Dispersive Estimates forTime-Dependent Wave Operators and

Applications to Nonlinear WaveEvolution

Eduard KirrUniv. of Illinois at Urbana-Champaign

Mathematical and Physical Models of Nonlinear Optics,

IMA Workshop, Minneapolis, Nov. 2016.

Collaborators: A. Zarnescu (U. Sussex, UK), O. Mızrak (Mersin

U., Turkey), R. Skulkhu (Mahidol U., Thailand).

1

An example

i∂tu(t, x) = (−∆ + V (x))u+ γ|u|αu, t ∈ (0,∞), x ∈ Rn

u(0, x) = u0 ∈ H1

where n ≥ 2 and

• V (x) is real valued, V ∈ L1 and

|V (x)| ≤C

(1 + |x|)3+, lim|x|→∞

|∇V (x)| = 0;

• −∆ + V as a self-adjoint operator on L2 has exactly onee-value E0 < 0 and no zero energy resonance.

The case of two or more e-values can also be treated.2

Assumptions for the nonlinearity I

i∂tu(t, x) = (−∆ + V (x))u+ γ|u|αu

• γ ∈ R;

• α0(n) < α < 4n−2,

where

α0(n) =2− n+

√n2 + 12n+ 4

2n=

√2 for n = 2,

1 for n = 3,

is called the Strauss limit.

Note that α0(n) < α < 4/n is part of the subcritical regime.

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Theorem 1. If max‖u0‖H1, ‖u0‖Lα+2α+1 is sufficiently small then

u(t) approaches a periodic solution (ground state) as t→∞.

More precisely there exists E,ψE(x), θ(t)t→∞→ 0 such that

u(t) = e−it(E+θ(t))ψE(x) + r(t)

‖r(t)‖Lp ≤

Cp

(1+|t|)n(12−

1p)

if 2 ≤ p ≤ 2n2+n−nα,

Cp

(1+|t|)nα2 −1

if 2n2+n−nα < p < 2n

n−2.

Note that the decay rate of the radiative part in Lp spaces is thesame as for the free Schrodinger equation up to

p =2n

2 + n− nαwere, in subcritical regimes α < 4/n, it saturates.

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Historical Remarks

• For supercritical nonlinearities, with localized initial data,such results were first obtained by Soffer-Weinstein ’90, Pillet-Wayne ’97, and extended to n = 1 by Buslaev-Perelman ’92and Buslaev-Sulem ’03.

• For critical and supercritical nonlinearities, with initial data inenergy space H1, first results are due to Gustafson-Nakanishi-Tsai ’04 in dimension n = 3 and extended to n = 1,2 byMizumachi ’07.

• For subcritical nonlinearities in dimensions n ≥ 2 see Kirr-Zarnescu ’07, ’09, Kirr-Mızrak ’09 and Skulkhu ’12.

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Assumptions for the nonlinearity II

The theorem can be extended to more general nonlinearitiesg(u) :

• g : C 7→ C is C1 (over the real structure of C),

• g(eiθu) = eiθg(u), g(u) = g(u),

• g(0) = 0 and |g′(s)| ≤ C(|s|α1 + |s|α2) where

α0(n) < α1 ≤ α2 <4

n− 2, s ∈ R,

and to initial data near large Solitary Waves (Kirr-Zarnescu inprepapration).

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General Hamiltonian Formulation

∂tu = JDE(u)

with energy E invariant under the group of symmetries T (θ) which

commutes with the skew-adjoint operator J, becomes equivalent

to the nonlinear Schrodinger equation under the definitions:

E(u) =1

2

∫Rn|∇u|2dx+

1

2

∫RnV |u|2dx+

γ

α+ 2

∫Rn|u|α+2dx,

J =

[0 −11 0

]T (θ)u = eiθu.

However many other wave type equations have the same struc-

ture: Dirac, Klein-Gordon, Korteweg-de-Vries, etc.

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Coherent Structures

In general they are (one parameter) solutions of the form

u(t) = T (Et)ψE

where ψE must satisfy:

JDE(ψE) = JEDQ(ψE),

where Q(u) = 12〈Bu, u〉 is the conserved quantity induced by the

symmetries T (θ) via Noether’s Theorem i.e., B is a bounded

operator such that JB extends T ′(0). In our particular example

they exist and are called bound-states u(t, x) = eiEtψE(x) where:

F (ψE, E) = (−∆ + V )ψE + γ|ψE|αψE − EψE = 0. (1)

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Existence of Bound States

Follows from a local bifurcation argument. The Frechet deriva-

tive of F (over real structure) at φ ∈ H2 and E ∈ R is:

DFψ(φ,E)[η] = L(φ,E)[η] = (−∆+V−E)η+γα+ 2

2|φ|αη+γ

α

2|φ|α−2φ2η

L(0,E) has nontrivial kernel at E0 and nontrivial solutions of (1)

bifurcate from (0, E0) :

ψE = aψ0 + h(a), 〈ψ0, h(a)〉 = 0

E = E(|a|) = O(|a|α)

where h : a ∈ C : |a| < δ 7→ H2(Rn), h(a) = O(|a|α+1) is C2.

Elliptic regularity and comparison theorems: ψE(x) ∈ H3 is ex-

ponentially decaying (in absolute value) as |x| → ∞.9

The Manifold of Bound States

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The Decomposition of the Dynamics

Projection onto the manifold of coherent states:

u(t) = T (E(t)t+ θ(t))ψE(t) + r(t)

leads to the equation for the correction term:

∂tr(t) = JLE(t)r(t) +G(E(t), r(t)) (2)

where LE is the linearization of DE(u)−EDQ(u) i.e., the Hessianof E(u)−EQ(u), at ψE and G contains only quadratic and higherorder terms in r. The time decay of r(t) is then obtained from aDuhamel formula for equation (2):

r(t) = W (t)r(0) +∫ t

0W (t− s)[JLE(s) − JLE0

]r(s)ds

+∫ t

0W (t− s)G(E(s), r(s))ds

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The parameters (E(t), θ(t)) i.e., the projection onto the coherent state man-ifold, must satisfy the modulation equations i.e.,

〈v, r(t)〉 = 0, ∀v ∈⋃n∈N

ker ((JLE0)∗)n .

Otherwise there will be no decay of r(t).

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OR we can use the actual, time dependent linearization for

which:

r(t) = Ω(t,0)r(0) +∫ t

0Ω(t, s)G(E(s), r(s))ds,

G contains only quadratic and higher order terms in r, and the

projection ψE(t) is such that:

〈v, r(t)〉 = 0, ∀v ∈⋃n∈N

ker((JLE(t)

)∗)ni.e.,

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the correction r(t) is in the invariant subspace of the linearization

JLE(t) which complements the tangent space to the coherent

state manifold at ψE(t). The projection onto this subspace will

be later called Pt.

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The Duhamel Formula

Consider a fixed “linearization”, say at (0, E0), denote H = −∆+

V :

r(t) ∼ e−iHtr(0)− iγ∫ t

0e−iH(t−s)(α+ 1)|ψE|αr(s)ds

− . . .− iγ∫ t

0e−iH(t−s)|r|αr(s)ds. (3)

which can immediately use standard dispersive estimates:

‖e−iHtP⊥‖Lp′ 7→Lp ≤C

|t|n(12−

1p)

because

r(t) = P⊥r(t), P⊥e−iHt = e−iHtP⊥.

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Closing the estimates for r(t)

Bootstrapping is used in this case:

Fix 0 ≤ T <∞ denote

Mp,q(T ) = sup|t|≤T

(1 + |t|)q‖r(t)‖Lp, q ≤ n(1/2− 1/p)

take the Lp norm in the Duhamel formula, multiply with (1+ |t|)qand pass to the supremum. For p =∞ one gets:

M∞,q ≤ Cε+ linear term + · · ·

+ C(1 + |t|)q∫ |t|

0

1

|t− s|n2‖r(s)‖α+1

Lα+1ds.

Via interpolation:

‖r(s)‖α+1Lα+1 ≤M

22,0

(M∞,q

(1 + |s|)q

)α−1

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Now

(1 + |t|)q∫ |t|

0

1

|t− s|n2

ds

(1 + |s|)q(α−1)≤

C(1+|t|)q(1+|t|)q(α−1)−1/2 if n = 1

C(1+|t|)q(1+|t|)q(α−1)−0 if n = 2

C(1+|t|)q(1+|t|)q(α−1) if n ≥ 3

Bootstrapping requires

α ≥ 3 if n = 1,α > 2 if n = 2,α ≥ 2 if n ≥ 3.

To optimize the estimate for this term one uses

p = α+ 2, q = n

(1

2−

1

α+ 2

)in which case one gets α > α0(n) and:

M(T ) ≤ Cε+ linear and other terms + CM(T )α+1

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i.e., without the other terms,

M(T )[1− CM(T )α+1

]≤ Cε

Choose ε such that

(2Cε)α+1 <1

2C

Cεα+1 ≤1

2Then the set

S =T ≥ 0 : CM(T )α+1 ≤

1

2

is nonempty, closed and open in [0,∞), i.e. one has

(1 + |t|)n(12−

1α+2)‖r(t)‖Lα+2

bounded for all times!

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But the same procedure leads to a growing in time linear term:

M(T ) ≤ Cε+ C(1 + |t|)q∫ |t|

0

1

|t− s|q‖ψE‖αLα+1‖r(s)‖Lα+2ds+ · · ·

≤ Cε+ CM(T )(1 + |t|)q

(1 + |t|)2q−1+ · · ·

where q = n(1/2− 1/(α+ 2)) < 1.

Way around: use weighted (localized) in space estimates:

‖(1+|x|2)−σ/2r(t)‖L2 ∼1

(1 + |t|)1+0, denote w− = (1+|x|2)−σ/2

which is reasonable because the linear dynamics satisfies it for σlarge enough:

‖w−e−iHtP⊥w−‖L2 7→L2 ∼

(1 + |t|)−3/2 if n is odd(1 + |t|)−1 log−2(2 + |t|) if n is even

,

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Weighted estimates for r(t)

w−r(t) ∼ w−e−iHtw−w+r(0)− iγ

∫ t0w−e

−iH(t−s)w−w2+|ψE|

αw−r(s)ds

− . . .− iγ∫ t

0w−e

−iH(t−s)|r|αr(s)ds.

Now take the L2 norm (in space). Note that the last termrequires reversion to Lp estimates and α > 4/n for integrable intime decay.

The ”bad interaction” between the linear term and the fully non-linear term persist even when one tries to optimize all estimatesat the same time!

The cure: do not treat any linear term as if it were nonlinear,i.e. use the correct, time dependent linearization.

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The Duhamel formula for the full linear (and time dependent)

operator:

r(t) ∼ Ω(t,0)r(0)−iγ∫ t

0Ω(t, s)[(α+1)αψE(s)α−1r(s)2+|r(s)|αr(s)]ds

where

∂tΩ(t, s) = −iLψE(t)Ω(t, s)

Ω(s, s) = I

avoids the linear term and allows low power nonlinearities close

to the Strauss limit at the expense of obtaining estimates of the

form (see page 14 for the definition of Pt):

‖Ω(t, s)Pt‖Lp′ 7→Lp ≤C

|t− s|n(12−

1p).

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Estimates for the linear propagator

Recall

i∂tΩ(t, s)Pt = (H︷ ︸︸ ︷

−∆ + V )P⊥RtPtΩ(t, s)

+γα+ 2

2|ψE|αΩ(t, s)Pt + γ

α

2|ψE|α−2ψ2

EΩ(t, s)Pt.

Where Rt is a near identity transformation between the ranges of

Pt and P⊥. Then the integral form of the equation for the linear

propagator is:

Ω(t, s)Pt ∼ e−iH(t−s)Pc−iγ(α+1)∫ tse−iH(t−τ)Pc|ψE(τ)|

αΩ(τ, s)Pτdτ.

Estimates require: localization, smallness of the scatterer |ψE(τ)|α

and near integrable dispersive estimates for H.

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Estimates for the linear propagator

• localization: otherwise the scatterer |ψE(t)|α could behavelike a long range potential; comes free for linearization atsolitary waves.

• smallness: otherwise radiation could be trapped near thescatterer via a parametric resonance phenomena; comes freefor small solitary waves and can be obtained by moving tothe slowly varying variable in the modulation equations.

• near integrable dispersive estimates for H : otherwise con-structive interference may occur after radiation is scatteredfrom |ψE(t)|α; present for Schrodinger operators in n ≥ 2space dimensions.

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Weighted estimates for the linear propagator

Contraction principle for the following fixed point problem:

< x >−σ Ω(t, s) < x >−σ∼< x >−σ e−iH(t−s) < x >−σ

+∫ ts< x >−σ e−iH(t−τ) < x >−σ < x >σ |ψE(τ)|

α < x >σ︸ ︷︷ ︸small

< x >−σ Ω(τ, s) < x >−σ dτ,

implies non-trapping.

For non-constructive interference use T (t, s) = P⊥[Ω(t, s)−e−iH(t−s)]which satisfies:

T (t, s) ∼ −iγ(α+ 1)∫ tse−iH(t−τ)P⊥|ψE|αe−iH(τ−s)P⊥

− iγ(α+ 1)∫ tse−iH(t−τ)P⊥|ψE|αT (τ, s)dτ.

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Non-weighted estimates for the linear propagator

In space dimension n = 2 for p > 2 :

‖T (t, s‖Lp′ 7→Lp ∼ ‖

∫ tse−iH(t−τ)Pc|ψE|αe−iH(τ−s)Pcdτ‖Lp′ 7→Lp

≤∫ ts

(t− τ)2/p−1(τ − s)2/p−1dτ ∼ (t− s)4/p−1

If the decay estimates are allowed to switch from 2/p−1 for the

free Schrodinger operator to 4/p − 1 for Ω(t, s), then estimates

for the nonlinear terms require supercritical nonlinearities. To

fix use p =∞. This requires removal of singularities at τ = t, s,

via a generalized Fourier Multiplier method, which needs Fourier

Transform of |ψE|α in L1.

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Conclusions

• There is a general way to obtain dispersive estimates for

small, localized, time-dependent perturbations of sufficiently

dispersive operators. This can greatly simplify and improve

the analysis of the nonlinear dynamics.

• There are many examples still waiting to be studied.

• New ideas are required for the case of weakly dispersive equa-

tions.

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