VIRIEL, MORAWETZ, AND INTERACTION MORAWETZ INEQUALITIES

J. COLLIANDER, M. KEEL, G. STAFFILANI, H. TAKAOKA, AND T. TAO

Abstract. We collect a number of monotonicity formulae for the non-linear Schrödinger equation.

1. Introduction

We consider the nonlinear Schrödinger equation

iu_{t} + Δu = F′(|u|^{2})u,  (1)

or in tensorial notation

iu_{t} + u_{kk} = F′(|u|^{2})u

(the Roman indices j, k will always be summing over spatial indices 1, …, n). Our considerations shall be in arbitrary dimension R^{n+1}, at least initially. We are primarily interested in the defocusing case F (x) := \frac{2}{p+1}x^{(p+1)/2} (so that F′(|u|^{2})u = |u|^{p−1}u), although the focusing case F (x) := −\frac{2}{p+1}x^{(p+1)/2} is of course also interesting.

We need the companion function

G(x) := F′(x)x − F (x);

in the above examples we see that G = \frac{p−1}{2}F, so for the purpose of this discussion we can treat F and G as being essentially the same function (we always assume p > 1).

In this note we discuss several of the conserved quantities and monotone quantities for this equation. We begin with the conserved quantities by introducing the stress energy tensor.

We define the^{1} mass density

T_{00} := |u|^{2}

J.E.C. is supported in part by an N.S.F. Postdoctoral Research Fellowship.

M.K. is supported in part by N.S.F. Grant DMS 9801558.

G.S. is supported in part by N.S.F. Grant DMS 9800879 and by a grant from Hewlett and Packard and the Sloan Foundation.

H.T. is supported in part by J.S.P.S. Grant No. 13740087.

T.T. is a Clay Prize Fellow and is supported in part by grants from the Packard Foundation.

1You might be asking why T_{00} is the mass density when it is more usually the energy density in wave equations. The reason for this has to do with Einstein’s famous equation E = mc^{2},

the mass current / momentum density

T_{0j} := T_{j0} := 2Im(u_{j}\overline{u})

and the momentum current

T_{jk} = T_{kj} := 4Re(u_{k}\overline{u}_{j}) − δ_{jk}Δ(|u|^{2}) + 2δ_{jk}G(|u|^{2}).

Note that once we remove the Laplacian term Δ(|u|^{2}) term from the momentum current T_{jk} then T_{jk} becomes positive definite.

We have the fundamental divergence-free property of the stress-energy tensor:

Lemma 1.1. We have

∂_{t}T_{00} + ∂_{j}T_{0j} = 0
∂_{t}T_{k0} + ∂_{j}T_{kj} = 0.

Proof This is a direct calculation, see for instance [17].

This already gives two conserved quantities: the total mass

∫_{R_{n}} T_{00} = ∫_{R_{n}} |u(t, x)|^{2} dx

and the total momentum

∫_{R_{n}} T_{0j} = 2Im ∫_{R_{n}} u_{j}(t, x)\overline{u(t, x)} dx.

There is also a conservation law coming from the energy density

e_{0}(t, x) := \frac{1}{2}|∇u|^{2} + \frac{1}{2}F (|u|^{2}) = \frac{1}{2}(u_{j}\overline{u_{j}} + F (|u|^{2}))

and energy current

e_{j}(t, x) := Im(u_{jk}\overline{u_{k}} + F′(|u|^{2})u\overline{u_{j}}),

namely

∂_{t}e_{0} + ∂_{j}e_{j} = 0,

and thus we have conservation of total energy

∫ \frac{1}{2}|∇u|^{2} + \frac{1}{2}F (|u|^{2}).  (2)

Unfortunately this conservation law does not lead to nearly as many monotone quantities as the mass and momentum densities, because the energy current is not itself a conserved density.

From Lemma 1.1 we have

∂_{tt}T_{00} = ∂_{jk}T_{jk}.

Now let a(t, x) be an arbitrary real-valued function on R^{n+1}, and define the viriel potential V_{0}^{a} corresponding to a as

V_{0}^{a}(t) := ∫ a(t, x)T_{00}(t, x) dx.

as well as the shift in frequency in ±c^{2} when approximating the Klein-Gordon equation by the Schrödinger equation.

We define the Morawetz action M_{0}^{a} corresponding to a by M_{0}^{a} := ∂_{t}V_{0}^{a}, thus from the above conservation laws and integration by parts we have

M_{0}^{a}(t) = ∂_{t}V_{0}^{a}(t) = ∫ a_{t}T_{00} + a_{j}T_{0j} dx  (3)

and

∂_{t}M_{0}^{a}(t) = ∂_{tt}V_{0}^{a}(t) = ∫ a_{tt}T_{00} + 2a_{tj}T_{0j} + a_{jk}T_{jk} dx.

We can expand this out (doing more integration by parts on the Laplacian term Δ(|u|^{2})) as

∂_{tt}V_{0}^{a}(t) = ∫ (a_{tt} − ΔΔa)|u|^{2}
− 4 ∫ a_{tj}Im(u\overline{u_{j})}
+ 4 ∫ a_{jk}Re(u_{k}\overline{u}_{j})
+ 2 ∫ (Δa)G(|u|^{2}).

In the time dependent case a(t, x) ≠ a(x), the a_{tj} terms present a problem, but we can treat those terms by completing the square, indeed we have

Q^{−2}|∇a_{t}|^{2}|u|^{2} − 4a_{tj}Im(u\overline{u_{j})} + 4Q^{2}|∇u|^{2}Re(u_{k}\overline{u}_{j}) = |Q^{−1}(∇a_{t})u − i2Q∇u|^{2}

for any positive quantity Q = Q(t, x), thus

∂_{tt}V_{0}^{a}(t) = ∫ (a_{tt} − ΔΔa − Q^{−2})|u|^{2}
+ ∫ |Q^{−1}(∇a_{t})u − i2Q∇u|^{2}
+ 4 ∫ (a_{jk} − Q^{2}δ_{jk})Re(u_{k}\overline{u}_{j})
+ 2 ∫ (Δa)G(|u|^{2}).

The game is then to choose Q and a so that as many as possible of these terms are non-negative (up to manageable errors).

We remark that in the time independent case a(t, x) = a(x) one should just choose Q = 0, and the viriel identity simplifies to

\begin{aligned}∂_{tt}V_{0}^{a}(t) = & ∫ −ΔΔa|u|^{2} \\ & + 4 ∫ a_{jk}Re(u_{k}\overline{u}_{j}) \\ & + 2 ∫ (Δa)G(|u|^{2}). \end{aligned}  (4)

while from (3) we have

∂_{t}V_{0}^{a}(t) = M_{0}^{a}(t) = 2Im ∫ a_{j}u_{j}(t, x)\overline{u(t, x)} dx  (5)

Example 0: Conservation of mass. If we set a(t, x) := 1, then V_{0}^{a}(t) is just the total mass V_{0}^{a}(t) = ∫R^{3} |u(t, x)|^{2} dx, and (4) just says that ∂_{tt}V_{0}^{a} = 0, which is consistent with conservation of mass. Indeed in this case we have from (5) that ∂_{t}V_{0}^{a} = M_{0}^{a} = 0.

Example 1: Center of mass. If we set a(t, x) := x, then V_{0}^{a}(t) is the center^{2} of mass V_{0}^{a}(t) = ∫R^{3} x|u(t, x)|^{2} dx, and (4) tells us that ∂_{tt}V_{0}^{a} = 0, thus V_{0}^{a} depends linearly on time. Indeed from (5) we have

∂_{t}V_{0}^{a} = M_{0}^{a} = 2 ∫_{R_{3}} u_{j}(t, x)\overline{u(t, x)}

thus the rate of change of center of mass is equal to the total momentum, which is in turn constant in time.

Example 2: The Viriel identity[8]. If we set a(t, x) := |x|^{2} = x_{j}x_{j}, then Δa = 2n and ΔΔa = 0, so from (5)

∂_{t}V_{0}^{a}(t) = 2 ∫ x_{j}T_{0j} dx = 2 ∫ x_{j}u_{j}\overline{u} dx

while from (4)

∂_{tt}V_{0}^{a}(t) = ∫ 8|∇u|^{2} + 4nG(|u|^{2}) dx.

Thus in the defocusing case the double time derivative of V_{0}^{a}(t) is positive; thus the expectation of |x|^{2} is convex in time. In the focusing case with p ≥ 1 + \frac{4}{n}, this quantity is bounded from above by a multiple of the energy (2), so when the energy is negative and V_{0}^{a}(t) is finite one necessarily has blowup in finite time.

Example 3: The Morawetz identity [14], [13]. If we set a(t, x) := |x|, then a_{j} = x_{j}/|x| and

a_{jk} = \frac{δ_{jk}}{|x|} − \frac{x_{j}x_{k}}{|x|^{3}} ,

so that (when n ≠ 1)

Δa = \frac{n − 1}{|x|}.

Thus by (5)

M_{0}^{a}(t) = 2 ∫ \frac{x_{j}}{|x|}T_{0j} dx = 2 ∫ \frac{x_{j}}{|x|}u_{j}\overline{u} dx = O(∥u(t)∥_{Ḣ^{1/2}(R^{n})}^{2}

(the latter bound being easily obtained when n ≥ 3 by interpolation and Hardy’s inequality ∥\frac{u}{|x|}∥_{L^{2}(R^{n})} ≲ ∥u∥_{Ḣ^{1}(R^{n})}) while from (4) we have

∂_{t}M_{0}^{a}(t) = −(n − 1) ∫ (Δ\frac{1}{|x|})|u|^{2}

+ 4 ∫ \frac{|∇ /_{0}u|^{2}}{|x|} dx

+ 2(n − 1) ∫ \frac{G(|u|^{2})}{|x|} dx,

2Strictly speaking, one has to normalize the total mass ∫R_{3} |u(t, x)|^{2} to equal 1 to obtain this interpretation.

where

|∇ /_{y}u|^{2} := |∇u|^{2} − |\frac{x − y}{|x − y|} · ∇u|^{2}.

In the three-dimensional case n = 3, we have −Δ(\frac{1}{|x|}) = 4πδ, and so

∂_{t}M_{0}^{a}(t) = 8π|u(t, 0)|^{2} + 4 ∫ \frac{|∇ /_{0}u|^{2} + G(u)}{|x|} dx

while

M_{0}^{a}(t) = 2 ∫ Im(\overline{u}\frac{x}{|x|} · ∇u) dx.

This gives us the standard Morawetz inequality (in the defocusing case)

∫_{0}^{T} |u(t, 0)|^{2} dt + ∫_{0}^{T} ∫ \frac{|∇ /_{0}u|^{2} + G(u)}{|x|} dx ≲ sup_{0≤t≤T} ∥u(t)∥2 ; Ḣ^{1/2}.

This identity can be used for instance to demonstrate scattering results for defocusing Schrödinger equations (see [13], [9]). In higher dimensions n ≥ 4, we obtain a similar inequality because

−ΔΔa = \frac{(n − 1)(n − 3)}{|x|^{3}}

is a positive measure. But in dimensions n = 1, 2 the distribution −ΔΔa is quite nasty (it isn’t a measure, for instance, and certainly not a positive measure).

Example 4: The Bourgain-Grillakis truncation of Morawetz’s inequality[1], [10], [2]. The Morawetz inequality is difficult to apply directly to the energy-critical NLS when p = 1 + \frac{4}{n−2} because the right-hand side is not scale invariant. However this can be fixed by a truncation argument as follows. Let R > 0, and let η be a bump function adapted to the ball |x| ≤ R which equals 1 on the ball |x| ≤ R/2. We set a(t, x) := |x|η; for |x| ≤ R/2 we see that a and its derivatives are the same as in Example 2, while in the region R/2 ≤ |x| ≤ R we have the bounds

a_{j} = O(1); a_{jk}, Δa = O(R^{−1}); ΔΔa = O(R^{−3}).

Of course when |x| > R all of these quantities vanish. We thus have

∂_{t}M_{0}^{a}(t) = −(n − 1) ∫_{|x|≤R/2}(Δ\frac{1}{|x|})|u|^{2}

+ 4 ∫_{|x|≤R/2} \frac{|∇ /_{0}u|^{2}}{|x|} dx

+ 2(n − 1) ∫_{|x|≤R/2} \frac{G(|u|^{2})}{|x|} dx

+ O(∫_{|x|∼R} \frac{|u|^{2}}{R^{3}} + \frac{|∇u|^{2} + G(u)}{R} dx).

If we specialize to the defocusing case in three dimensions n = 3 and try to control things by the energy E defined in (2)

∂_{t}M_{0}^{a}(t) = 8π|u(t, 0)|^{2}

+ 4 ∫_{|x|≤R/2} \frac{|∇ /_{0}u|^{2}}{|x|} dx

+ 4 ∫_{|x|≤R/2} \frac{G(|u|^{2})}{|x|} dx

+ O(R^{−1}E).

Meanwhile, from Sobolev and Hölder we have

M_{0}^{a}(t) ≲ ∫_{|x|≲R} |u||∇u| ≲ RE,

and so if we integrate by parts on a time interval I we obtain

∫_{I} |u(t, 0)|^{2} dt + ∫_{I} ∫_{|x|≤R/2} \frac{|∇ /_{0}u|^{2} + G(|u|^{2})}{|x|} dxdt ≲ R^{−1}|I|E + RE.

It is now clear that the optimal relationship between R and I is R ∼ |I|^{1/2}, which is what Bourgain and Grillakis use for the energy-critical NLS:

∫_{I} |u(t, 0)|^{2} dt + ∫_{I} ∫_{|x|≲|I|^{1/2}} \frac{|∇ /_{0}u|^{2} + G(|u|^{2})}{|x|} dxdt ≲ |I|^{1/2}E.

Example 5: Local smoothing estimates. This is a variant of the Morawetz inequality, first observed by Sjölin [16], and Vega [18] (and related to the Kato smoothing effect [12] for non-linear Schrödinger equations). Take a(x) := ⟨x⟩ := (1 + |x|^{2})^{1/2}. Thus function is strictly convex, indeed

a_{jk} = \frac{δ_{jk}}{⟨x⟩} − \frac{x_{j}x_{k}}{⟨x⟩^{3}}

(δ_{jk} being the Kronecker delta function) which is positive definite:

a_{jk}(x)\overline{u_{j}(t, x)}u_{k}(t, x) ≥ c\frac{|∇u(t, x)|^{2}}{⟨x⟩^{3}} .  (6)

In particular we have

Δa = \frac{n − 1}{⟨x⟩} + \frac{1}{⟨x⟩^{3}}

and a further calculation shows

−ΔΔa = \frac{(n − 1)(n − 3)}{⟨x⟩^{3}} + \frac{6(n − 3)}{⟨x⟩^{5}} + \frac{15}{⟨x⟩^{7}}

which is positive when n ≥ 3. Thus by (4) we have

∂_{t}M_{0}^{a}(t) ≥c ∫ \frac{|∇u|^{2}}{⟨x⟩^{3}} dx

+ c ∫ \frac{G(|u|^{2})}{⟨x⟩} dx,

and thus by arguing as with the Morawetz inequality we see that

∫_{0}^{1} ∫_{R_{3}} \frac{|∇u(t, x)|^{2}}{⟨x⟩^{3}} dxdt + ∫R ∫_{R_{3}} \frac{|u(t, x)|^{2}}{⟨x⟩^{7}} dxdt ≤ C∥u(0)∥2 ; Ḣ^{1/2}.

In particular, we have the local smoothing estimate^{3}

∫_{0}^{T} ∫_{|x|≤1} |∇u(t, x)|^{2} + |u(t, x)|^{2} dxdt ≤ C sup_{0≤t≤T} ∥u(t)∥2 ; Ḣ^{1/2}.

By rescaling by any R (and readjusting T accordingly, we obtain

∫_{0}^{T} ∫_{|x|≤R} |∇u(t, x)|^{2} dxdt ≤ CR sup_{0≤t≤T} ∥u(t)∥2 ; Ḣ^{1/2}

and in particular that

∫_{0}^{T} ∫_{R_{3}} \frac{|∇u(t, x)|^{2}}{⟨x⟩^{1+ε}} dxdt ≤ C_{ε} sup_{0≤t≤T} ∥u(t)∥2 ; Ḣ^{1/2}

for any ε > 0. Unfortunately this ε is sharp; if one sets u(0) to be a bump function then the L^{2} mass of ∇u(t, x) is concentrated in the region ⟨x⟩ ≲ ⟨t⟩ and so ∫ ∫ \frac{|∇u(t,x)|^{2}}{⟨x⟩} dxdt will diverge logarithmically.

This example has been extended to general manifolds with no trapped geodesics, see [3], [7]. The estimate is in fact false if the trapped geodesic assumption is dropped [7].

Example 6: The Nakanishi-Morawetz identity [15]. Let λ := \sqrt{t^{2} + x^{2}}. If we set a(t, x) := λ, then we observe the identities

a_{t} = \frac{t}{λ}
a_{j} = \frac{x_{j}}{λ}
a_{tj} = −\frac{tx_{j}}{λ^{3}}
a_{tt} = \frac{1}{λ} − \frac{t^{2}}{λ^{3}} = \frac{|x|^{2}}{λ^{3}}
a_{jk} = \frac{δ_{jk}}{λ} − \frac{x_{j}x_{k}}{λ^{3}} = \frac{δ_{jk}t^{2}}{λ^{3}} + \frac{δ_{jk}|x|^{2} − x_{j}x_{k}}{λ^{3}}
Δa = \frac{n}{λ} − \frac{|x|^{2}}{λ^{3}} .

In particular we observe that

Δa ≥ \frac{t^{2}}{λ^{3}}  (7)

and

ΔΔa = O(λ^{−3}).  (8)

3One can also obtain this estimate by translating the Morawetz inequality obtained previously by an amount O(1) and then averaging over all such translations.

Thus

M_{0}^{a}(t) = ∫ \frac{t}{λ}|u|^{2} + 2Im(\overline{u}\frac{x}{λ} · ∇u) dx

and (if we let Q be a positive quantity to be chosen later)

∂_{t}M_{0}^{a}(t) = ∫ (\frac{|x|^{2}}{λ^{3}} − ΔΔa − \frac{t^{2}|x|^{2}}{λ^{6}Q^{2}} )|u|^{2}

+ ∫ | − \frac{tx}{Qλ^{3}} u − i2Q∇u|^{2}

+ 4 ∫ (\frac{t^{2}}{λ^{3}} − Q^{2})|∇u|^{2} + \frac{|∇ /_{0}u|^{2}|x|^{2}}{λ^{3}}

+ 2 ∫ (Δa)G(|u|^{2}).

If we choose Q^{2} := \frac{t^{2}}{λ^{3}} , then there is some cancellation, and we simplify to

∂_{t}M_{0}^{a}(t) = − ∫ ΔΔa|u|^{2}

+ ∫ \frac{|xu + 2it∇u|^{2}}{λ^{3}}

+ 4 ∫ \frac{|∇ /_{0}u|^{2}|x|^{2}}{λ^{3}}

+ 2 ∫ (Δa)G(|u|^{2})

which we rearrange (using (7), (8)) as

∫ \frac{|xu + 2it∇u|^{2}}{λ^{3}} + \frac{|∇ /_{0}u|^{2}|x|^{2}}{λ^{3}} + \frac{G(u)t^{2}}{λ^{3}} ≲ ∂_{tt}V_{0}^{a}(t) + O(λ^{−3} ∫ |u|^{2}).

If we integrate this on 1 ≤ t ≤ T and control things by the energy E and the L^{2} mass M, then we obtain Nakanishi’s Morawetz inequality

∫_{1}^{T} ∫ \frac{|xu + 2it∇u|^{2}}{λ^{3}} + \frac{|∇ /_{0}u|^{2}|x|^{2}}{λ^{3}} + \frac{G(u)t^{2}}{λ^{3}} dxdt ≲ C(E, M).

Note that this inequality works in all dimensions, including n = 1, 2, and was used by Nakanishi [15] to establish scattering for L^{2} supercritical defocusing non-linear Schrödinger equations in those dimensions. It also precludes the existence of a soliton in the cone |x| ≲ |t| since t^{2}/λ^{3} ∼ 1/t in this cone, and so \frac{G(u)t^{2}}{λ^{3}} would be logarithmically divergent if there was a soliton.

2. Interaction inequalities

Let a(x) be as before; for simplicity we do not assume that a(x) depends on time. Now we consider the viriel interaction potential V^{a} corresponding to a as

V^{a}(t) := ∫ ∫ T_{00}(t, x)T_{00}(t, y)a(x − y) dxdy.

Note that the symmetry in x and y allows us to reduce to the case when a is even, although it will not be necessary to do so here.

VIRIEL AND MORAWETZ INEQUALITIES 9

We then define the Morawetz interaction potential M^{a} := ∂_{t}V^{a} as before, thus by the conservation laws and integration by parts

M^{a}(t) = ∂_{t}V^{a}(t) = ∫ ∫ (T_{00}(t, x)T_{0j}(t, y) − T_{0j}(t, x)T_{00}(t, y))a_{j}(x − y) dxdy. Note that if a is even then a_{j} is odd, and we can use symmetry to equate the two terms in this expression. Now we compute double derivatives: ∂_{t}M^{a}(t) = 2 ∫ ∫ (T_{00}(t, x)T_{jk}(t, y)−2T_{0j}(t, x)T_{0k}(t, y)+T_{jk}(t, x)T_{00}(t, y))a_{jk}(x−y). We now try to simplify the integrand

(T_{00}(t, x)T_{jk}(t, y) − 2T_{0j}(t, x)T_{0k}(t, y) + T_{jk}(t, x)T_{00}(t, y))a_{jk}(x − y) (9)

appearing above; we shall do so in terms of the quantities

A_{j}(t, x, y) := u(t, x)\overline{u_{j}(t, y)} + u_{j}(t, x)\overline{u(t, y)}

and

B_{j}(t, x, y) := u(t, x)u_{j}(t, y) − u_{j}(t, x)u(t, y).

Now we consider (9). This expression splits into the sum of the nonlinear terms

2Δa(x − y)(|u(t, x)|^{2}G(|u(t, y)|^{2}) + G(|u(t, x)|^{2})|u(t, y)|^{2})  (10)

the Laplacian terms

− 2Δa(x − y)(|u(t, x)|^{2}Δ_{y}(|u(t, y)|^{2}) + Δ_{x}(|u(t, x)|^{2})|u(t, y)|^{2})  (11)

and the u∇uu∇u terms

4a_{jk}(x − y)(|u(t, x)|^{2}Re(u_{k}(t, y)\overline{u_{j}(t, y)}) − 2Im(u(t, x)\overline{u_{j}(t, x)})Im(u(t, y)\overline{u_{j}(t, y)})
+ Re(u_{k}(t, x)\overline{u_{j}(t, x)})|u(t, y)|^{2}).

To shorten the notation we omit the t dependence. We expand the previous as

2a_{jk}(x − y)(u(x)\overline{u(x)}(u_{k}(y)\overline{u}_{j}(y) + \overline{u_{k}(y)}u_{j}(y))
+ (u(x)\overline{u_{j}(x)} − \overline{u(x)}u_{j}(x))
(u(y)\overline{u_{k}(y)} − \overline{u(y)}u_{k}(y))
+ (u_{k}(x)\overline{u}_{j}(x) + \overline{u_{k}(x)}u_{j}(x))u(y)\overline{u(y)}).

This can be (rather miraculously) factorized as

2a_{jk}(x − y)(A_{j}(t, x, y)\overline{A_{k}(t, x, y)} + B_{j}(t, x, y)\overline{B_{k}(t, x, y)}).

This can be verified by expanding both expressions as a sum of eight terms, and using the symmetry a_{jk} = a_{kj} to swap j and k whenever necessary.

Combining all these considerations, we thus obtain the identity

\begin{aligned}∂_{t}M^{a}(t) & = ∫ ∫ |u(t, x)|^{2}|u(t, y)|^{2}(−4ΔΔa)(x − y) dxdy \\ & + 4 ∫ ∫ (|u(t, x)|^{2}G(|u(t, y)|^{2}) + G(|u(t, x)|^{2})|u(t, y)|^{2})Δa(x − y) dxdy \\ & + 4 ∫ ∫ (A_{j}(t, x, y)\overline{A_{k}(t, x, y)} + B_{j}(t, x, y)\overline{B_{k}(t, x, y)})a_{jk}(x − y) dxdy \end{aligned} (12)

(compare this with (4)). In the special case when a is non-strictly convex, then the third summand is positive, and we thus have

∂_{t}M^{a}(t) ≥ ∫ ∫ |u(t, x)|^{2}|u(t, y)|^{2}(−4ΔΔa)(x − y) dxdy

+ 4 ∫ ∫ (|u(t, x)|^{2}G(|u(t, y)|^{2}) + G(|u(t, x)|^{2})|u(t, y)|^{2})Δa(x − y) dxdy.

Example 7: The interaction Morawetz inequality [4], [?]. We set a(x − y) = |x − y| and n = 3, so we obtain

∂_{t}M^{a}(t) ≥ 16π ∫ ∫ |u(t, x)|^{2}|u(t, y)|^{2}δ(x − y) dxdy

+ 4 ∫ ∫ (|u(t, x)|^{2}G(|u(t, y)|^{2}) + G(|u(t, x)|^{2})|u(t, y)|^{2})\frac{n − 1}{|x|} dxdy

In the defocusing case the second term is non-negative, hence

∂_{t}M^{a}(t) ≥ 16π ∫ |u(t, x)|^{4} dx.

Meanwhile

M^{a}(t) = 4 ∫ ∫ |u(t, x)|^{2}Im(u(t, y)\overline{\frac{x − y}{|x − y|} · ∇u(t, x)}),

and so we obtain the interaction Morawetz inequality

∫_{0}^{T} ∫ |u(t, x)|^{4} dxdt ≲ ∥u(0)∥_{2}^{2} sup_{0≤t≤T} ∥u(t)∥2 ; Ḣ^{1/2}.

This gives a more direct way to establish scattering for certain defocusing Schrödinger equations, see [4], [5].

It would be interesting to see if some of the other previous examples could also be extended to the interaction setting. In particular, an interaction inequality in dimensions n = 1 and n = 2 would be of significant interest. In the energy-critical case p = 1+ \frac{4}{n−2} it would be of interest to replace this inequality by one in which the right-hand side depended only on the energy (as in the Bourgain-Grillakis version of the Morawetz inequality). Some progress in this direction is in [6].

3. Classical analogues

We now present a classical analogue of the above analysis in the linear case F ≡ 0.

Instead of a wave u, we have a collection of “billiard balls” or “particles” P_{1}, …, P_{N} such that each billiard ball P_{α} has (constant) mass m_{α}, position x_{α}, and velocity v_{α}, with the latter two evolving by Newton’s laws of motion

ẋ_{α}(t) = v_{α}(t)
v̇_{α}(t) = 0

with the Schrödinger equation

iψ_{t} = \frac{ℏ}{2}Δψ

with 0 < ℏ ≪ 1 by the Gaussian beam solution

ψ := (ℏ(t + iT))^{−n/2} ∑_{α=1}^{N} m_{α}^{1/2} exp(\frac{i(x − x_{α}(t))^{2}}{2ℏ(t + iT)} + \frac{iv_{α} · x}{ℏ} − \frac{i|v_{α}|^{2}t}{2ℏ})

which is coherent for times |t| ≲ T (for each such time, ψ(t) is localized in \sqrt{ℏT} - neighbourhoods of x_{α}(t) for α = 1, …, N, with the “particle” at x_{α} having an L^{2} mass of m_{α}, a spatial frequency of v_{α}/ℏ, and a time frequency of |v_{α}|^{2}/2ℏ, hence a momentum of m_{α}v_{α} and an energy of \frac{1}{2}m_{α}|v_{α}|^{2} by Planck’s laws). But I do not know how to do a similar relation for the nonlinear Schrodinger equation and a classical Newtonian equation (the obvious thing would be to add a potential energy term to Newton’s equations, but this corresponds to adding a linear perturbation V ψ to the Schrödinger equation, not a non-linear perturbation).

The heuristics of semi-classical analysis assert that any classical observable of the form

⟨P ⟩_{c} := ∑_{α} m_{α}P (x_{α}, v_{α})

corresonds to the quantum observable

⟨P ⟩_{q} := ⟨ψ, P (x, −iℏ∇_{x})ψ⟩_{L^{2}(R^{n})}

where P (x, −iℏ∇_{x}) is the pseudo-differential operator with symbol P (x, ℏξ). (This heuristic is somewhat imprecise because there is some freedom to quantize, i.e. to select a given pseudo-differential operator for each observable). Observe that Newton’s laws show that the classical observable ⟨P ⟩_{c} evolves according to the equation

∂_{t}⟨P ⟩_{c} = ⟨{H, P }⟩_{c}

where H is the classical Hamiltonian H(x, v) = \frac{1}{2}|v|^{2}, and \{H, P \} is the Poisson bracket

\{H, P \}(x, v) := \frac{∂H}{∂v} · \frac{∂P}{∂x} − \frac{∂H}{∂x} · \frac{∂P}{∂v} .

Meanwhile, the quantum observable ⟨P ⟩_{q} can be seen (for P self-adjoint) to evolve according to the Heisenberg equation

∂_{t}⟨P ⟩_{q} = ⟨\frac{i}{ℏ}[H, P]⟩_{q}  (13)

where H is the quantum Hamiltonian H(x, −iℏ∇_{x}) = \frac{ℏ^{2}}{2} Δ, and [H, P] is the Lie bracket [H, P] := HP − P H. The correspondence between the classical world and the quantum world is then captured by the statement from the theory of pseudodifferential operators that the symbol of the Lie bracket of two quantum operators is, to top order only, given by the Poisson bracket of the corresponding classical functions. Of course the lower order terms of the quantum operators have no classical analogue, which is why the quantum equation can exhibit significantly non-classical behavior when ℏ is large.

The above observables were single particle quantities. One can also define two particle observables

⟨P ⟩_{c} := ∑_{α} ∑_{β} m_{α}m_{β}P (x_{α}, x_{β}, v_{α}, v_{β})

and their corresponding quantum analogues

⟨P ⟩_{q} := ⟨ψ ⊗ ψ, P (x, y, −iℏ∇_{x}, −iℏ∇_{y})ψ ⊗ ψ⟩_{L^{2}(R^{n}×R^{n})}

where ψ ⊗ ψ(t, x, y) := ψ(t, x)ψ(t, y) is the tensor product of ψ with itself; notice that this function itself evolves according to the Schrödinger equation

i∂_{t}(ψ ⊗ ψ) = \frac{ℏ}{2}Δ_{R^{n}×R^{n}}(ψ ⊗ ψ).

This tensor product property is very special to the Schrödinger equation (and the closely related heat equation); no such simple relationship exists for other models such as wave or Klein-Gordon equations. One again has analgoues of the above evolution equations (one basically enlarges phase space by a factor of two, and replaces the single-particle Hamiltonian H(x, v) = \frac{1}{2}|v|^{2} with the two particle Hamiltonian H(x_{1}, x_{2}, v_{1}, v_{2}) = \frac{1}{2}|v_{1}|^{2} + \frac{1}{2}|v_{2}|^{2}).

Given a potential function a(t, x), we can then define the classical viriel potential

V_{0}^{a}(t) := ∑_{α} m_{α}a(t, x_{α})

with the corresponding classical Morawetz action

M_{0}^{a}(t) := ∂_{t}V_{0}^{a}(t) = ∑_{α} ∇a(t, x_{α}) · v_{α} + a_{t}(t, x_{α})

with derivative

∂_{t}M_{0}^{a}(t) = ∑_{α} ∇^{2}a(t, x_{α}) · (v_{α} ⊗ v_{α}) + 2∇a_{t}(t, x_{α}) · v_{α} + a_{tt}(t, x_{α})

In particular, if a(t, x) = a(x) is time independent and weakly convex (so that ∇^{2}a is positive semi-definite everywhere, and also (∇a(t, x_{α}) − ∇a(t, x_{β})) · (x_{α} − x_{β}) is always non-negative by the mean-value theorem) then M_{0}^{a} is an increasing quantity. (Compare this with the quantum case, where one only gets monotonicity of M_{0}^{a} if one adds the additional assumption −ΔΔa ≥ 0; note that weak convexity already implies that Δa ≥ 0).

Similarly, if a is even, we can define the classical virial interaction potential

V^{a}(t) := ∑_{α} ∑_{β} a(t, x_{α} − x_{β})

with classical Morawetz interaction potential

M^{a}(t) := ∂_{t}V^{a}(t) = ∑_{α} ∑_{β} ∇a(t, x_{α} − x_{β}) · (v_{α} − v_{β}) + a_{t}(t, x_{α} − x_{β})

with derivative

∂_{t}M^{a}(t) = ∑_{α} ∑_{β} ∇^{2}a(t, x_{α} − x_{β}) · ((v_{α} − v_{β}) ⊗ (v_{α} − v_{β})) + 2∇a_{t}(t, x_{α} − x_{β}) · (v_{α} − v_{β}) + a_{tt}(t, x_{α} − x_{β})

+ ∑_{α,β,γ} F′(|x_{α} − x_{γ}|^{2})(∇a(t, x_{α} − x_{β}) − ∇a(t, x_{γ} − x_{β})) · (x_{α} − x_{γ})

(I used some symmetry games with the last summand, exploiting the assumption that a is even). Again, if a is time independent and weakly convex then the classical Morawetz interaction is increasing (in the quantum case, we again need the additional assumption −ΔΔa ≥ 0).

It’s instructive to review the earlier Examples in the classical framework, using the standard quantization heuristics. Of course, these heuristics are only justifiable in the free case F = 0; I don’t understand nonlinear quantization at all.

One might wonder if the Morawetz interaction inequality

∫_{0}^{T} ∫ |u(t, x)|^{4} dxdt ≲ ∥u(0)∥_{2}^{2} sup_{0≤t≤T} ∥u(t)∥2 ; Ḣ^{1/2}

has a classical analogue. It is a little dangerous to do so because the left-hand side is a lower order term, which is not supposed to have a classical interpretation. Nevertheless if one is willing to proceed in a very non-rigorous fashion one can still get something like an interpretation. The L^{4} norm can be rewritten as

∫_{0}^{T} ∫ ∫ |u(t, x)|^{2}|u(t, y)|^{2}δ(x − y) dxdydt

and as a naive guess one might expect that this quantity should measure the number of collisions:

∫_{0}^{T} ∑_{α,β} m_{α}m_{β}δ(x_{α}(t) − x_{β}(t)).

This isn’t quite right, because the diagonal terms α = β would give an infinite contribution. A better classical analogue would be

COLLISIONS := ∑_{t,α,β:x_{α}(t)=x_{β}(t)} m_{α}m_{β}|v_{α}(t) − v_{β}(t)|;

one weights each collision by the relative velocity of the colliding particles. (If you like, each collision should give off a sound with volume m_{α}m_{β}|v_{α}(t)−v_{β}(t)|, and so COLLISIONS measures how noisy the collection of particles are from bumping into each other.) In particular, the diagonal terms α = β give no contribution. The classical analogue of the interaction Morawetz inequality is then

COLLISIONS ≤ 4(∑_{α} m_{α})(∑_{α} m_{α}|v_{α}|).

This is proven by considering the classical Morawetz interaction potential

M(t) = ∑_{α,β} m_{α}m_{β}(v_{α} − v_{β}) · \frac{x_{α} − x_{β}}{|x_{α} − x_{β}|}

(the diagonal terms α = β are zero and so can be excluded), which is clearly bounded in magnitude by 2(∑_{α} m_{α})(∑_{α} m_{α}|v_{α}|), and observing that M(t) is monotone increasing, and jumps up by m_{α}m_{β}|v_{α} − v_{β}| every time particles P_{α} and P_{β} collide.

4. Positive commutators

The Heisenberg equation (13) is the starting point for a connection with another viewpoint of monotonicity formulae (at least in the linear setting F ≡ 0), namely the method of positive commutators. Recall that if u be a (Schwartz) solution to the Schrödinger equation iu_{t} + Δu = 0 and P is a time-independent self-adjoint linear operator on R^{n}, then we have the Heisenberg equation

∂_{t}⟨Au(t), u(t)⟩_{L^{2}(R^{n})} = ⟨iAΔu, u⟩_{L^{2}(R^{n})} + ⟨Au, iΔu⟩_{L^{2}(R^{n})}
= ⟨i[A, Δ_{M}]u(t), u(t)⟩_{L^{2}(R^{n})}.

Note that both sides of this equation are real if A is self-adjoint. From the fundamental theorem of calculus we see in particular that

Re ∫_{0}^{T} ⟨i[A, Δ]u(t), u(t)⟩_{L^{2}(R^{n})} dt ≤ 2 sup_{t=0,T} |Re⟨Au(t), u(t)⟩_{L^{2}(R^{n})}|. (14)

Note that we may omit the real parts on both sides if A is self-adjoint. The strategy of the method of positive commutators is to select a self-adjoint operator A so that the operator i[A, Δ_{M}] is positive (semi-)definite, possibly modulo a lower order error term, and then apply (14) to obtain a non-trivial spacetime bound on u. From the pseudo-differential calculus and the sharp Garding inequality, one only expects to be able to do this if the symbol a(x, ξ) of A increases along the bicharacteristic flow of Δ (i.e. geodesic flow in phase space). At first glance this method only seems able to obtain quadratic estimates on u, but we will can obtain quartic estimates (and in particular the L^{4} bound in the interaction Morawetz inequality) by replacing u by the tensor product U(t, x, y) := u(t, x)u(t, y).

To apply (14) we of course need to compute some commutators.

Lemma 4.1. Let a(x) be a real-valued tempered distribution on M, which of course can be thought of as a multiplier operator (af)(x) := a(x)f(x) on Schwartz functions. Then we have the single commutator identity

i[a, Δ] = −2ia_{j}∂_{j} − i(Δa)  (15)

and the double commutator identity

− [[a, Δ], Δ] = −4∂_{k}a_{jk}∂_{j} − (ΔΔa).  (16)

Proof The identity (15) follows directly from the Leibnitz rule Δ(af) = aΔf + 2a_{j}f_{j} + (Δa)f. Now we prove (16). Observe that both sides of (16) are secondorder self-adjoint operators with real coefficients. Also, by applying both sides to the constant function 1 we see that the constant terms of (16) match. Thus it will suffice to show that the principal symbols of both sides match, since this would mean that the difference of the left-hand side and right-hand side is a self-adjoint first-order operator with real coefficients and no constant term, which is necessarily zero.

Let us now ignore all terms of first or zeroth order. Inserting (15), we see that (16) becomes

2[a_{j}∂_{j}, ∂_{kk}] + l.o.t. = −4a_{jk}∂_{jk} + l.o.t.  (17)

But this can be easily verified by expanding the commutator using the Leibnitz rule and discarding all terms of first or lower order.

From this Lemma and (14) (and an integration by parts) we thus have

4 ∫_{0}^{T} ∫_{R_{n}} a_{jk}\overline{u_{j}}u_{k} dxdt + ∫_{0}^{T} ∫_{R_{n}}(−ΔΔa)|u|^{2} dxdt ≤ C sup_{t=0,T} | ∫_{R_{n}} a_{j}u_{j}\overline{u} dx|

(18)

which is of course just a reformulation of (4), and can be used to give the oneparticle estimates (at least in the linear case F ≡ 0).

This formulation also can be used to give the interaction estimates. We apply (18) to the Schrödinger wave U(t, x, y) := u(t, x)u(t, y) and the multiplier a(x, y) = a(x−y) to obtain

4 ∫_{0}^{T} ∫_{R_{n}} ∫_{R_{n}} a_{αβ}(x − y)\overline{U_{α}}U_{β} dxdydt

+ ∫_{0}^{T} ∫_{R_{n}} ∫_{R_{n}}(−4ΔΔa)(x − y)|u(t, x)|^{2}|u(t, y)|^{2} dxdydt

≤ C sup_{t=0,T} | ∫_{R_{n}} ∫_{R_{n}} a_{α}(x − y)U_{α}\overline{U} dxdy|

where α, β range over the 2n indices of the product manifold R^{n} × R^{n} (which also explains the factor of 4 in ΔΔa). One can simplify this using the translationinvariant form a(x, y) = a(x, y) of the multiplier to obtain what is basically a reformulation of (12). The interaction Morawetz inequality in particular is just a reflection of the (weak) convexity of the metric function |x − y|, as well as the positivity of −ΔΔ|x − y|, which is a positive scalar multiple of the Dirac delta δ(x − y).

This positive commutator formulation of the interaction Morawetz inequality lends itself very well for extending the above results to other manifolds than R^{n}; see [11].

5. Klein-Gordon Morawetz?

It is an interesting question (raised by Jalal Shatah) whether all these computations have analogues in, say, the non-linear wave/Klein-Gordon equation

−ϕ_{tt} + Δϕ = F′(|ϕ|^{2})ϕ

where we allow F (x) to contain linear terms m^{2}x (which would then correspond to a mass term such as m^{2}ϕ in the above equation). For simplicity let us take ϕ to be real. Certainly we still have an energy density

T_{00} := \frac{1}{2}|∇ϕ|^{2} + \frac{1}{2}|ϕ_{t}|^{2} + \frac{1}{2}F (|ϕ|^{2})

and an energy current / momentum density

T_{0j} := T_{j0} := −ϕ_{t}ϕ_{j}

and a momentum current

T_{jk} = T_{kj} := ϕ_{j}ϕ_{k} + \frac{1}{2}δ_{jk}(|ϕ_{t}|^{2} − |∇ϕ|^{2} − F (|ϕ|^{2}))

with the same conservation laws

∂_{t}T_{00} + ∂_{j}T_{0j} = 0
∂_{t}T_{k0} + ∂_{j}T_{kj} = 0

This allows us, formally at least, to do many of the same calculations for Klein-Gordon as we did for Schrödinger. For instance, one can perform Example 3 all over again with little change. The main problem is that T_{jk} has no good positivity properties; recall in the Schrödinger case that we made T_{jk} into a positive definite matrix by removing a Laplacian term Δ(|u|^{2}), which could be treated by integration by parts. In the Klein-Gordon case I see no easy way to make T_{jk} positive, except perhaps in the exceptional case of one spatial dimension. This has defeated any attempts so far to generate a Klein-Gordon interaction Morawetz inequality, though perhaps it is just a matter of being more clever with integration by parts.

6. Appendix: Schrödinger as nonrelativistic limit of Klein-Gordon

We shall need three positive parameters: the Planck constant ℏ, the mass m, and the speed of light c. From these parameters we can form the rest energy E := mc^{2}, the frequency E =: ℏν and the wavelength λ := cν.

The free Schrödinger equation is given by

iψ_{t} = \frac{ℏ}{2m}Δψ

while the free Klein-Gordon equation is given by

−ϕ_{tt} + c^{2}Δϕ = ν^{2}ϕ = \frac{m^{2}c^{4}}{ℏ^{2}} ϕ.

The relationship between the two is as follows: if we start with the Klein-Gordon equation and write

ϕ = e^{itE/ℏ}ψ  (19)

then ψ obeys the equation

iψ_{t} + \frac{ℏ}{2mc^{2}} ψ_{tt} = \frac{ℏ}{2m}Δψ.

Thus in the nonrelativistic limit c → ∞ (holding ℏ and m fixed), we (formally, at least) recover the Schrödinger equation.

Let ϕ solve the Klein-Gordon equation. We define the energy density

T_{00} = \frac{ℏ^{2}}{2mc^{2}} |ϕ_{t}|^{2} + \frac{ℏ^{2}}{2m}|∇ϕ|^{2} + \frac{mc^{2}}{2} |ϕ|^{2}

and the momentum density

T_{0j} := \frac{ℏ^{2}}{mc^{2}} Re(\overline{ϕ_{t}}ϕ_{j}).

We have the energy conservation law

∂_{t}T_{00} = c^{2}∂_{j}T_{0j}

and the momentum conservation law

∂_{t}T_{0j} = ∂_{k}T_{jk}

where T_{jk} is the stress tensor

T_{jk} := \frac{ℏ^{2}}{m} Re(\overline{ϕ_{j}}ϕ_{k}) + \frac{ℏ^{2}}{2mc^{2}} δ_{jk}(|ϕ_{t}|^{2} − |∇ϕ|^{2} − \frac{m^{2}c^{4}}{ℏ^{2}} |ϕ|^{2}).

Using (19), we have the expansions

T_{00} = mc^{2}|ψ|^{2} + \frac{ℏ^{2}}{m} |∇ϕ|^{2} − \frac{ℏ^{2}}{2m}∂_{j}Re(\overline{ψ}ψ_{j}) + \frac{ℏ^{2}}{2mc^{2}} ∂_{t}Re(\overline{ψ}ψ_{t})

and

T_{0j} = ℏIm(\overline{ψ}ψ_{j}) + \frac{ℏ^{2}}{mc^{2}} Re(\overline{ψ_{t}}ψ_{j}).

Thus in the nonrelativistic limit c → ∞, the energy ∫ T_{00} becomes the rest energy E ∫ |ψ|^{2} plus the kinetic energy \frac{ℏ^{2}}{m} ∫ |∇ϕ|^{2}, while the momentum similarly becomes ℏ ∫ Im(\overline{ψ}ψ_{j}). The stress becomes

T_{jk} ≈ \frac{ℏ^{2}}{m} Re(\overline{ψ_{j}}ψ_{k}) + ℏδ_{jk}Im(\overline{ψ}ψ_{t}).

If one only looks at the highest order terms (viewed in terms of powers of c, setting ℏ and m to 1) and then normalizes out the speed of light (by then setting c = 1), we recover the Schrödinger densities and currents for mass and momentum; note how the E = mc^{2} relationship turns energy into mass up to top order.

7. Acknowledgements

These notes grew out of discussions and joint work with Jim Colliander, Mark Keel, Gigliola Staffliani, and Hideo Takaoka, as well as later discussions and joint work with with Andrew Hassell and Jared Wunsch. We also thank Cathleen Morawetz and Jalal Shatah for useful comments.

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