. Majda and J. A. Biello, The nonlinear interaction of barotropic and equatorial baroclinic Rossby waves, J. Atmospheric Sci., 60 (2003), 1809-1821.  doi: 10.1175/1520-0469(2003)060<1809:TNIOBA>2.0.CO;2.

[35]

L. Molinet, A note on ill posedness for the KdV equation, Differential Integral Equations, 24 (2011), 759-765. 

[36]

L. Molinet, Sharp ill-posedness results for the KdV and mKdV equations on the torus, Adv. Math., 230 (2012), 1895-1930.  doi: 10.1016/j.aim.2012.03.026.

[37]

T. Oh, Diophantine conditions in global well-posedness for coupled KdV-type systems, Electron. J. Differential Equations, (2009), 48 pp.

[38]

T. Oh, Diophantine conditions in well-posedness theory of coupled KdV-type systems: Local theory, Int. Math. Res. Not. IMRN, 18 (2009), 3516-3556.  doi: 10.1093/imrn/rnp063.

[39]

L. Rosier, Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain, ESAIM Control Optim. Cal. Var., 2 (1997), 33-55.  doi: 10.1051/cocv:1997102.

[40]

L. Rosier and B.-Y. Zhang, Control and stabilization of the korteweg-de Vries equation: Recent progress, J. Syst. Sci. Complex, 22 (2009), 647-682.  doi: 10.1007/s11424-009-9194-2.

[41]

D.-L. Russell and B.-Y. Zhang, Exact controllability and stabilizability of the Korteweg-de Vries equation, Trans. Amer. Math. Soc., 348 (1996), 3643-3672.  doi: 10.1090/S0002-9947-96-01672-8.

[42]

J. C. Saut and R. Temam, Remarks on the Korteweg-de Vries equation, Israel J. Math., 24 (1976), 78-87.  doi: 10.1007/BF02761431.

[43]

J.-C. Saut and N. Tzvetkov, On a model system for the oblique interaction of internal gravity waves, M2AN Math. Model. Numer. Anal., Special issue for R. Temam's 60th birthday 34 (2000), 501–523. doi: 10.1051/m2an:2000153.

[44]

A. Sjöberg, On the Korteweg-de Vries equation: Existence and uniqueness, Department of Computer Sciences, Uppsala University, Uppsala, Sweden, 1967.

[45]

A. Sjöberg, On the Korteweg-de Vries equation: Existence and uniqueness, J. Math. Anal. Appl., 29 (1970), 569-579.  doi: 10.1016/0022-247X(70)90068-5.

[46]

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

[47]

T. Tao, Multilinear weighted convolution of $L^2$-functions, and applications to nonlinear dispersive equations, Amer. J. Math., 123 (2001), 839-908.  doi: 10.1353/ajm.2001.0035.

[48]

L. Tartar, Interpolation non linéaire et régularité, J. Functional Analysis, 9 (1972), 469-489.  doi: 10.1016/0022-1236(72)90022-5.

[49]

R. Temam, Sur un problème non linéaire, J. Math. Pures Appl., 48 (1969), 159-172. 

[50]

M. Tsutsumi and T. Mukasa, Parabolic regularizations for the generalized Korteweg-de Vries equation, Funkcial. Ekvac., 14 (1971), 89-110. 

[51]

M. Tsutsumi, T. Mukasa and R. Iino, On the generalized Korteweg-de Vries equation, Proc. Japan Acad., 46 (1970), 921-925. 

[52]

B.-Y. Zhang, Analyticity of solutions of the generalized Kortweg-de Vries equation with respect to their initial values, SIAM J. Math. Anal., 26 (1995), 1488-1513.  doi: 10.1137/S0036141093242600.

[53]

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Figure 1.  Range of $ s $ and $ b $ when $ s<-\frac{3}{4} $
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Table 1.  Main Results
Case $ r=\frac{a_{2}}{a_{1}} $ Coefficients $ b_{ij} $, $ c_{ij} $ and $ d_{ij} $ $ s $
(1) $r <0$ $(c_{ij})=0$, $d_{11}=d_{12}$ and $d_{21}=d_{22}$
Otherwise 		$s\geq -\frac{13}{12}$
$s>-\frac{3}{4}$
(2) $0 <r <\frac{1}{4}$ $c_{12}=d_{21}=d_{22}=0$
Otherwise 		$s>-\frac{3}{4}$
$s\geq 0$
(3) $r=\frac{1}{4}$ $c_{21}=d_{11}=d_{12}=0$
Otherwise 		$s\geq 0$
$s\geq\frac{3}{4}$
(4) $\frac{1}{4} <r <1$ arbitrary $s\geq 0$
(5) $r=1$ $b_{12}=b_{21}=0$, $d_{11}=d_{12}$ and $d_{21}=d_{22}$
$b_{12}=b_{21}=0$, $d_{11}\neq d_{12}$ or $d_{21}\neq d_{22}$ 		$s>-\frac{3}{4}$
$s>0$
(6) $1 <r<4$ arbitrary $s\geq 0$
(7) $r=4$ $c_{12}=d_{21}=d_{22}=0$
Otherwise 		$s\geq 0$
$s\geq\frac{3}{4}$
(8) $r>4$ $c_{21}=d_{11}=d_{12}=0$
Otherwise 		$s>-\frac{3}{4}$
$s\geq 0$
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Case $ r=\frac{a_{2}}{a_{1}} $ Coefficients $ b_{ij} $, $ c_{ij} $ and $ d_{ij} $ $ s $
(1) $r <0$ $(c_{ij})=0$, $d_{11}=d_{12}$ and $d_{21}=d_{22}$
Otherwise 		$s\geq -\frac{13}{12}$
$s>-\frac{3}{4}$
(2) $0 <r <\frac{1}{4}$ $c_{12}=d_{21}=d_{22}=0$
Otherwise 		$s>-\frac{3}{4}$
$s\geq 0$
(3) $r=\frac{1}{4}$ $c_{21}=d_{11}=d_{12}=0$
Otherwise 		$s\geq 0$
$s\geq\frac{3}{4}$
(4) $\frac{1}{4} <r <1$ arbitrary $s\geq 0$
(5) $r=1$ $b_{12}=b_{21}=0$, $d_{11}=d_{12}$ and $d_{21}=d_{22}$
$b_{12}=b_{21}=0$, $d_{11}\neq d_{12}$ or $d_{21}\neq d_{22}$ 		$s>-\frac{3}{4}$
$s>0$
(6) $1 <r<4$ arbitrary $s\geq 0$
(7) $r=4$ $c_{12}=d_{21}=d_{22}=0$
Otherwise 		$s\geq 0$
$s\geq\frac{3}{4}$
(8) $r>4$ $c_{21}=d_{11}=d_{12}=0$
Otherwise 		$s>-\frac{3}{4}$
$s\geq 0$
Table 2.  LWP Results
Case Coefficient $ a_2 $ $ s $
(1) $ a_2\in(-\infty, 0)\cup \{1\} \cup \{4, \infty\} $ $ s>-\frac34 $
(2) $ a_2\in(0, 1)\cup(1, 4) $ $ s\geq 0 $
(3) $ a_2=4 $ $ s\geq\frac{3}{4} $
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Case Coefficient $ a_2 $ $ s $
(1) $ a_2\in(-\infty, 0)\cup \{1\} \cup \{4, \infty\} $ $ s>-\frac34 $
(2) $ a_2\in(0, 1)\cup(1, 4) $ $ s\geq 0 $
(3) $ a_2=4 $ $ s\geq\frac{3}{4} $
Table 3.  GWP Results
Case Coefficient $ a_2 $ $ s $
(1) $ a_2=1 $ $ s>-\frac34 $
(2) $ a_2\not\in\{1, 4\} $ $ s\geq 0 $
(3) $ a_2=4 $ $ s\geq1 $
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Case Coefficient $ a_2 $ $ s $
(1) $ a_2=1 $ $ s>-\frac34 $
(2) $ a_2\not\in\{1, 4\} $ $ s\geq 0 $
(3) $ a_2=4 $ $ s\geq1 $
Table 4.  LWP Results
Case Coefficients $ a_1 $ and $ c_{12} $ $ s $
(1) $ a_1\in(-\infty, 0)\cup(0, \frac14) $ $ s>-\frac34 $
(2) $ a_1\in(\frac14, 1)\cup(1, \infty) $ $ s\geq 0 $
(3) $ a_1=1 $ $ s>0 $
(4) $ a_1=\frac14 $ $ s\geq\frac{3}{4} $
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Case Coefficients $ a_1 $ and $ c_{12} $ $ s $
(1) $ a_1\in(-\infty, 0)\cup(0, \frac14) $ $ s>-\frac34 $
(2) $ a_1\in(\frac14, 1)\cup(1, \infty) $ $ s\geq 0 $
(3) $ a_1=1 $ $ s>0 $
(4) $ a_1=\frac14 $ $ s\geq\frac{3}{4} $
Table 5.  GWP Results
Case Coefficients $ a_1 $ and $ c_{12} $ $ s $
(1) $ a_1\not\in\{\frac14, 1\} $, $ c_{12}>0 $ $ s\geq 0 $
(2) $ a_1=\frac14 $, $ c_{12}>0 $ $ s\geq 1 $
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Case Coefficients $ a_1 $ and $ c_{12} $ $ s $
(1) $ a_1\not\in\{\frac14, 1\} $, $ c_{12}>0 $ $ s\geq 0 $
(2) $ a_1=\frac14 $, $ c_{12}>0 $ $ s\geq 1 $
Table 6.  LWP Results
Case $ \rho_1 $, $ \rho_2 $ and $ \sigma_{i} (1\leq i\leq 4) $ $ s $
(1) $ \sigma_3=0 $, $ \rho_1=1 $ $ s>-\frac34 $
(2) $ \rho_2\sigma_3^2>1 $ $ s>-\frac34 $
(3) $ \rho_2\sigma_3^2<1 $, (1.13) fails $ s\geq 0 $
(4) $ \rho_2\sigma_3^2<1 $, (1.13) holds $ s\geq\frac{3}{4} $
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Case $ \rho_1 $, $ \rho_2 $ and $ \sigma_{i} (1\leq i\leq 4) $ $ s $
(1) $ \sigma_3=0 $, $ \rho_1=1 $ $ s>-\frac34 $
(2) $ \rho_2\sigma_3^2>1 $ $ s>-\frac34 $
(3) $ \rho_2\sigma_3^2<1 $, (1.13) fails $ s\geq 0 $
(4) $ \rho_2\sigma_3^2<1 $, (1.13) holds $ s\geq\frac{3}{4} $
Table 7.  GWP Results
Case $ \rho_1 $, $ \rho_2 $ and $ \sigma_{i} (1\leq i\leq 4) $ $ s $
(1) $ \rho_2\sigma_3^2\neq 1 $, (1.13) fails $ s\geq 0 $
(2) $ \rho_2\sigma_3^2\neq 1 $, (1.13) holds $ s\geq 1 $
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Case $ \rho_1 $, $ \rho_2 $ and $ \sigma_{i} (1\leq i\leq 4) $ $ s $
(1) $ \rho_2\sigma_3^2\neq 1 $, (1.13) fails $ s\geq 0 $
(2) $ \rho_2\sigma_3^2\neq 1 $, (1.13) holds $ s\geq 1 $
Table 8.  Bilinear Estimates
Type $ r<0 $ $ 0<r<\frac{1}{4} $ $ r=\frac{1}{4} $ $ r>\frac{1}{4} $, $ r\neq 1 $ $ r=1 $
(D1): (3.4) $ s>-\frac{3}{4} $ $ s>-\frac{3}{4} $ $ s\geq \frac{3}{4} $ $ s\geq 0 $ $ s>-\frac{3}{4} $
(D2): (3.5) $ s>-\frac{3}{4} $ $ s\geq \frac{3}{4} $ $ s\geq 0 $ $ s>-\frac{3}{4} $
(ND1): (3.6) $ s>-\frac{3}{4} $ $ s>-\frac{3}{4} $ $ s\geq \frac{3}{4} $ $ s\geq 0 $ $ s>0 $
(ND2): (3.7) $ s>-\frac{3}{4} $ $ s>-\frac{3}{4} $ $ s\geq \frac{3}{4} $ $ s\geq 0 $ $ s>0 $
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Type $ r<0 $ $ 0<r<\frac{1}{4} $ $ r=\frac{1}{4} $ $ r>\frac{1}{4} $, $ r\neq 1 $ $ r=1 $
(D1): (3.4) $ s>-\frac{3}{4} $ $ s>-\frac{3}{4} $ $ s\geq \frac{3}{4} $ $ s\geq 0 $ $ s>-\frac{3}{4} $
(D2): (3.5) $ s>-\frac{3}{4} $ $ s\geq \frac{3}{4} $ $ s\geq 0 $ $ s>-\frac{3}{4} $
(ND1): (3.6) $ s>-\frac{3}{4} $ $ s>-\frac{3}{4} $ $ s\geq \frac{3}{4} $ $ s\geq 0 $ $ s>0 $
(ND2): (3.7) $ s>-\frac{3}{4} $ $ s>-\frac{3}{4} $ $ s\geq \frac{3}{4} $ $ s\geq 0 $ $ s>0 $
Table 9.  Sharpness of Bilinear Estimates
Type $ r<0 $ $ 0<r<\frac{1}{4} $ $ r=\frac{1}{4} $ $ r>\frac{1}{4} $, $ r\neq 1 $ $ r=1 $
(D1): (3.4) $ s<-\frac{3}{4} $ $ s<-\frac{3}{4} $ $ s< \frac{3}{4} $ $ s< 0 $ $ s<-\frac{3}{4} $
(D2): (3.5) $ s<-\frac{3}{4} $ $ s< \frac{3}{4} $ $ s< 0 $ $ s<-\frac{3}{4} $
(ND1): (3.6) $ s<-\frac{3}{4} $ $ s<-\frac{3}{4} $ $ s< \frac{3}{4} $ $ s< 0 $ $ s<0 $
(ND2): (3.7) $ s<-\frac{3}{4} $ $ s<-\frac{3}{4} $ $ s<\frac{3}{4} $ $ s< 0 $ $ s<0 $
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Type $ r<0 $ $ 0<r<\frac{1}{4} $ $ r=\frac{1}{4} $ $ r>\frac{1}{4} $, $ r\neq 1 $ $ r=1 $
(D1): (3.4) $ s<-\frac{3}{4} $ $ s<-\frac{3}{4} $ $ s< \frac{3}{4} $ $ s< 0 $ $ s<-\frac{3}{4} $
(D2): (3.5) $ s<-\frac{3}{4} $ $ s< \frac{3}{4} $ $ s< 0 $ $ s<-\frac{3}{4} $
(ND1): (3.6) $ s<-\frac{3}{4} $ $ s<-\frac{3}{4} $ $ s< \frac{3}{4} $ $ s< 0 $ $ s<0 $
(ND2): (3.7) $ s<-\frac{3}{4} $ $ s<-\frac{3}{4} $ $ s<\frac{3}{4} $ $ s< 0 $ $ s<0 $
Table 10.  Troubles and Critical Indexes ($ r = \frac{{\alpha}_2}{{\alpha}_1} $)
$ r<0 $ $ 0<r<\frac{1}{4} $ $ r=\frac{1}{4} $ $ r>\frac{1}{4} $, $ r\neq 1 $ $ r=1 $
(D1): (3.4) (T2)
$-\frac{3}{4}$ 		(T2)
$-\frac{3}{4}$ 		(T1)+(T2)
$\frac{3}{4}$ 		(T1)
$ 0$ 		(T2)
$-\frac{3}{4}$
(D2): (3.5) None
$ -\frac{13}{12}$ 		(T2)
$-\frac{3}{4}$ 		(T1)+(T2)
$ \frac{3}{4}$ 		(T1)
$ 0$ 		(T2)
$-\frac{3}{4}$
(ND1): (3.6) (T3)
$-\frac{3}{4}$ 		(T2) or (T3)
$-\frac{3}{4}$ 		(T1)+(T2)
$ \frac{3}{4}$ 		(T1)
$ 0$ 		(T2)+(T3)
$0$
(ND2): (3.7) (T3)
$-\frac{3}{4}$ 		(T2) or (T3)
$-\frac{3}{4}$ 		(T1)+(T2)
$ \frac{3}{4}$ 		(T1)
$ 0$ 		(T2)+(T3)
$0$
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$ r<0 $ $ 0<r<\frac{1}{4} $ $ r=\frac{1}{4} $ $ r>\frac{1}{4} $, $ r\neq 1 $ $ r=1 $
(D1): (3.4) (T2)
$-\frac{3}{4}$ 		(T2)
$-\frac{3}{4}$ 		(T1)+(T2)
$\frac{3}{4}$ 		(T1)
$ 0$ 		(T2)
$-\frac{3}{4}$
(D2): (3.5) None
$ -\frac{13}{12}$ 		(T2)
$-\frac{3}{4}$ 		(T1)+(T2)
$ \frac{3}{4}$ 		(T1)
$ 0$ 		(T2)
$-\frac{3}{4}$
(ND1): (3.6) (T3)
$-\frac{3}{4}$ 		(T2) or (T3)
$-\frac{3}{4}$ 		(T1)+(T2)
$ \frac{3}{4}$ 		(T1)
$ 0$ 		(T2)+(T3)
$0$
(ND2): (3.7) (T3)
$-\frac{3}{4}$ 		(T2) or (T3)
$-\frac{3}{4}$ 		(T1)+(T2)
$ \frac{3}{4}$ 		(T1)
$ 0$ 		(T2)+(T3)
$0$
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