Thesis Defense: Study of Long Time Behaviour of Solutions of the - - PowerPoint PPT Presentation

Introduction Growth of Sobolev Norms Multi-solitons Exceptional 2 solitons Conclusion

Thesis Defense: Study of Long Time Behaviour of Solutions of the Zakharov-Kuznetsov Equations

Fr´ ed´ eric Valet

under supervision of Rapha¨

el Cˆ

• te

Universit´ e de Strasbourg

July, 15th 2020

1 / 27

Introduction Growth of Sobolev Norms Multi-solitons Exceptional 2 solitons Conclusion

1

Introduction What kind of plasma? The Zakharov-Kuznetsov equations

2

Growth of Sobolev Norms The cascade phenomenon Theorem Perspectives on the growth of Sobolev norms

3

Multi-solitons What is a (multi)-soliton? Theorem

4

Exceptional 2 solitons Introduction Conjecture

5

Conclusion

2 / 27

Introduction Growth of Sobolev Norms Multi-solitons Exceptional 2 solitons Conclusion What kind of plasma? The Zakharov-Kuznetsov equations

Introduction

A plasma is a ”soup” of electrons and ions. What kind of plasma are we considering?

3 / 27

Introduction Growth of Sobolev Norms Multi-solitons Exceptional 2 solitons Conclusion What kind of plasma? The Zakharov-Kuznetsov equations

EARTH

4 / 27

Introduction Growth of Sobolev Norms Multi-solitons Exceptional 2 solitons Conclusion What kind of plasma? The Zakharov-Kuznetsov equations

EARTH 60 km 90 km 120 km 103 km IONOSPHERE

4 / 27

Introduction Growth of Sobolev Norms Multi-solitons Exceptional 2 solitons Conclusion What kind of plasma? The Zakharov-Kuznetsov equations

EARTH 60 km 90 km 120 km 103 km IONOSPHERE SOLAR RADIATION

4 / 27

Introduction Growth of Sobolev Norms Multi-solitons Exceptional 2 solitons Conclusion What kind of plasma? The Zakharov-Kuznetsov equations

EARTH 60 km 90 km 120 km 103 km IONOSPHERE SOLAR RADIATION ions electrons

4 / 27

Introduction Growth of Sobolev Norms Multi-solitons Exceptional 2 solitons Conclusion What kind of plasma? The Zakharov-Kuznetsov equations

EARTH 60 km 90 km 120 km 103 km IONOSPHERE SOLAR RADIATION ions electrons Temperature (K) / Pressure (Pa) 200 / 2 220 / 10−2 1270 / 10−4

4 / 27

Introduction Growth of Sobolev Norms Multi-solitons Exceptional 2 solitons Conclusion What kind of plasma? The Zakharov-Kuznetsov equations

EARTH 60 km 90 km 120 km 103 km IONOSPHERE SOLAR RADIATION ions electrons Temperature (K) / Pressure (Pa) 200 / 2 220 / 10−2 1270 / 10−4 Magnetic field (0.25-0.65 G)

4 / 27

Introduction Growth of Sobolev Norms Multi-solitons Exceptional 2 solitons Conclusion What kind of plasma? The Zakharov-Kuznetsov equations

The environment we consider satisfies: Cold plasma (soup of electrons and ions, Telectrons >> Tions), 1 Low pressure, Electrostatic (magnetic field not oscillating). Long wave, small amplitude limit.

1Hsu and Heelis 2017 at 840km above the ground 5 / 27

Introduction Growth of Sobolev Norms Multi-solitons Exceptional 2 solitons Conclusion What kind of plasma? The Zakharov-Kuznetsov equations

The environment we consider satisfies: Cold plasma (soup of electrons and ions, Telectrons >> Tions), Low pressure, Electrostatic (magnetic field not oscillating). Long wave, small amplitude limit. We obtain the Euler-Poisson system:    ∂tn + ∇((n + 1)v) = 0, ∂tv + (v · ∇) v + ∇φ + ae ∧ v = 0 ∆φ − eφ + (1 + n) = 0, with n the deviation of density of ions with respect to 1, v the velocity of ions, φ the electric potential, a the measure of electromagnetic field in the first direction e =T (1, 0, 0).

5 / 27

Introduction Growth of Sobolev Norms Multi-solitons Exceptional 2 solitons Conclusion What kind of plasma? The Zakharov-Kuznetsov equations

The environment we consider satisfies: Cold plasma (soup of electrons and ions, Telectrons >> Tions), Low pressure, Electrostatic (magnetic field not oscillating). Long wave, small amplitude limit. Euler-Poisson system can be simplified a, at the main order, by the Zakharov-Kuznetsov equation in 3D: ∂tu + ∂1(∆u + u2) = 0, (ZK3D) with u(t, x) ∈ R the deviation of numbers of ions to 1, t ∈ It, x ∈ Rd, d = 3; ∂i: derivative in the ith direction, ∆ the Laplacian.

aKuznetsov and Zakharov 1974, Lannes, Linares, and Saut 2013, Han-Kwan

2013

5 / 27

Introduction Growth of Sobolev Norms Multi-solitons Exceptional 2 solitons Conclusion What kind of plasma? The Zakharov-Kuznetsov equations

Other considered equations: ∂tu + ∂1(∆u + u2) = 0, with x ∈ R2, (ZK2D)

6 / 27

Introduction Growth of Sobolev Norms Multi-solitons Exceptional 2 solitons Conclusion What kind of plasma? The Zakharov-Kuznetsov equations

Other considered equations: ∂tu + ∂1(∆u + u2) = 0, with x ∈ R2, (ZK2D) and ∂tu + ∂1(∆u + u3) = 0, with x ∈ R2. (mZK2D)

6 / 27

Introduction Growth of Sobolev Norms Multi-solitons Exceptional 2 solitons Conclusion What kind of plasma? The Zakharov-Kuznetsov equations

Other considered equations: ∂tu + ∂1(∆u + u2) = 0, with x ∈ R2, (ZK2D) and ∂tu + ∂1(∆u + u3) = 0, with x ∈ R2. (mZK2D) Cauchy problems in Sobolev spaces Hs:

6 / 27

Introduction Growth of Sobolev Norms Multi-solitons Exceptional 2 solitons Conclusion What kind of plasma? The Zakharov-Kuznetsov equations

Other considered equations: ∂tu + ∂1(∆u + u2) = 0, with x ∈ R2, (ZK2D) and ∂tu + ∂1(∆u + u3) = 0, with x ∈ R2. (mZK2D) Cauchy problems in Sobolev spaces Hs: (ZK3D): LWP for s > − 1

2 (Herr and Kinoshita 2020);

GWP s ≥ 0 (Herr and Kinoshita 2020).

6 / 27

Introduction Growth of Sobolev Norms Multi-solitons Exceptional 2 solitons Conclusion What kind of plasma? The Zakharov-Kuznetsov equations

Other considered equations: ∂tu + ∂1(∆u + u2) = 0, with x ∈ R2, (ZK2D) and ∂tu + ∂1(∆u + u3) = 0, with x ∈ R2. (mZK2D) Cauchy problems in Sobolev spaces Hs: (ZK3D): LWP for s > − 1

2 (Herr and Kinoshita 2020);

GWP s ≥ 0 (Herr and Kinoshita 2020). (ZK2D): LWP for s > − 1

4 (Kinoshita 2019a);

GWP s > − 1

13 (Shan, Wang, and Zhang 2020).

6 / 27

Introduction Growth of Sobolev Norms Multi-solitons Exceptional 2 solitons Conclusion What kind of plasma? The Zakharov-Kuznetsov equations

Other considered equations: ∂tu + ∂1(∆u + u2) = 0, with x ∈ R2, (ZK2D) and ∂tu + ∂1(∆u + u3) = 0, with x ∈ R2. (mZK2D) Cauchy problems in Sobolev spaces Hs: (ZK3D): LWP for s > − 1

2 (Herr and Kinoshita 2020);

GWP s ≥ 0 (Herr and Kinoshita 2020). (ZK2D): LWP for s > − 1

4 (Kinoshita 2019a);

GWP s > − 1

13 (Shan, Wang, and Zhang 2020).

(mZK2D): LWP for s ≥ 1

4 (Kinoshita 2019b);

blow up possible (Farah, Holmer, Roudenko, and Yang 2018).

6 / 27

Introduction Growth of Sobolev Norms Multi-solitons Exceptional 2 solitons Conclusion The cascade phenomenon Theorem Perspectives on the growth of Sobolev norms

Dispersive equations: each frequency moves at a different velocity. (ZK2D) is non-linear: low frequencies can move to higher frequencies: At which velocity? ⇒ Cascade phenomenon. Goal: find an upper bound of this velocity.

7 / 27

Introduction Growth of Sobolev Norms Multi-solitons Exceptional 2 solitons Conclusion The cascade phenomenon Theorem Perspectives on the growth of Sobolev norms

Dispersive equations: each frequency moves at a different velocity. (ZK2D) is non-linear: low frequencies can move to higher frequencies: At which velocity? ⇒ Cascade phenomenon. Goal: find an upper bound of this velocity. Studied by Bourgain (Bourgain 1993b), (Bourgain 1993a) and Staffilani (Staffilani 1997) for (KdV) and (NLS): ∂tu + ∂x

• ∂2

xu + u2

= 0, x ∈ R, u(t, x) ∈ R, (KdV) and i∂tu + ∆u + αu|u|2 = 0, x ∈ Rd, u(t, x) ∈ C, d = 1, 2, 3. (NLS) A way to study this velocity is by studying the Hs norm of the solution. Here, we study (ZK2D).

7 / 27

Introduction Growth of Sobolev Norms Multi-solitons Exceptional 2 solitons Conclusion The cascade phenomenon Theorem Perspectives on the growth of Sobolev norms

Theorem (Cˆ

• te,V. ;19)

Let an integer s ≥ 2, and an initial condition u0 ∈ Hs(R2). Let u the solution of (ZK2D) with the initial condition u0, and A := sup

t≥0

u(t)H1. Then u ∈ C(R, Hs), and for any β > s−1

2 , there exists a constant

C = C(s, β, A) such that: ∀t ∈ R, u(t)Hs ≤ C(1 + |t|)β(1 + u0Hs).

8 / 27

Introduction Growth of Sobolev Norms Multi-solitons Exceptional 2 solitons Conclusion The cascade phenomenon Theorem Perspectives on the growth of Sobolev norms

Theorem (Cˆ

• te,V. ;19)

Let an integer s ≥ 2, and an initial condition u0 ∈ Hs(R2). Let u the solution of (ZK2D) with the initial condition u0, and A := sup

t≥0

u(t)H1. Then u ∈ C(R, Hs), and for any β > s−1

2 , there exists a constant

C = C(s, β, A) such that: ∀t ∈ R, u(t)Hs ≤ C(1 + |t|)β(1 + u0Hs). Remarks: Concerning the Hs-norm: only the Hs-norm of the IC u0Hs. Dependency of the parameters: the H1 norm appears, but bounded by the energy and the mass (conserved quantities). Lower bound of any solution?

8 / 27

Introduction Growth of Sobolev Norms Multi-solitons Exceptional 2 solitons Conclusion The cascade phenomenon Theorem Perspectives on the growth of Sobolev norms

An estimate on a small time interval due to the Cauchy problem on the Hs-norm, then iterate to cover the time interval: ∀t ∈ [t0 − T(u(t0)H1), t0 + T(u(t0)H1)], u(t)Hs ≤ u(t0)Hs + C(u(t0)H1)(1 + u(t0)1−ǫ

Hs ).

9 / 27

Introduction Growth of Sobolev Norms Multi-solitons Exceptional 2 solitons Conclusion The cascade phenomenon Theorem Perspectives on the growth of Sobolev norms

An estimate on a small time interval due to the Cauchy problem on the Hs-norm, then iterate to cover the time interval: ∀t ∈ [t0 − T(u(t0)H1), t0 + T(u(t0)H1)], u(t)Hs ≤ u(t0)Hs + C(u(t0)H1)(1 + u(t0)1−ǫ

Hs ).

Difficulties: bilinear estimates to deal with the non-linearity.

9 / 27

Introduction Growth of Sobolev Norms Multi-solitons Exceptional 2 solitons Conclusion The cascade phenomenon Theorem Perspectives on the growth of Sobolev norms

Definition X s,b holds for the Bourgain space: X s,b =

• u ∈ F(R1+d); u2

X s,b :=

• |ξ|sτ − ξ1|ξ|2b|ˆ

u|2dτdξ < ∞

• .

Proposition (Cˆ

• te, V. 2019)

Let 0 < δ < 1

12, b′ := − 1 2 + 2δ, b := 1 2 + δ. There exist a constant C,

independent of δ, such that for all − 1

2 + 6δ < s < 0 the following

estimate holds: ∀u, v ∈ X s,b, uvX s,b′ ≤ CuX s,bvX s,b. → need to deal differently depending on the interaction of the different

• frequencies. (Molinet and Pilod 2015)

The proof does not adapt to the 3D case: problem of dimension.

10 / 27

Introduction Growth of Sobolev Norms Multi-solitons Exceptional 2 solitons Conclusion The cascade phenomenon Theorem Perspectives on the growth of Sobolev norms

Perspectives on the problem of Sobolev norms

Growth of polynomial norms for (ZK2D); any lower bound for a sequence

• f times tk → +∞: u(tk)Hs ≥ exp(c(ln(ln(tk))

1 2 )? Very resonant

solution, and an unbounded solution.

11 / 27

Introduction Growth of Sobolev Norms Multi-solitons Exceptional 2 solitons Conclusion What is a (multi)-soliton? Theorem

1

Introduction What kind of plasma? The Zakharov-Kuznetsov equations

2

Growth of Sobolev Norms The cascade phenomenon Theorem Perspectives on the growth of Sobolev norms

3

Multi-solitons What is a (multi)-soliton? Theorem

4

Exceptional 2 solitons Introduction Conjecture

5

Conclusion

12 / 27

Introduction Growth of Sobolev Norms Multi-solitons Exceptional 2 solitons Conclusion What is a (multi)-soliton? Theorem

Definition of a soliton

A soliton is a wave solution of a PDE, keeping its form along the time, moving at a constant velocity c in one direction, with high decay at infinity.

13 / 27

Introduction Growth of Sobolev Norms Multi-solitons Exceptional 2 solitons Conclusion What is a (multi)-soliton? Theorem

Example

ions electrons EARTH

14 / 27

Introduction Growth of Sobolev Norms Multi-solitons Exceptional 2 solitons Conclusion What is a (multi)-soliton? Theorem

Example

ions electrons EARTH

14 / 27

Introduction Growth of Sobolev Norms Multi-solitons Exceptional 2 solitons Conclusion What is a (multi)-soliton? Theorem

Example

ions electrons EARTH

14 / 27

Introduction Growth of Sobolev Norms Multi-solitons Exceptional 2 solitons Conclusion What is a (multi)-soliton? Theorem

A soliton moving at a velocity c is defined by u(t, x) = Qc(x − cte1), where Qc is a ground state: it is a positive function, in H1, and satisfies: −cQc + ∆Qc + Qp

c = 0.

(1) It is a non-trivial local minimizer of: E(u) =

• Rd c|u|2 + |∇u|2 −

1 p + 1|u|p+1dx. Qc thus exists, it is unique, and thus is rotationally invariant and exponentially decreasing. u is thus a soliton.

15 / 27

Introduction Growth of Sobolev Norms Multi-solitons Exceptional 2 solitons Conclusion What is a (multi)-soliton? Theorem

On the multi-solitons

Here, (ZK) holds for (ZK3D), (ZK2D) and (mZK2D). Definition A solution u of (ZK) in dimension d is a multi-soliton (or K-soliton) at +∞, if there exists a time T0, K distinct velocities (ck)1≤k≤K ∈ R∗

+ K

and K shifts (yk)1≤k≤K ∈ (Rd)K such that u is defined on [T0, +∞), and by denoting R the sum of K decoupled solitons: R(t, x) :=

K

• k=1

Qck

• x − ckte1 − yk

, the solution converges to R in long time: lim

t→+∞ u(t) − R(t)H1 = 0.

(2)

16 / 27

Introduction Growth of Sobolev Norms Multi-solitons Exceptional 2 solitons Conclusion What is a (multi)-soliton? Theorem

Here, (ZK) holds for (ZK3D), (ZK2D) and (mZK2D). Theorem (V. 20) Let K ∈ N∗, K distinct velocities 0 < c1 < · · · < cK and K shifts (yk)k. There exists a multi-soliton of (ZK) associated with those velocities and shifts, denoted by R∗ and defined on a time interval [T0, +∞). It is unique in H1 in the sense of (2). Furthermore, R∗ ∈ C∞([T0, +∞) × Rd) and there exists a constant δ > 0, and for all s ≥ 1, a constant As such that: ∀t ≥ T0, R∗(t) − R(t)Hs ≤ Ase−δt.

17 / 27

Introduction Growth of Sobolev Norms Multi-solitons Exceptional 2 solitons Conclusion What is a (multi)-soliton? Theorem

Here, (ZK) holds for (ZK3D), (ZK2D) and (mZK2D). Theorem (V. 20) Let K ∈ N∗, K distinct velocities 0 < c1 < · · · < cK and K shifts (yk)k. There exists a multi-soliton of (ZK) associated with those velocities and shifts, denoted by R∗ and defined on a time interval [T0, +∞). It is unique in H1 in the sense of (2).Furthermore, R∗ ∈ C∞([T0, +∞) × Rd) and there exists a constant δ > 0, and for all s ≥ 1, a constant As such that: ∀t ≥ T0, R∗(t) − R(t)Hs ≤ Ase−δt.

17 / 27

Introduction Growth of Sobolev Norms Multi-solitons Exceptional 2 solitons Conclusion What is a (multi)-soliton? Theorem

Ideas of the proof 1

First goal: To control the H1 norm. Let a sequence of times (Sn)n → +∞, and a sequence of solutions (un)n on a time interval [Tn, Sn], with final condition: un(Sn) = R(Sn). Use of local masses, local energies, and monotonicity of adequate linear combinations of those quantities. Use of the coercivity of the linearised operator L = −c + ∆ + pQp−1

c

• n

a subspace H of H1: ∀u ∈ H, un(t)2

H1 ≤ −CLun(t), un(t).

1(Martel 2005) 18 / 27

Introduction Growth of Sobolev Norms Multi-solitons Exceptional 2 solitons Conclusion What is a (multi)-soliton? Theorem

Second goal: To find an upper bound of the Hs-norm for s ≥ 4. Requires new ideas ; more difficult than for (KdV): v(t) := u(t) − R(t). where u holds for un. We want:

• d

dt

• v2

˙ Hs

• + O(l.o.t.)
• vH1 + v2−ǫ

˙ Hs vǫ H1 + v3 ˙ Hs.

Avoid v2

Hs at any cost!

We use a compensation in the time derivative, dealt with by monotonicity .

19 / 27

Introduction Growth of Sobolev Norms Multi-solitons Exceptional 2 solitons Conclusion What is a (multi)-soliton? Theorem

Second goal: To find an upper bound of the Hs-norm for s ≥ 4. Requires new ideas ; more difficult than for (KdV): v(t) := u(t) − R(t). where u holds for un. We want:

• d

dt

• v2

˙ Hs + c

• Rd(Ds−1v)2η
• + O(l.o.t.)
• vH1 + v2−ǫ

˙ Hs vǫ H1 + v3 ˙ Hs.

Avoid v2

Hs at any cost!

We use a compensation in the time derivative, dealt with by monotonicity .

19 / 27

Introduction Growth of Sobolev Norms Multi-solitons Exceptional 2 solitons Conclusion What is a (multi)-soliton? Theorem

Ideas of the proof

Function η: x1 and η satisfies: |∇R| ≤ ∂1η.

20 / 27

Introduction Growth of Sobolev Norms Multi-solitons Exceptional 2 solitons Conclusion What is a (multi)-soliton? Theorem

Ideas of the proof

Uniqueness: R∗ the previous built solution, and another multi-soliton φ; we need to estimate: z(t) := R∗(t) − φ(t), and prove that: z(t)H1 ≤ Ce−γt sup

t′>t

z(t′)H1. We use the property of monotonicity, and the control of the H5 norm of the error because of terms like: ∂1 (∆R − ∆R∗)L∞ .

21 / 27

Introduction Growth of Sobolev Norms Multi-solitons Exceptional 2 solitons Conclusion Introduction Conjecture

Exceptional 2-solitons

Multisolitons converge to a sum of K solitons with distinct veolicities ck. What happens if 2 velocities are equal?

22 / 27

Introduction Growth of Sobolev Norms Multi-solitons Exceptional 2 solitons Conclusion Introduction Conjecture

Exceptional 2-solitons

Multisolitons converge to a sum of K solitons with distinct veolicities ck. What happens if 2 velocities are equal? We propose here a construction to prove existence of a particular 2 solitons with strong interaction, solution of the equation ∂tu + ∂1 (∆u + f (u)) = 0, f (u) = |u|p−1u, with 2 < p < 3, and in dimension d = 2. p = 2: area of interaction is not localized enough ; p > 2. p < 3: stable case (d = 2), should be easier.

22 / 27

Introduction Growth of Sobolev Norms Multi-solitons Exceptional 2 solitons Conclusion Introduction Conjecture

z2(t0)

t = t0

z2(t1)

t = t1

z2(t2)

t = t2

Expected centers of mass of ”solitons” at times t0 < t1 < t2. Goal: find the velocity on the second axis (red arrows). Equivalently: find the distance z2(t) between the two ”solitons” (orange line).

23 / 27

Introduction Growth of Sobolev Norms Multi-solitons Exceptional 2 solitons Conclusion Introduction Conjecture

Known result for subcritical (gKdV)

Theorem (Nguyen, 2017) On the previous defined equation in dimension d = 1 with f (u) = |u|p−1u, where 2 < p < 5, there exists a solution u and a constant c = c(p) satisfying:

• u(t) −

2

• i=1

(−1)iQ(· − t − (−1)i log(ct))

• H1

→

t→+∞ 0.

Remark: c =

• 8(p − 1)

5 − p (2p + 2)

1 p−1

QL2 > 0.

24 / 27

Introduction Growth of Sobolev Norms Multi-solitons Exceptional 2 solitons Conclusion Introduction Conjecture

Expected result

Conjecture There exists a solution R∗ defined on [T0, +∞), and a constant cp, satisfying:

• R∗(t) −

2

• i=1

(−1)iQ

• · − te1 + (−1)i

2

• ln(cpt) + 1

2 ln(ln(t))

• e2
• H1(R2)

→

t→+∞ 0.

25 / 27

Introduction Growth of Sobolev Norms Multi-solitons Exceptional 2 solitons Conclusion

Conclusion and perspectives

Polynomial growth of the norms for (ZK2D). Existence and uniqueness of multi-solitons. 2-solitons with high interactions, leading the long time behaviour. Might be unique.

26 / 27

Introduction Growth of Sobolev Norms Multi-solitons Exceptional 2 solitons Conclusion

Thank you!

27 / 27