Benjamin-Ono equation
[Thanks to Nikolay Tzvetkov and Felipe Linares for help with this section - Ed]
The generalized Benjamin-Ono equation BO_a is the scalar equation
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) [[references.html#Bj1967 Bj1967]], references.html#On1975 On1975, which models one-dimensional internal waves in deep water. Both of these equations are completely integrable (see e.g. references.html#AbFs1983 AbFs1983, references.html#CoiWic1990 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 references.html#Po1991 Po1991
- For s > 3/2 this is in references.html#Io1986 Io1986
- For s > 3 this is in references.html#Sau1979 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 references.html#Sau1979 Sau1979, references.html#GiVl1989b GiVl1989b, references.html#GiVl1991 GiVl1991, references.html#Tom1990 Tom1990
- Global well-posedness in H^s for s >= 1 [Ta-p]
- For s >= 3/2 this is in references.html#Po1991 Po1991
- For smooth solutions this is in references.html#Sau1979 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 – 3a/8 [KnKoe-p]
- For s >= 3/4 (2-a) this is in references.html#KnPoVe1994b KnPoVe1994b
- GWP is known when s >= (a+1)/2 when a > 4/5, from the conservation of the Hamiltonian references.html#KnPoVe1994b KnPoVe1994b
- The LWP results are obtained by energy methods; it is known that pure iteration methods cannot work references.html#MlSauTz2001 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 references.html#KnPoVe1994b 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 references.html#KnPoVe1994b KnPoVe1994b.
- For k>4, LWP for small data in H^s, s >=3/4 was obtained in references.html#KnPoVe1994b 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 references.html#Li1999 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 references.html#GuoTan1992 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 references.html#OttSud1982 OttSud1982.
