
具有强迫项Dullin-Gottwald-Holm,方程整体耗散解的存在性
Global Dissipative Solutions of the Dullin-Gottwald-Holm Equation with a Forcing
本文借鉴Bressan和Constantin于2007年提出的新特征线法,利用具有强迫项的Dullin-Gottwald-Holm方程的平衡律和一些新的估计,证明了该方程在H1(R)中整体耗散解的存在性.
In terms of Bressan and Constantin's arguments in 2007, by exploiting the balance law and some estimates, we prove the existence of global dissipative solutions for the Dullin-Gottwald-Holm equation with a forcing term in H1(R).
Dullin-Gottwald-Holm方程 / 耗散解 / 平衡律 / 新特征线法 {{custom_keyword}} /
Dullin-Gottwald-Holm equation / dissipative solutions / balance law / new characteristic method {{custom_keyword}} /
[1] Bressan A., Constantin A., Global dissipative solutions of the Camassa-Holm equation, Anal. Appl., 2007, 5:1-27.
[2] Bressan A., Chen G., Zhang Q., Uniqueness of conservative solutions to the Camassa-Holm equation via characteristics, Discr. Cont. Dyn. Syst., 2015, 35:25-42.
[3] Bressan A., Chen G., Zhang Q., Unique conservative solutions to a variational wave equation, Arch. Rat. Mech. Anal., 2015, 217:1069-1101.
[4] Bressan A., Constantin A., Global conservative solutions of the Hunter-Saxton equation, SIAM J. Math. Anal., 2005, 37:996-1026.
[5] Bressan A., Constantin A., Global conservative solutions to the Camassa-Holm equation, Arch. Rat. Mech. Anal., 2007, 183:215-239.
[6] Constantin A., Lannes D., The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal., 2009, 192:165-186.
[7] Constantin A., Escher J., Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 1998, 181:229-243.
[8] Constantin A., Escher J., Particle trajectories in solitary water waves, Bull. Amer. Math. Soc., 2007, 44:423-431.
[9] Constantin A., McKean H. P., A shallow water equation on the circle, Comm. Pure Appl. Math., 1999, 52:949-982.
[10] Constantin A., Molinet L., Orbital stability of solitary waves for a shallow water equation, Physica D, 2001, 157:75-89.
[11] Chen G., Shen Y., Existence and regularity of solutions in nonlinear wave equations, Discr. Cont. Dyn. Syst., 2015, 35:3327-3342.
[12] Danchin R., A few remarks on the Camassa-Holm equation, Differential and Integral Equations, 2000, 14:953-988.
[13] Dullin H. R., Gottwald G. A., Holm D. D., An integrable shallow water equation with linear and nonlinear dispersion, Physical Review Letters, 2001, 87:194501.
[14] Kato T., Quasi-linear equations of evolution, with applications to partial differential equations, Spectral Theory and Differential Equations, Springer, Berlin 1975:25-70.
[15] Krishnan E. V., Khan Q. J. A., Lie group of transformations for a KdV-Boussinesq equation, Czechoslovak Journal of Physics, 2003, 53:99-105.
[16] Li B., Zhu S. H., Leng L. H., Uniqueness of global weak solutions to the Dullin-Gottwald-Holm equation with a forcing term (in Chinese), Journal of Sichuan Normal University (Natural Science Edition), 2018, 2:159-168.
[17] Li Y., Olver P., Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation, Differential Equations, 2000, 162:27-63.
[18] Liu Y., Global existence and blow-up solutions for a nonlinear shallow water equation, Mathematische Annalen, 2006, 335:717-735.
[19] Tian L. X., Gui G., Liu Y., On the well-posedness problem and the scattering problem for the Dullin-Gottwald-Holm equation, Communications in Mathematical Physics, 2005, 257:667-701.
[20] Xin Z., Zhang P., On the weak solutions to a shallow water equation, Comm. Pure Appl. Math., 2000, 53:1411-1433.
[21] Xin Z., Zhang P., On the uniqueness and large time behavior of the weak solutions to a shallow water equation, Comm. Partial Differential Equations, 2002, 27:1815-1844.
[22] Yin Z. Y., Global existence and blow-up for a periodic integrable shallow water equation with linear and nonlinear dispersion, Dynamics of Continuous Discrete and Impulsive Systems, 2005, 12:129.
[23] Zhou Y., Blow-up of solutions to the DGH equation, Journal of Functional Analysis, 2007, 250:227-248.
[24] Zhou Y., Guo Z., Blow up and propagation speed of solutions to the DGH equation, Discrete and Continuous Dynamical Systems Series B, 2009, 12:657-670.
[25] Zhu S. H., Existence and uniqueness of global weak solutions of the Camassa-Holm equation with a forcing, Discr. Cont. Dyn. Syst., 2016, 36:5201-5221.
国家自然科学基金基金资助项目(11871138,11771314)
/
〈 |
|
〉 |