59
Views
33
CrossRef citations to date
0
Altmetric
Original Articles

The dissipation of nonlinear dispersive waves: the case of asymptotically weak nonlinearity

The case of asymptotically weak

Pages 1665-1693 | Published online: 08 May 2007

References

  • J. P. Albert, D. B. Dix Asymptotic behavior of solutions of linear dispersive-dissipative equations, in preparation
  • Amick , C.J , Bona , J.L and Schonbek , M.E . 1989 . Decay of Solutions of some Nonlinear Wave Equations . J . Diff. Eq , 81 : 1 – 49 .
  • Bergh , J and Löfström , J . 1976 . “ Interpolation Spaces ” . Berlin : Springer Verlag .
  • Biler , P . 275 . Asymptotic Behavior i n Time of Solutions to some equations Generalizing the Korteweg-de Vries - Burgers Equation . Bull. Polish Acad. Sci. Math , 32 ( 1984 ) : 281
  • F. M. Christ, M. I. Weinstein, Dispersion of Small Amplitude Solutions of the Generalized Korteweg-de Vries Equation, J. Funct. Anal. to appear
  • Dix , D.B . 1988 . Decay of Solutions to the Benjamin-Ono-Burgers Equation . Ph.Dthesis . 1988 . University of Chicago
  • 1991 . Temporal Asymptotic Behavior of Solutions of the Benjamin-Ono-Burgers Equation . J . Diff. Eq , 90 238 – 287 .
  • Uniqueness and Nonuniqueness in the Initial-Value Problem for Burgers' Equation, in preparation
  • Escobedo , M and Zuazua , E . 1989 . Comportement asymptotique des solutions d'une equation de convection-diffusion . C. R. Acad. Sci. Paris , 309 329 – 334 .
  • A. Friedman Partial Differential Equations, Holt, Rinehart, Winston, New York, 1969
  • Herz , C.S . 1968 . Lipschitz Spaces and Bemstein's Theorem on Absolutely Convergent Fourier Transforms . J . Math. Mech , 18 283 – 323 .
  • Hopf , E . 1950 . The Partial Differential Equationut+ uux = µuxx . Comm. Pure Appl.Math , 3 201 – 230 .
  • Il'in , A.M and Ole[ibreve]nik , O.A . 1985 . Behavior of the Solutions of the Cauchy Problem for Certain Quasilinear Equations for Unbounded Increase of the Time . Dokl. Akad. Nauk. SSSR , 120 25 – 28 . Amer. Math. Soc. Transl. 42 1964, 19-23
  • 1960 . Asymptotic behavior of solutions ofthe Cauchy problem for some quasilinear equations for large values of time . Mat. Sbornik , 51 191 – 216 .
  • Kajikiya , R and Miyakawa , T . 1986 . On L2 Decay of Weak Solutions of the Navier-Stokes Equations in Rn . Math. Z , 192 135 – 148 .
  • Kato , T . 1983 . On the Cauchy Problem for the Korteweg-de Vries Equation . Stud. Appl. Math. Adv. Math. Suppl, Stud , 8 93 – 123 . Generalized
  • Kaup , D.J and Newell , A.C . 1978 . An exact solution for a derivative nonlinear Schrödinger equation . J. Math. Phys , 19 798 – 801 .
  • Lax , P.D . 1957 . Hyperbolic Systems of Conservation Laws II . Comm. Pure Appl. Math , 10 537 – 566 .
  • Naumkin , P.I . 1987 . Whitham's Equation with a Singular Kernel . U.S.S.R. Comput.Maths.Phys , 27 209 – 211 .
  • Naumkin , P.I and Shishmarev , I.A . 1990 . Asymptotic Behavior of the Solutions of the Whitham's Equation for Large Time . Matematicheskoe Modelirovanie , 2 75 – 88 . Russian
  • Olver , F.W.J . 1974 . “ Asymptotics and Special Functions ” . San Diego : Academic Press .
  • Pego , R.L . 1986 . Stability in Systems of Conservation Laws with Dissipation . Lect. Appl. Math , 23 : 345 – 357 .
  • Petersen , B.E . 1983 . “ Introudction to the Fourier Transform and Pseudo-differential Operators ” . Boston : Pitman Publishing Inc .
  • Saut , J.C . 1979 . Sur quelques généralisation de l'equation de Korteweg-de Vries . J. Math. Pures et appl , 58 : 21 – 61 .
  • Schonbek , M.E . 1980 . Decay of Solutions to Parabolic Conservation Laws . Comm. Pde , 7 : 449 – 473 .
  • 1986 . Large Time Behaviour of Solutions to the Navier-Stokes Equations . Comm. P.D.E , 11 : 733 – 763 .
  • Stein , E.M . 1970 . “ Singular Integrals and Differentiability Properties of Functions ” . Princeton, NJ : Princeton Univ. Press .
  • Strauss , W.A . 1974 . Dispersion of Low Energy Waves for Two Conservative Equations . Arch. Rational. Mech. Anal , 55 : 86 – 92 .
  • 1981 . Nonlinear scattering at low energy . J. Funct. Anal , 41 : 281 – 293 .
  • Triebel , H . 1983 . “ Theory of Function Spaces ” . Basel : Birkhäuser Verlag .
  • L. Zang Cauchy Problem for a Nonlinear Parabolic Systems with Singular Integrals Differential Terms, Preprint
  • Decay Estimations for the Solutions of Some Nonlinear Evolutionary Equations, Preprint
  • Zhou , Y and Guo , B . 1984 . On the System of the Generalized Korteweg-de Varies Equtions . Acta. Math. Sinica , 27 : 154 – 176 . Chinese
  • 1986 . Existence of Global Weak Solutions for Generalized Korteweg-de Vries Systems with Several Variables . Scientia Sinica , 29 : 375 – 390 .

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.