Hostname: page-component-77c89778f8-7drxs Total loading time: 0 Render date: 2024-07-17T17:59:34.175Z Has data issue: false hasContentIssue false
  • English
  • Français

Analysis of 2 + 1 diffusive–dispersive PDE arising in river braiding

Published online by Cambridge University Press:  16 February 2016

SALEH TANVEER  and
CHARIS TSIKKOU
SALEH TANVEER
Affiliation:
Mathematics Department, The Ohio State University, Columbus, OH 43210, USA email: tanveer@math.ohio-state.edu
CHARIS TSIKKOU
Affiliation:
Mathematics Department, West Virginia University, Morgantown, WV 26505, USA email: tsikkou@math.wvu.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We present local existence and uniqueness results for the following 2 + 1 diffusive–dispersive equation due to P. Hall arising in modelling of river braiding:

$$\begin{equation*} u_{yyt} - \gamma u_{xxx} -\alpha u_{yyyy} - \beta u_{yy} + \left ( u^2 \right )_{xyy} = 0 \end{equation*}$$
for (x,y) ∈ [0, 2π] × [0, π], t > 0, with boundary condition uy=0=uyyy at y=0 and y=π and 2π periodicity in x, using a contraction mapping argument in a Bourgain-type space Ts,b. We also show that the energy ∥u(·, ·, t)∥2L2 and cumulative dissipation ∫0t∥uy (·, ·, s)∥L22dt are globally controlled in time t.

Keywords

Initial/boundary value problemdispersive–diffusive PDEriver braidingBourgain-type space
Type
Papers
Information
European Journal of Applied Mathematics , Volume 27 , Issue 5 , October 2016 , pp. 756 - 780
Copyright
Copyright © Cambridge University Press 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Angelopoulos, Y. (2013) Well-posedness and ill-posedness results for the Novikov-Veselov equation, preprint, ArXiv:1307.4110.Google Scholar
[2] Angulo, J., Matheus, C. & Pilod, D. (2009) Global well-posedness and non-linear stability of periodic traveling waves for a Schrödinger-Benjamin-Ono system. Commun. Pure Appl. Anal. 8 (3), 815–844. MR2476660CrossRefGoogle Scholar
[3] Bourgain, J. (1993) Fourier transform restriction phenomena for certain lattice subsets and application to nonlinear evolution equations I. Schrödinger equations. Geom. Funct. Anal. 3 (2), 107–156. MR1209299 (95d:35160a)CrossRefGoogle Scholar
[4] Bourgain, J. (1993) Fourier transform restriction phenomena for certain lattice subsets and application to nonlinear evolution equations II. The KdV equation. Geom. Funct. Anal. 3 (3), 209–262. MR1215780 (95d:35160b)Google Scholar
[5] Bourgain, J. (1993) On the Cauchy problem for the Kadomstev-Petviashvili equation. Geom. Funct. Anal. 3 (4), 315–341. MR1223434 (94d:35142)CrossRefGoogle Scholar
[6] Chen, R., Liu, Y. & Zhang, P. (2012) Local regularity and decay estimates of solitary waves for the rotation-modified Kadomtsev-Petviashvili equation. Trans. Am. Math. Soc. 364 (7), 3395–3425. MR2901218CrossRefGoogle Scholar
[7] Chen, W. & Li, J. (2007) On the low regularity of the modified Korteweg-de Vries equation with a dissipative term. J. Differ. Equ. 240 (1), 125–144. MR2349167 (2008g:35182)Google Scholar
[8] Chen, W. & Li, J. (2008) On the low regularity of the Benney-Lin equation. J. Math. Anal. Appl. 339 (2), 1134–1147. MR2377072 (2009e:35228)CrossRefGoogle Scholar
[9] Chen, W., Li, J. & Miao, C. (2007) On the well-posedness of the Cauchy problem for dissipative modified Korteweg-de Vries equations. Differ. Integral Equ. 20 (11), 1285–1301. MR2372427 (2008k:35401)Google Scholar
[10] Chen, W., Li, J. & Miao, C. (2008) On the low regularity of the fifth order Kadomtsev-Petviashvili I equation. J. Differ. Equ. 245 (11), 3433–3469. MR2460030 (2009j:35302)CrossRefGoogle Scholar
[11] Darwich, M. (2012) On the well-posedness for Kadomtsev-Petviashvili-Burgers I equation. J. Differ. Equ. 253 (5), 1584–1603. MR2927391CrossRefGoogle Scholar
[12] Esfahani, A. (2012) The ADMB-KdV equation in Anisotropic Sobolev spaces. Differ. Equ. Appl. 4 (3), 459–484. MR3012073Google Scholar
[13] Hadac, M. (2008) Well-Posedness for the Kadomtsev-Petviashvili II equation and generalisations. Trans. Am. Math. Soc. 360 (12), 6555–6572. MR2434299 (2009g:35265)CrossRefGoogle Scholar
[14] Hall, P. (2006) Nonlinear evolution equations and braiding of weakly transporting flows over gravel beds. Stud. Appl. Math. 117 (1), 27–69. MR2236577 (2007c:76034)Google Scholar
[15] Kenig, C. E., Ponce, G. & Vega, L. (1991) Well-posedness of the initial value problem for the Korteweg-de Vries equation. J. Am. Math. Soc. 4 (2), 323–347. MR1086966 (92c:35106)CrossRefGoogle Scholar
[16] Kenig, C. E., Ponce, G. & Vega, L. (1996) A bilinear estimate with applications to the KdV equation. J. Am. Math. Soc. 9 (2), 573–603. MR1329387 (96k:35159)CrossRefGoogle Scholar
[17] Kojok, B. (2007) Sharp well-posedness for Kadomtsev-Petviashvili-Burgers (KBII) equation in R 2 . J. Differ. Equ. 242 (2), 211–247. MR2363314 (2009b:35363)Google Scholar
[18] Molinet, L. & Ribaud, F. (2002) On the low regularity of the Kortewig-de Vries-Burgers equation. Int. Math. Res. Not. 37, 1979–2005. MR1918236 (2003e:35272)CrossRefGoogle Scholar
[19] Otani, M. (2006) Well-posedness of the generalized Benjamin-Ono-Burgers equations in Sobolev spaces of negative order. Osaka J. Math. 43 (4), 935–965. MR2303557 (2008d:35202)Google Scholar
[20] Pecher, H. (2012) Some new well-posedness results for the Klein-Gordon-Schrödinger system. Differ. Integral Equ. 25 (1–2), 117–142. MR2906550 (2012m:35310)Google Scholar
[21] Tao, T. (2006) Nonlinear Dispersive Equations: Local and Global Analysis}. CBMS Regional Conference Series in Mathematics, 106. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2006. xvi+373 pp. ISBN: 0-8218-4143-2.Google Scholar