Non-linear Schrodinger equations

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Overview

The free Schrodinger equation

i u_t + D u = 0

where u is a complex-valued function in R^{d+1}, describes the evolution of a free non-relativistic quantum particle in d spatial dimensions.  This equation can be modified in many ways, notably by adding a potential or an obstacle, but we shall be interested in non-linear perturbations such as

u_t - i D u = f(u, u, Du, Du)

where D denotes spatial differentiation.  In such full generality, we refer to this equation as a derivative non-linear Schrodinger equation (D-NLS).  If the non-linearity does not contain derivatives then we refer to this equation as a semilinear Schrodinger equation (NLS).  These equations (particularly the cubic NLS) arise as model equations from several areas of physics.

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Schrodinger estimates

Solutions to the linear Schrodinger equation and its perturbations are either estimated in mixed space-time norms L^q_t L^r_x or L^r_x L^q_t, or in X^{s,b} spaces, defined by

|| u ||_s,b = || <x>^s  <t-|x|²>^b \hat{u} ||₂.

Note that these spaces are not invariant under conjugation.

Linear space-time estimates in which the space norm is evaluated first are known as Strichartz estimates.  They are useful for NLS without derivatives, but are much less useful for derivative non-linearities.  Other linear estimates include smoothing estimates and maximal function estimates.    The X^{s,b} spaces are used primarily for bilinear estimates, although more recently multilinear estimates have begun to appear.  These spaces and estimates first appear in the context of the Schrodinger equation in [Bo1993], although the analogous spaces for the wave equation appeared earlier [RaRe1982], [Be1983] in the context of propogation of singularities.  See also [Bo1993b], [KlMa1993].

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Linear estimates

[More references needed here!]

On R^d:

• If f is in X^{0,1/2+}, then
• (Energy estimate) f is in L^¥_t L²_x
• (Strichartz estimates) f is in L^{2(d+2)/d}_x,t [Sz1977].
• More generally, f is in L^q_t L^r_x whenever 1/q+n/2r = n/4, r < ¥, and q > 2.
• The endpoint q=2, r = 2d/(d-2) is true for d³ 3 [KeTa1998].  When d=2 it fails even in the BMO case [Mo1998], although it still is true for radial functions [Ta2000b], [Stv-p]. In fact the estimates are true assuming for non-radial functions some additional regularity in the angular variable [Ta2000b], although there is a limit as to low little regularity one can impose [MacNkrNaOz-p].
• In the radial case there are additional weighted smoothing estimates available [Vi2001]
• When d=1 one also has f in L⁴_t L^¥_x.
• When d=1 one can refine the L² assumption on the data in rather technical ways on the Fourier side, see e.g. [VaVe2001].
• When d=1, the L^6_{t,x} estimate has a maximizer [Kz-p2]. This maximizer is in fact given by Gaussian beams, with a constant of 12^{-1/12} [Fc-p4]. Similarly when d=2 with the L^4 estimate, which is also given by Gaussian beams with a constant of 2^{-1/2}
• (Kato estimates) D^{1/2} f is in L²_{x,loc} L²_t [Sl1987], [Ve1988]
• When d=1 one can improve this to D^{1/2} f in L^¥_x L²_t
• (Maximal function estimates) In all dimensions one has D^{-s} f is in L²_x,loc L^¥_t for all s > 1/2.
• When d=1 one also has D^{-1/4} f in L⁴_x L^¥_t.
• When d=2 one also has D^{-1/2} f in L⁴_x L^¥_t.  The -1/2 can be raised to -1/2+1/32+e [TaVa2000b], with the corresponding loss in the L⁴ exponent dictated by scaling.  Improvements are certainly possible.
• Variants of some of these estimates exist for manifolds, see [BuGdTz-p]
• Fixed time estimates for free solutions:
• (Energy estimate) If f(0) is in L², then f(t) is also in L².
• (Decay estimate) If f(0) is in L¹, then f(t) has an L^¥ norm of O(t^{-d/2}).
• Interpolants between these two are very useful for proving Strichartz estimates and obtaining scattering.

On T:

• X^{0,3/8} embds into L⁴_x,t  [Bo1993] (see also [HimMis2001]).
• X^{0+,1/2+} embeds into L⁶_x,t [Bo1993].  One cannot remove the + from the 0+ exponent, however it is conjectured in  [Bo1993] that one might be able to embed X^{0,1/2+} into L^{6-}_x,t.

On T^d:

• When d >= 1, X^{d/4 - 1/2+,1/2+} embeds into L⁴_x,t (this is essentially in [Bo1993])
• The endpoint d/4 - 1/2 is probably false in every dimension.

Strichartz estimates are also available on more general manifolds, and in the presence of a potential. Inhomogeneous estimates are also available off

the line of duality; see [Fc-p2] for a discussion.

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Bilinear estimates

• On R² we have the bilinear Strichartz estimate [Bo1999]:

|| uv ||_{1/2+, 0} <~ || u ||_{1/2+, 1/2+}  || v ||_{0+, 1/2+}

• On R² [St1997], [CoDeKnSt-p], [Ta-p2] we have the sharp estimates

|| u v ||_{0, -1/2+} <~ || u ||_{-1/2+, 1/2+}  || v ||_{-1/2+, 1/2+}

|| u v ||_{-1/2-, -1/2+} <~ || u ||_{-3/4+, 1/2+}  || v ||_{-3/4+, 1/2+}

|| u v ||_{-1/2-, -1/2+} <~ || u ||_{-3/4+, 1/2+}  || v ||_{-3/4+, 1/2+}

|| u v ||_{-1/4+, -1/2+} <~ || u ||_{-1/4+, 1/2+}  || v ||_{-1/4+, 1/2+}

• On R [KnPoVe1996b] we have

|| u v ||_{-3/4-, -1/2+} <~ || u ||_{-3/4+, 1/2+}  || v ||_{-3/4+, 1/2+}

|| u v ||_{-3/4+, -1/2+} <~ || u ||_{-3/4+, 1/2+}  || v ||_{-3/4+, 1/2+}

|| u v ||_{-1/4+, -1/2+} <~ || u ||_{-1/4+, 1/2+}  || v ||_{-1/4+, 1/2+}

and [BkOgPo1998]

|| u v ||_{L^¥_t H^1/3_x} <~ || u ||_{0, 1/2+}  || v ||_{0, 1/2+}

Also, if u has frequency |x| ~ R and v has frequency |x| << R then we have (see e.g. [CoKeStTkTa-p4])

|| u v ||_1/2,0 <~ || u ||_{0, 1/2+}  || v ||_{0, 1/2+}

and similarly for uv, uv, uv.

• The s indices on the right cannot be lowered, but perhaps the s indices on the left can be raised in analogy with the R² estimates.  The analogues on T are also known [KnPoVe1996b]:

|| u v ||_{-1/2-, -1/2+} <~ || u ||_{-1/2+, 1/2+}  || v ||_{-1/2+, 1/2+}

|| u v ||_{-3/4+, -1/2+} <~ || u ||_{-1/2+, 1/2+}  || v ||_{-1/2+, 1/2+}

|| u v ||_{0, -1/2+} <~ || u ||_{0, 1/2+}  || v ||_{0, 1/2+}

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Trilinear estimates

• On R we have the following refinement to the L^6 Strichartz inequality [Gr-p2]:

|| u v w ||_{0, 0} <~ || u ||_{0, 1/2+}  || v ||_{-1/4, 1/2+} || w ||_{1/4, 1/2+}

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Multilinear estimates

• In R² we have the variant

|| u_1 ... u_n ||_{1/2+, 1/2+} <~ || u_1 ||_1+,1/2+ ... || u_n ||_1+,1/2+

where each factor u_i is allowed to be conjugated if desired.  See  [St1997b], [CoDeKnSt-p].

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Semilinear Schrodinger (NLS)

[Many thanks to Kenji Nakanishi with valuable help with the scattering theory portion of this section.  However, we are still missing many references and results, e.g. on NLS blowup. - Ed.]

The semilinear Schrodinger equation is

i u_t + D u + l |u|^{p-1} u = 0

for p>1.  (One can also add a potential term, which leads to many physically interesting problems, however the field of Schrodinger operators with potential is far too vast to even attempt to summarize here).  In order to consider this problem in H^s one needs the non-linearity to have at least s degrees of regularity; in other words, we usually assume

p is an odd integer, or p > [s]+1.

This is a Hamiltonian flow with the Hamiltonian

H(u) = ò |Ñ u|²/2 - l |u|^{p+1}/(p+1) dx

and symplectic form

{u, v} = Im ò u v dx.

From t