On a systematic understanding ofsmoothing estimates forwater wave equations
JSPS-DFG Japanese-German Graduate Externship
Kickoﬀ Meeting
Waseda University, June 18, 2014

Mitsuru Sugimoto (Nagoya Univ.)Based on the joint work withMichael Ruzhansky (Imperial College)
arXiv:math.AP/0612274

Equations of water waves• Korteweg-de Vries (shallow water wave)∂tu + ∂x3u + u∂xu = 0• Benjamin-Ono (deep water wave)∂tu − ∂x|Dx|u + u∂xu = 01

• Davey-Stewartson (shallow water wave of 2D)i∂tu − ∂x2u ± ∂y2u = c1|u|2u + c2u∂x2∆−1|u|2i∂tu − ∂x2u ± ∂y2u = c1|u|2u + c2u∂x2□−1|u|2*By further perturbation analysis....

2

• Dysthe (deep water wave of 2D)()

1 ∂u
1 ∂ 2u 1 ∂ 2u∂u2++−−u|u|2i∂t2 ∂x2 ∂y 2 4 ∂x2)(()
33i ∂ u∂ ui∂u∂u−6+ u u−u=32
8 ∂x
∂x∂y2
∂x
∂x
5i 2 ∂u∂ 2|u|2−1− |u|− u|D|2∂x∂x23

• Hogan (deep water wave of 2D)()∂ 2u∂ 2u∂u∂u+ cg+ p 2 + q 2 − γu|u|22i
∂t
∂x
∂y∂x
∂ 3u
∂ 3u
∂u
2= − ir 3 − is− iµ|u|
2∂x
∂x∂y
∂x
2 |u|2∂u∂+ iν|u|2− u|D|−1∂x∂x2(c, p, q, γ, r, s, µ, ν :real parameters)4• Shrira (3D packet of internal gravity wave)∂uωkk ∂ 2u
ωll ∂ 2u∂ 2ui+++ ωkl
22∂t2 ∂x2 ∂y∂x∂y[]
3333ωkkk ∂ uωkkl ∂ uωkll ∂ uωlll ∂ u−i+++3226 ∂x2 ∂x ∂y2 ∂x∂y6 ∂y 3()∂u∂u+iγu u−u=0∂s∂s∂ : derivative along a line, γ, ω : real parameters)( ∂s...
5

The equation we will consider

i∂tu + a(D1, D2)u = F (b(D1, D2)u),u(0, x) = φ(x),wherea(ξ1, ξ2) is a polynomial of order 3,b(ξ1, ξ2) is a function of growth order 1.
Dysthe, Hogan, Shrira equations are of this form.
6

Two important tools to show the well-posedness• Strichartz estimate
eita(D1,D2)φ(x)qLt (Lrx )≲ ∥φ∥L2• Smoothing estimate∗⟨x⟩−s|D|eita(D1,D2)φ(x)L2t,x≲ ∥φ∥L27

Historically, smoothing estimate was first shown to theequation ∂ u + ∂ 3u + u∂ u = 0,xtx
 u(0, x) = φ(x) ∈ L2(R).

The solution u = u(t, x) (t, x ∈ R) satisfies∫ T ∫ R−T −R|∂xu(x, t)|2 dxdt ≤ c(T, R, ∥φ∥L2 )(Kato 1983).
8

Normal forms
By linear chage of variables, polynomials a(ξ1, ξ2) of order 3 are redueced to one of the following normal forms:
ξ13, ξ1ξ22, ξ13 + ξ23, ξ13 − ξ1ξ22,
ξ13 + ξ22, ξ13 + ξ1ξ2, ξ1ξ22 + ξ12,
ξ13 + ξ23 + ξ1ξ2, ξ13 − ξ1ξ22 + ξ12 + ξ22(modulo polynomials of order 1)
Strichartz estimates are given for them except for thecase a(ξ1, ξ2) = ξ13, ξ1ξ22 (Ben-Artzi, Koch, Saut 2003).
9

What are known for smoothing estimates?
We consider smoothing estimates for solutionsu(t, x) = eita(Dx)φ(x)to general equations{(i∂t + a(Dx)) u(t, x) = 0u(0, x) = φ(x) ∈ L2(Rn)where a(ξ) are real-valued and dispersive in the following senses:
10Principal term only(H)a(ξ) = am(ξ),∇am(ξ) ̸= 0(ξ ̸= 0),

where principal term am(ξ) satisfies• am(ξ) ∈ C ∞(Rn \ 0),• am(λξ) = λmam(ξ) (λ > 0, ξ ̸= 0)
Example: a(ξ1, ξ2) = ξ13 + ξ23, ξ13 − ξ1ξ22 satisfy (H).
11

Principal term + Lower oredr terms:(L)•a(ξ) ∈ C ∞(Rn),∇am(ξ) ̸= 0 (ξ ̸= 0), ∇a(ξ) ̸= 0 (ξ ∈ Rn)•|∂ α(a(ξ) − am(ξ))| ≤ C⟨ξ⟩m−1−|α|•a(ξ) ∈ C ∞(Rn),•|∂ α(a(ξ) − am(ξ))| ≤ C⟨ξ⟩m−1−|α|(|ξ| ≥ 1)⇐⇒(L)|∇a(ξ)| ≥ C⟨ξ⟩m−1,(|ξ| ≥ 1)

Example: a(ξ) = ξ13 + ξ23 + ξ1 satisfies (L).
12

Theorem 1. Assume (H) or (L). Let m > 0 and lets > 1/2. Then we have⟨x⟩−s|Dx|(m−1)/2eita(Dx)φ(x)L2(Rt ×Rnx)≤ C∥φ∥L2(Rn)(Ruzhansky and S. 2012).
Remark. Any polynomial a(ξ) which satisfies the estimate in Theorem 1 has to be dispersive, that is∇am(ξ) ̸= 0(ξ ̸= 0).

(Hoshiro 2003)
13

Non-dispersive case
What happens if{(i∂t + a(Dx)) u(t, x) = 0u(0, x) = φ(x) ∈ L2(Rn)does not satisfy∇a(ξ) ̸= 0(ξ ∈ Rn) ?

We cannot have smoothing estimates (Hoshiro 2003).
14But such case appears naturally in equation of water
waves:
If fact, normal forms:
ξ13,

ξ1ξ22,

ξ13 + ξ22,

ξ13 + ξ23,

ξ13 − ξ1ξ22,

ξ13 + ξ1ξ2,

ξ1ξ22 + ξ12,

ξ13 + ξ23 + ξ1ξ2,ξ13 − ξ1ξ22 + ξ12 + ξ22

does not satisfies (H) nor (L).
15

Invariant estimate
We suggest an estimate which we expect to have fornon-dispersive equations:⟨x⟩−s|∇a(Dx)|1/2eita(Dx)φ(x)L2(Rt ×Rnx)≤ C∥φ∥L2(Rn)x(s > 1/2)

and let us call it invariant estimate.
This estimate has a number of advantages:
16• in the dispersive case ∇a(ξ) ̸= 0, it is equivalent tothe usual estimate (Theorem 1);• it is invariant under canonical transformations for theoperator a(Dx);• it does continue to hold for a variety of non-dispersive
operators a(Dx), where ∇a(ξ) may become zero onsome set and when the usual estimate fails;
17

Methods of approach

1. Comparison principle · · · comparison of the symbol
implies the comparison of estimate. (New idea)

2. Canonical Transformation · · · shift an equation toanother simple one. (Egorov’s theorem)

These are new method even for dispersive equations!
18Comparison Principle
Theorem 2 (1D case). Let f, g ∈ C 1(R) be real-valued
and strictly monotone. If σ, τ ∈ C 0(R) satisfy|σ(ξ)||τ (ξ)|≤|f ′(ξ)|1/2|g ′(ξ)|1/2
then we have∥σ(Dx)eitf (Dx)φ(x)∥L2(Rt) ≤ ∥τ (Dx)eitg(Dx)φ(x)∥L2(Rt)for all x ∈ R.
19

Theorem 3 (2D case). Let f (ξ, η), g(ξ, η) ∈ C 1(R2) bereal-valued and strictly monotone in ξ ∈ R for each fixedη ∈ R. If σ, τ ∈ C 0(R2) satisfy|σ(ξ, η)|fξ (ξ, η)

1/2≤|τ (ξ, η)|gξ (ξ, η)

1/2

then we haveσ(Dx, Dy )eitf (Dx,Dy )φ(x, y)

L2(Rt ×Ry )≤ ∥τ (Dx, Dy )eitg(Dx,Dy )φ(x, y)∥L2(Rt×Ry )for all x ∈ R.
20

Theorem 4 (Radially Symmetric case). Let f, g ∈ C 1(R+)
be real-valued and strictly monotone. If σ, τ ∈ C 0(R+)
satisfy|σ(ρ)||f ′(ρ)|1/2≤|τ (ρ)||g ′(ρ)|1/2

then we have∥σ(|Dx|)eitf (|Dx|)φ(x)∥L2(Rt)≤ ∥τ (|Dx|)eitg(|Dx|)φ(x)∥L2(Rt)for all x ∈ Rn.
21

Low dimensional model estimates
By the comparison principal, we can show the equivalence of low dimensional estimates of various type:In the 1D case, we have (l, m > 0).
√mm |Dx|(m−1)/2eit|Dx| φ(x) 2L (Rt )√l= l |Dx|(l−1)/2eit|Dx| φ(x)L2(Rt )

for all x ∈ R. Here supp φb ⊂ [0, +∞) or (−∞, 0].
22(1)In the 2D case, we have (l, m > 0)m−1(m−1)/2itD|D|
xy|Dy |eφ(x, y)

L2(Rt ×Ry )l−1(l−1)/2itD|D|
xyeφ(x, y)= |Dy |L2(Rt ×Ry )for all x ∈ R.

23(2)

On the other hand, in 1D case, we have
eitDx φ(x) = φ(x + t)
hence we have trivially
eitDx φ(x)L2(Rt )

= ∥φ∥L2(Rx)(3)

for all x ∈ R.
Using the equality (3), the right hand sides of (1) and(2) with l = 1 can be estimated, and we have:24• 1D Casem(m−1)/2
it|D|
x|Dx|
eφ(x)

L2 (Rt )≤ C∥φ∥L2(Rx)• 2D Casem−1(m−1)/2
itD|D|
xy|Dy |eφ(x, y)L2(Rt ×Ry )≤ C∥φ∥L2(R2 )x,yfor all x ∈ R.
The following is straightforward from these estimates:25

Proposition 1. Suppose m > 0 and s > 1/2. Then forn ≥ 1 we have−s⟨x⟩m|(m−1)/2it|Dn|Dn|eφ(x)L2 (Rt ×Rnx)≤ C∥φ∥L2(Rn)xand for n ≥ 2 we have−s⟨x⟩m−1(m−1)/2itD|D|
n1|Dn|eφ(x)L2 (Rt ×Rnx)≤ C∥φ∥L2(Rn),x

where Dx = (D1, . . . , Dn).
The first one gives the invariant estimates for the normalform a(ξ1, ξ2) = ξ1326

Canonical Transformation
Smoothing estimate for dispersive case (Theorem 1) canbe reduced to low dimensional model estimates (Proposition 1) by the Canonical transformation:
For the change of variable ψ : Rn \ 0 → Rn \ 0 satisfying
ψ(λξ) = λψ(ξ) for all λ > 0 and ξ ∈ Rn \ 0, we set
Iu(x) = F −1[(F u)(ψ(ξ))](x).
27Then we have the relationa(Dx) · I = I · σ(Dx),a(ξ) = (σ ◦ ψ )(ξ).

In dispersive case, we may replace a(D) byσ(D) = |Dn|m · · · if a(ξ) is ellipticσ(D) = D1|Dn|m−1 · · · if a(ξ) is non-elliptic
by canonical transformation!

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Summary for dispersive case
• Trivial estimate eitDx φ(x)⇓L2 (Rt )= ∥φ∥L2(Rx)(Comparison Principle)• Low dimensional model estimate⇓(Canonical Transform)• Smoothing estimates for dispersive equations
29

Secondary comparison
By using comparison principle again to the smoothing
estimates obtained from the comparison principle, we
can have new estimates.
This is a powerful tool to induce the invariant estimatesfor non-dispersive equations.

30

Radially symmetric case
From Theorem 1 with a(ξ) = |ξ|m, we obtainm−s(m−1)/2it|D|x⟨x⟩ |Dx|eφL2(Rt ×Rnx)≤ C∥φ∥L2(Rn).
x

If we set g(ρ) = ρm, τ (ρ) = ρ(m−1)/2, then we have√|τ (ρ)|/|g ′(ρ)|1/2 = 1/ m.
Hence by the comparison result for radially symmetriccase (Theorem 4), we have
31

Theorem 5. Let f ∈ C 1(R+) be real-valued and strictly
monotone. If σ ∈ C 0(R+) satisfy|σ(ρ)| ≤ |f ′(ρ)|1/2,then we have∥⟨x⟩−sσ(|Dx|)eitf (|Dx|)φ(x)∥L2(Rt×Rn) ≤ C∥φ∥L2(Rn)xxfor s > 1/232

A radial function a(ξ) = f (|ξ|) always satisfies|∇a(ξ)| = |f ′(|ξ|)|.
From the secondary comparison (Theorem 5), we obtain
Theorem 6. Suppose n ≥ 1 and s > 1/2. Let a(ξ) =f (|ξ|) and f ∈ C ω (R+) be real-valued. Then we have⟨x⟩−s|∇a(Dx)|1/2eita(Dx)φ(x)L2(Rt ×Rnx)≤ C∥φ∥L2(Rn).
x33

Example.
a(ξ) = (|ξ|2 − 1)2 is non-dispersive because∇a(ξ) = 4(|ξ|2 − 1)ξ = 0
if |ξ| = 0, 1.
But we have the invariant estimate by Theorem 6.

34

Non-radially symmetric case*Compare again to the low dimensional model estimatesm(m−1)/2
it|D|
x|Dx|
e
φ(x)

L2(Rt )m−1(m−1)/2itD|D|
xy
φ(x, y)|Dy |e≤ C∥φ∥L2(Rx)
L2(Rt ×Ry )≤ C∥φ∥L2(R2 )x,ythen we have:
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Theorem 7 (1D secondary comparison). Let f ∈ C 1(R)be real-valued and strictly monotone. If σ ∈ C 0(R) satisfies|σ(ξ)| ≤ |f ′(ξ)|1/2,
then we have∥⟨x⟩−sσ(Dx)eitf (Dx)φ(x)∥L2(Rt×Rx) ≤ C∥φ(x)∥L2(Rx)for s > 1/2.
36

Theorem 8 (2D secondary comparison). Let f ∈ C 1(R2)
be real-valued and f (ξ, η) be strictly monotone in ξ ∈ R
for every fixed η ∈ R. If σ ∈ C 0(R2) satisfies|σ(ξ, η)| ≤ |∂f /∂ξ(ξ, η)|1/2,
then we have⟨x⟩−sσ(Dx, Dy )eitf (Dx,Dy )φ(x, y)

L2(Rt ×R2x,y )≤ C∥φ(x, y)∥L2(R2 )x,yfor s > 1/2.
37

Normal forms:• a(ξ) = ξ13 + ξ22
By 1D secondary comparison (Theorem 7), we have
⟨x1⟩

−s3
itD
1
|D1|eφ(x)L2(Rt ×R2x)⟨x2⟩−s|D2|1/2eit3D2 φ(x)≤ C∥φ∥L2(R2)

2L2(Rt ×R2x)x≤ C∥φ∥L2(R2)xfor s > 1/2.
38

Hence by ⟨x⟩−s ≤ ⟨xk ⟩−s (k = 1, 2) we have
⟨x⟩

−s()
1/2
|D1| + |D2|eita(Dx)φ(x) 2
L (R ×R2)
tx≤ C∥φ∥L2(R2)xand hence have⟨x⟩−s|∇a(Dx)|1/2eita(Dx)φ(x)L2(Rt ×R2x)≤ C∥φ∥L2(R2).
x39• a(ξ) = ξ12 + ξ1ξ22
By 2D secondary comparison (Theorem 8), we have fors > 1/2⟨x1⟩−s|2D1 + D22|1/2eita(D1, D2)φ(x)⟨x2⟩−s|D1D2|1/2eita(D1, D2)φ(x)L2 (Rt ×R2x)

L2(Rt ×R2x)≤ C∥φ∥L2(R2),x≤ C∥φ∥L2(R2),x

hence we have similarly⟨x⟩−s|∇a(Dx)|1/2eita(Dx)φ(x)L2(Rt ×R2x)≤ C∥φ∥L2(R2).
x40• a(ξ) = ξ1ξ22
By 2D secondary comparison (Theorem 8), we have fors > 1/2⟨x1⟩−s|D2|eita(D1, D2)φ(x)≤ C∥φ∥L2(R2),x
L2(Rt ×R2x)⟨x2⟩−s|D1D2|1/2eita(D1, D2)φ(x) 2≤ C∥φ∥L2(R2),x
L (Rt ×R2)
x

hence we have similarly⟨x⟩−s|∇a(Dx)|1/2eita(Dx)φ(x)L2(Rt ×R2x)≤ C∥φ∥L2(R2).
x41

Non-dispersive case controlled by Hessian
We will show that in the non-dispersive situation therank of ∇2a(ξ) still has a responsibility for smoothingproperties.
First let us consider the case when dispersiveness (L) istrue only for large ξ:(L′)|∇a(ξ)| ≥ C⟨ξ⟩m−1(|ξ| >> 1),|∂ α(a(ξ) − am(ξ))| ≤ C⟨ξ⟩m−1−|α|(|ξ| >> 1)
42

Theorem 9. Suppose n ≥ 1, m ≥ 1, and s > 1/2. Leta ∈ C ∞(Rn) be real-valued and assume that it has finitely
many critical points. Assume (L′) and∇a(ξ) = 0 ⇒ det ∇2a(ξ) ̸= 0.
Then we have⟨x⟩−s|∇a(Dx)|1/2eita(Dx)φ(x)L2(Rt ×Rnx)≤ C∥φ∥L2(Rn).
xExample: a(ξ) = ξ13 + ξ1ξ2, ξ13 + ξ23 + ξ1ξ2,
ξ13 − ξ1ξ22 + ξ12 + ξ22 satisfies assumptions in Theorem 9.
43Outline of proof: * ∇a(ξ) ̸= 0 ⇒ dispersive.
* ∇a(ξ) = 0 ⇒ by Morse’s lemmaa(ξ) = (σ ◦ ∃ψ)(ξ),σ(η) = non-degenerate quadratic form,and σ satisfies dispersiveness (H).
Hence the estimate can be reduced to the dispersivecase!
44⊚ Next we consider the case when a(ξ) is homogeneous(of oder m). Then, by Euler’s identity, we have∇a(ξ) =1ξ∇2a(ξ)
m−1(ξ ̸= 0),

hence∇a(ξ) = 0 ⇒ det ∇2a(ξ) = 0(ξ ̸= 0).

Therefore assumption in Theorem 10 does not make any
sense in this case, but we can have the following result
if we use the idea of canonical transform wisely:
45

Theorem 10. Suppose n ≥ 2 and s > 1/2.
C ∞(Rn \ 0) be real-valued and satisfya(λξ) = λ2a(ξ)

Let a ∈

(λ > 0, ξ ̸= 0).

Assume that∇a(ξ) = 0 ⇒ rank ∇2a(ξ) = n − 1(ξ ̸= 0).

Then we have⟨x⟩−s|∇a(Dx)|1/2eita(Dx)φ(x)L2(Rt ×Rnx)≤ C∥φ∥L2(Rn).
x46ξ12ξ22
2 + · · · + ξ 2 satisfies the asExample. a(ξ) = 2+ξn3
ξ1 + ξ22
sumptions in Theorem 11.
In the case n = 2, this is an illustration of a smoothing
estimate for the Cauchy problem for an equation likei∂tu + D12D22∆−1u = 0(A mixture of Davey-Stewartson and Benjamin-Ono typeequations).
47

Conclusions
Invariant estimate is true at least for• radially symmetric a(ξ) = f (|ξ|), f ∈ C ω (R+),• polynomials a(ξ1, ξ2) of order 3.
• non-dispersive a(ξ) controlled by its Hessian.
48