## Miscellaneous equations

Vlasov-Maxwell(VM) equations

[More contributions on this section would be appreciated. - Ed.]

The VM equations are given by

f_t + v . grad_x f + (E + v x B) . grad_v f = 0
div E = 4 pi rho
div B = 0
E_t = curl B - 4 pi j
B_t = - curl E

where f(t,x,v) is the particle density (and is non-negative), j(t,x) = int v f(t,x,v) dv is the current density, rho(t,x) = int f(t,x,v) dv is the charge density, and v = v / (1 + |v|^2)^{1/2} is the relativistic velocity.  The vector fields E(t,x) and B(t,x) represent the electromagnetic field.  x and v live in R^3 and t lives in R.  This equation is a coupled wave and conservation law system, and models collision-less plasma at relativistic velocities.

• Assuming that the particle density remains compactly supported in the velocity domain for all time, GWP in C^1 was proven in [GsSr1986b] (see also [GsSr1986], [GsSr1987].
• An alternate proof of this result is in [KlSt2002].  A stronger result (which only imposes compact support conditions on the initial data, not on all time) regarding solutions to Vlasov-Maxwell which are purely outgoing (no incoming radiation) is in [Cal-p].
• The velocity demain hypothesis can be removed in the "2.5 dimensional model" where the x_3 dependence is trivial but the v_3 dependence is not [GsScf1990].
• Further results are in [GsSch1988], [Rei1990], [Wol1984], [Scf1986]

The non-relativistic limit of Vlasov-Maxwell is Vlasov-Poisson, in which the electromagnetic field (E + v x B) is replaced by
nabla Delta^{-1} 4 pi rho.  Considerably more is known for the existence theory of this equation.

The KP equations are given by

(ut + uxxx + uux)x + l uyy = 0

where u=u(x,y,t) is a real-valued function of two space variables and one time variable, and l is a constant scalar.  When l = 0 the evolution is trivial in the y variable and the equation collapses to the KdV equation.  When l < 0 the equation is known as the KP-I equation, and when l > 0 the equation is the KP-II equation.  These equations are a model for shallow long waves in the x direction, with some mild dispersion in the y direction [KdPv1970], [PvYn1989].  Generally speaking, KP-I is a good model when surface tension is strong, and KP-II is a good model when surface tension is weak.

KP (or the 2D Boussinesq equations) can also arise as a model for the (somewhat unphysical) 3D water wave equation in the shallow case, similar to how KdV arises from the 2D water wave equation.  Under further limiting assumptions (slowly varying amplitude, weak non-linearity) one can obtain the Davey-Stewatson equation (formally, at least).  KP-I also arises as a model for sound waves in ferromagnetic media [TrFl1985].

The equations are completely integrable, and thus have an infinite number of conserved quantities.  However for the LWP and GWP theory the most important conserved quantities are the L2 norm

ò u2 dx dy

and the Hamiltonian

ò  ux2 - u3/3 - l ((partialx)-1 uy)2  dx dy.

The next conserved quantity contains terms which resemble the L^2 norm of u_xx or of {partial_x)^{-2} u_yy.  (Note that one y derivative has the same scaling as two x derivatives.)

Explicit solutions can be obtained by inverse scattering [AxPgSac1997], [FsSng1992], [Zx1990], although this does not appear to be directly helpful to the low-regularity LWP and GWP theory.

The Cauchy problem is usually studied in two-parameter Sobolev spaces Hx^{s1} Hy^{s2}.  However, the presence of inverse derivatives such as (partialx)^{-1} necessitates some technical modifications to these spaces at low frequencies which we will not detail here.

Despite the apparent similarity of KP-I and KP-II, the two equations behave very differently, both qualitatively and quantitatively.  KP-I has the advantage of having a Hamiltonian which is positive definite in the leading order terms, however it contains resonances which make an X^{s,b}-based analysis somewhat delicate.  The KP-II equation does not have any non-trivial resonances, but its Hamiltonian does not have a definite sign.

Just as KdV can be generalized to gKDV-k, KP can be generalized to gKP-k, by replacing uux with uk ux (and u3/3 with uk+1/(k+1) in the Hamiltonian).  The value k=4/3 is critical in the sense that the potential energy term in the Hamiltonian can be controlled by the other two terms, and thus one expects blow up in general for long times when k > 4/3.  This is supported by numerics [BnDgKar1986], [WgAbSe1994], [Wc1987].

• Scaling is s1 + 2s2 + 1/2 = 0.
• GWP is known for data in a space roughly like (s1,s2) = (2,0), which is small in a certain weighted space [CoKnSt-p3].  Examples from [MlSauTz-p2] show that something like this type of additional condition is necessary.
• For data in a space roughly like (2,0) intersect (-2,2) and no weight condition this is in [Kn-p]
• For data in a space which is roughly like (s1,s2) = (3,0) intersect (-2,2) this is in [MlSauTz-p3].
• For small smooth data this was achieved by inverse scattering techniques in [FsSng1992], [Zx1990]
• On T, Global weak L2 solutions were obtained for small L2 data in [Scz1987] and for large L2 data in [Co1996]. Assuming a (3,0) regularity at least, these global weak solutions are unique [Scz1987].  (The analogous uniqueness result on R is in [MlSauTz-p3]; H^1 global weak solutions were constructed in [Tom1996].)
• LWP in the energy space (which is essentially (1,0) intersect (-1,1)) assuming also that yu is in L^2 [CoKeSt-p2].  Note that the latter property is preserved by the flow.  A technical refinement to Besov spaces is also available [CoKeSt-p2]; see also [CoKeSt-p3].
• For (s1,s2) = (3/2+, 1/2+) this is in [MlSauTz-p2], however a certain technical condition at low frequencies has to be imposed (similarly for the results below).  Note that without any such restriction the flow map is not even C^2 in standard Sobolev spaces [MlSauTz-p2], [MlSauTz-p3]
• A LWP result in a space roughly like (3/2+) intersect (-1,1) is in [Kn-p].
• For (s1,s2)=(2+,2+) this is in [IoNu1998]
• For (s1,s2) = (3,3) this is in [IsMjStb1995], [Uk1989], [Sau1993]
• LWP and GWP in the energy space ((1,0) intersect (-1,1)) without any localization condition is still an important unsolved problem.
• If one considers the fifth-order KP-I equation (replace uxxx by uxxxxx) then one has GWP in the energy space (when both the L2 norm and Hamiltonian are finite) [SauTz2000].  This has been extended to the partly periodic case (x,y) in T x R in [SauTz-p].  The corresponding problems for R x T and T x T remain open.
• On T x T one has LWP for (s1,s2) = (3,3) [IsMjStb1994]
• "Lump" soliton solutions exist for KP-I (and more generally for gKP-k-I for k = 1,2,3, but no solitons exist for k ³ 4 [WgAbSe1994], [Sau1993], [Sau1995], where solitons are understood to have at least some decay at infinity).  When k > 4/3 these solitons are not orbitally stable [WgAbSe1994], [LiuWg1997], and in fact blowup solutions can be demonstrated to exist from a virial identity argument [Liu2001] (see also [TrFl1985], [Sau1993]).  For 2 < k < 4 one in fact has strong orbital instability [Liu2001].  For 1 £ k < 4/3 one has orbital stability [LiuWg1997], [BdSau1997].

• Scaling is s1 + 2s2 + 1/2 = 0.
• GWP for s1 > -1/14, s2 = 0 [IsMj2003].
• For s1 > -1/64 this is also in [IsMj2001].
• GWP for s1 > -1/78, s2 = 0 [Tk-p] assuming a moment condition.
• A similar result, with a slightly stricter constraint on s1 but no moment condition, was obtained in [Tz-p].
1. For s1 = s2 ³ 0 this was proven in [Bo1993c], and this argument also applies to the periodic setting.  Heuristically this result is indicated by the local smoothing estimates in [Sau1993].
• LWP for s1 > -1/3, s2 = 0 [TkTz-p4], [IsMj2001]
• For s1 > -1/4, s2 = 0 this was shown in [Tk-p2]
• For s1 > -e, s2 = 0 and small data this was shown in [Tz1999].
• For s1 = s2 ³ 0 this was proven in [Bo1993c], and this argument also applies to the periodic setting.
• For s1, s2 ³ 3 this is in [Uk1989]
• Related results are in [IoNu1998], [IsMjStb2001].
• Weak solutions in a weighted L2 space were constructed in [Fa1990].
• For s1 < -1/3 the natural bilinear estimate fails [TkTz-p4].
• Remark: Unlike KP-I, KP-II does not admit soliton solutions.

The KP-II equation can be generalized to three dimensions (replace partial_yy with partial_yy + partial_zz), with s_1 regularity in the x direction and s_2 in the y,z directions.  Scaling is now s_1 + 2s_2 – ½ = 0.  In isotropic spaces, local well-posedness in H^s with s > 3/2 was established assuming the low frequency condition that partial_x^{-1} u is also in H^s [Tz1999].  Anisotropically, local well-posedness in the space s_1 > 1, s_2 > 0 was established in [IsLopMj-p].

The Zakharov system

The Zakharov system consists of a complex field u and a real field n which evolve according to the equations

i u_t +  D u = un
Box n = -(|u|2)xx

thus u evolves according to a coupled Schrodinger equation, while n evolves according to a coupled wave equation.  We usually place the initial data u(0) in H^{s0}, the initial position n(0) in H^{s1}, and the initial velocity nt(0) in H^{s1-1} for some real s0, s1.

This system is a model for the propagation of Langmuir turbulence waves in an unmagnetized ionized plasma [Zk1972].  Heuristically, u behaves like a solution cubic NLS, smoothed by 1/2 a derivative.   If one sends the speed of light in Box to infinity, one formally recovers the cubic nonlinear Schrodinger equation.  Local existence for smooth data – uniformly in the speed of light! - was established in [KnPoVe1995b] by energy and gauge transform methods; this was generalized to non-scalar situations in [Lau-p], [KeWg1998].

An obvious difficulty here is the presence of two derivatives in the non-linearity for n.  To recover this large loss of derivatives one needs to use the separation between the paraboloid t = x2 and the light cone |t| = |x|.

There are two conserved quantities: the L2 norm of u

ò |u|2

and the energy

ò |ux|2 + |n|2/2 + |D-1x nt|2/2 + n |u|2.

The non-quadratic term n|u|2 in the energy becomes difficult to control in three and higher dimensions.  Ignoring this part, one needs regularity in (1,0) to control the energy.

Zakharov systems do not have a true scale invariance, but the critical regularity is (s0,s1) = ((d-3)/2, (d-2)/2).  In dimensions d³4 LWP is known on Rd with an e of this value [GiTsVl1997].  For the lower dimensional cases, see below.

Zakharov on R and T

• Zakharov systems do not have a true scale invariance, but the critical regularity is (s0,s1) = (-1,-3/2).
• LWP on R for (0,-1/2), or more generally whenever -1/2 < s0 - s1 £ 1 and 2s0³ s1 + 1/2 ³ 0 [GiTsVl1997]
• On T (with irrational period) the same result holds but one must place the additional restriction 0 £ s0 - s1 [Tk-p3].
• For general period, a slightly weaker version of this (assuming weighted pointwise bounds on Fourier co-efficients, and establishing an invariant Gibbs measure), is in [Bo1994]
• GWP on R for (9/10+,0) [Pe-p].
• In the energy class (1,0) this was shown in [GiTsVl1997].

2D Zakharov

• Zakharov systems do not have a true scale invariance, but the critical regularity is (s0,s1) = (-1/2,-1).
• LWP for (s0,s1) = (1/2,0) [GiTsVl1997]
• GWP for small (1,0) data [BoCo1996]; the smallness is needed to control the nonquadratic portion of the energy.
• As long as the H1 norm remains bounded (which is automatic for small data), the higher H^s norms of u grow by at most |t|(s-1)+ [CoSt-p]
• Explicit blowup solutions have been constructed with a blowup rate of t-1 in H1 norm [GgMe1994], [GgMe1994b].  This is optimal in the sense that no slower blowup rate is possible [Me1996b]

3D Zakharov

• Zakharov systems do not have a true scale invariance, but the critical regularity is (s0,s1) = (0,-1/2).
• LWP for (1/2,0) [GiTsVl1997]
• For (s0,s1) = (1,0) this was proven in [BoCo1996].
• GWP for small smooth data is in [OzTs1994]

Other wave-Schrodinger systems

The Zakharov system is not the only wave-Schrodinger system studied.  Another system of interest is the ``Yukawa-type'' system

i u_t + D u = -A u
Box A = m2 A + |u|2

for d=3.  A represents the meson field, while u is the nucleon field.

Global well posedness in the energy class (H1, H1 x L2) is in [Bch1984], [BlChd1978], [FuTs1978], [HaWl1987].  Modified wave operators were constructed for large energy data at infinity in [GiVl-p2].

With positive mass m=1, global well-posedness can be pushed to (Hs, Hm x Hm-1) whenever 1 ³ s,m > 7/10 and s+m > 3/2 [Pe-p2].

A generalized Zakharov system with a magnetic component was studied in [KeWg1998], with local existence of smooth solutions obtained.

Another such system is the Davey-Stewartson system [DavSte1974] in 2 spatial dimensions, a complex field u, and a real field phi:

i u_t + c_0 u_xx + u_yy = c_1 |u|^2 u + c_2 u phi_x
phi_xx + c_3 phi_yy = partial_x ( |u|^2 )

The field phi depends elliptically on u when c_3 is positive and thus one usually only specifies the initial data for u, not phi.  This equation is a modification of the cubic nonlinear Schrodinger equation and is completely integrable in the cases (c_0, c_1, c_2, c_3) = (-1,1,-2,1) (DS-I) and (1,-1,2,-1) (DS-II).   When c_3 > 0 the situation becomes a nonlinear Schrodinger equation with non-local nonlinearity, and can be treated by Strichartz estimates [GhSau1990]; for c_3 < 0 the situation is quite different but one can still use derivative NLS techniques (i.e. gauge transforms and energy methods) to obtain some local existence results [LiPo1993].  Further results are in [HaSau1995].

The Davey-Stewartson system is a special case of the Zakharov-Schulman system

i u_t + L_1 u = phi u

L_2 phi = L_3( |u|^2 )

where L_1, L_2, L_3 are various constant coefficient differneital operators; these describe the interactions of small amplitude, high frequency waves with acoustic waves [ZkShl1980].  Using energy methods and gauge transformations, local existence for smooth data was established in [KnPoVe1995b]; see also [GhSau1992].

The Ishimori system [Im1984] has a complex field u and a real field phi in two dimensions, and has the form

iu_t + u_xx - a u_yy = 2 u (u_x^2 - u_y^2) / (1 + |u|^2) - i b(phi_x u_y - phi_y u_x)
phi_xx + a' phi_yy = 8 Im(u_x u_y) / (1 + |u|^2)^2

The case (a,a') = (+1,-1) is studied in [Sy1992].  The case (a,a') = (-1,1) is studied in [HySau1995], [Ha-p], [KnPoVe2000]; in this case one has LWP for small data in the space H^4 intersect L^2( (x^2 + y^2)^4 dx dy) [KnPoVe2000].

Higher order dispersive systems

One can study more general dispersive equations of the general form

u_t + P(nabla) u = F(u, nabla u, ...)

where P(nabla) is an anti-selfadjoint constant coefficient operator, and F involves fewer derivatives on u than P and contains only quadratic and higher terms, and may possibly contain non-local operations such as Hilbert transforms, Hartree-type potentials, or Riesz transforms.  Such equations arise as various approximations to wave equations, see e.g. [Dy1979], [Hog1985].  Smoothing effects for the linear part of the equation were established in [BenKocSau2003], [Hs-p].  Nonlinear local existence in the analytic category was established in [Bd1993].  For smooth but not analytic data some local existence results have been established in [Tar1995], [Tar1997], [Ci-p].