Benjamin-Ono equation

Benjamin-Ono equation

[Thanks to and Felipe Linares for help with this section - Ed]

The generalized Benjamin-Ono equation is the scalar equation

$\partial _{t}u+D_{x}^{1+a}\partial _{x}u+u\partial _{x}u=0.$

where D_x = sqrt{-Delta} is the positive differentiation operator. When a=1 this is [#kdv KdV]; when a=0 this is the Benjamin-Ono equation (BO) [[Bibliography#Bj1967|Bj1967]], On1975, which models one-dimensional internal waves in deep water. Both of these equations are completely integrable (see e.g. AbFs1983, CoiWic1990), though the intermediate cases 0 < a < 1 are not.

When a=0, scaling is s = -1/2, and the following results are known:

• LWP in H^s for s >= 1 [Ta-p]
  • For s >= 9/8 this is in [KnKoe-p]
  • For s >= 5/4 this is in [KocTz-p]
  • For s >= 3/2 this is in Po1991
  • For s > 3/2 this is in Io1986
  • For s > 3 this is in Sau1979
  • For no value of s is the solution map uniformly continuous [KocTz-p2]
    • For s < -1/2 this is in [BiLi-p]
• Global weak solutions exist for L^2 data Sau1979, GiVl1989b, GiVl1991, Tom1990
• Global well-posedness in H^s for s >= 1 [Ta-p]
  • For s >= 3/2 this is in Po1991
  • For smooth solutions this is in Sau1979

When 0 < a < 1, scaling is s = -1/2 - a, and the following results are known:

• LWP in H^s is known for s > 9/8 \u2013 3a/8 [KnKoe-p]
  • For s >= 3/4 (2-a) this is in KnPoVe1994b
• GWP is known when s >= (a+1)/2 when a > 4/5, from the conservation of the Hamiltonian KnPoVe1994b
• The LWP results are obtained by energy methods; it is known that pure iteration methods cannot work MlSauTz2001
  • However, this can be salvaged by combining the H^s norm || f ||_{H^s} with a weighted Sobolev space, namely || xf ||_{H^{s - 2s_*}}, where s_* = (a+1)/2 is the energy regularity.[CoKnSt-p4]

One can replace the quadratic non-linearity uu_x by higher powers u^{k-1} u_x, in analogy with KdV and gKdV, giving rise to the gBO-k equations (let us take a=0 for sake of discussion).The scaling exponent is 1/2 - 1/(k-1).

• For k=3, one has GWP for large data in H^1 [KnKoe-p] and LWP for small data in H^s, s > ½ [MlRi-p]
  • For small data in H^s, s>1, LWP was obtained in KnPoVe1994b
  • With the addition of a small viscosity term, GWP can also be obtained in H^1 by complete integrability methods in [FsLu2000], with asymptotics under the additional assumption that the initial data is in L^1.
  • For s < ½, the solution map is not C^3 [MlRi-p]
• For k=4, LWP for small data in H^s, s > 5/6 was obtained in KnPoVe1994b.
• For k>4, LWP for small data in H^s, s >=3/4 was obtained in KnPoVe1994b.
• For any k >= 3 and s < 1/2 - 1/k the solution map is not uniformly continuous [BiLi-p]

The KdV-Benjamin Ono (KdV-BO) equation is formed by combining the linear parts of the KdV and Benjamin-Ono equations together.It is globally well-posed in L^2 Li1999, and locally well-posed in H^{-3/4+} [KozOgTns] (see also [HuoGuo-p] where H^{-1/8+} is obtained).Similarly one can generalize the non-linearity to be k-linear, generating for instance the modified KdV-BO equation, which is locally well-posed in H^{1/4+} [HuoGuo-p].For general gKdV-gBO equations one has local well-posednessin H^3 and above GuoTan1992.One can also add damping terms Hu_x to the equation; this arises as a model for ion-acoustic waves of finite amplitude with linear Landau damping OttSud1982.