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 ote Universit´ e de Strasbourg July, 15th 2020 1 / 27
Introduction Growth of Sobolev Norms Multi-solitons Exceptional 2 solitons Conclusion Introduction 1 What kind of plasma? The Zakharov-Kuznetsov equations Growth of Sobolev Norms 2 The cascade phenomenon Theorem Perspectives on the growth of Sobolev norms Multi-solitons 3 What is a (multi)-soliton? Theorem Exceptional 2 solitons 4 Introduction Conjecture Conclusion 5 2 / 27
Introduction Growth of Sobolev Norms What kind of plasma? Multi-solitons The Zakharov-Kuznetsov equations Exceptional 2 solitons Conclusion Introduction A plasma is a ”soup” of electrons and ions. What kind of plasma are we considering? 3 / 27
Introduction Growth of Sobolev Norms What kind of plasma? Multi-solitons The Zakharov-Kuznetsov equations Exceptional 2 solitons Conclusion EARTH 4 / 27
Introduction Growth of Sobolev Norms What kind of plasma? Multi-solitons The Zakharov-Kuznetsov equations Exceptional 2 solitons Conclusion 10 3 km IONOSPHERE 120 km 90 km 60 km EARTH 4 / 27
Introduction Growth of Sobolev Norms What kind of plasma? Multi-solitons The Zakharov-Kuznetsov equations Exceptional 2 solitons Conclusion SOLAR RADIATION 10 3 km IONOSPHERE 120 km 90 km 60 km EARTH 4 / 27
Introduction Growth of Sobolev Norms What kind of plasma? Multi-solitons The Zakharov-Kuznetsov equations Exceptional 2 solitons Conclusion SOLAR RADIATION ions electrons 10 3 km IONOSPHERE 120 km 90 km 60 km EARTH 4 / 27
Introduction Growth of Sobolev Norms What kind of plasma? Multi-solitons The Zakharov-Kuznetsov equations Exceptional 2 solitons Conclusion SOLAR RADIATION Temperature (K) ions electrons / Pressure (Pa) 10 3 km IONOSPHERE 1270 / 10 − 4 120 km 220 / 10 − 2 90 km 200 / 2 60 km EARTH 4 / 27
Introduction Growth of Sobolev Norms What kind of plasma? Multi-solitons The Zakharov-Kuznetsov equations Exceptional 2 solitons Conclusion SOLAR RADIATION Temperature (K) ions electrons / Pressure (Pa) 10 3 km IONOSPHERE 1270 / 10 − 4 Magnetic field (0.25-0.65 G) 120 km 220 / 10 − 2 90 km 200 / 2 60 km EARTH 4 / 27
Introduction Growth of Sobolev Norms What kind of plasma? Multi-solitons The Zakharov-Kuznetsov equations Exceptional 2 solitons Conclusion The environment we consider satisfies: Cold plasma (soup of electrons and ions, T electrons >> T ions ), 1 Low pressure, Electrostatic (magnetic field not oscillating). Long wave, small amplitude limit. 1 Hsu and Heelis 2017 at 840km above the ground 5 / 27
Introduction Growth of Sobolev Norms What kind of plasma? Multi-solitons The Zakharov-Kuznetsov equations Exceptional 2 solitons Conclusion The environment we consider satisfies: Cold plasma (soup of electrons and ions, T electrons >> T ions ), Low pressure, Electrostatic (magnetic field not oscillating). Long wave, small amplitude limit. We obtain the Euler-Poisson system: ∂ t n + ∇ (( n + 1) v ) = 0 , ∂ t v + ( v · ∇ ) v + ∇ φ + ae ∧ v = 0 ∆ φ − e φ + (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 What kind of plasma? Multi-solitons The Zakharov-Kuznetsov equations Exceptional 2 solitons Conclusion The environment we consider satisfies: Cold plasma (soup of electrons and ions, T electrons >> T ions ), 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 3 D : ∂ t u + ∂ 1 (∆ u + u 2 ) = 0 , (ZK 3 D ) with u ( t , x) ∈ R the deviation of numbers of ions to 1, t ∈ I t , x ∈ R d , d = 3; ∂ i : derivative in the i th direction, ∆ the Laplacian. a Kuznetsov and Zakharov 1974, Lannes, Linares, and Saut 2013, Han-Kwan 2013 5 / 27
Introduction Growth of Sobolev Norms What kind of plasma? Multi-solitons The Zakharov-Kuznetsov equations Exceptional 2 solitons Conclusion Other considered equations: ∂ t u + ∂ 1 (∆ u + u 2 ) = 0 , with x ∈ R 2 , (ZK 2 D ) 6 / 27
Introduction Growth of Sobolev Norms What kind of plasma? Multi-solitons The Zakharov-Kuznetsov equations Exceptional 2 solitons Conclusion Other considered equations: ∂ t u + ∂ 1 (∆ u + u 2 ) = 0 , with x ∈ R 2 , (ZK 2 D ) and ∂ t u + ∂ 1 (∆ u + u 3 ) = 0 , with x ∈ R 2 . (mZK 2 D ) 6 / 27
Introduction Growth of Sobolev Norms What kind of plasma? Multi-solitons The Zakharov-Kuznetsov equations Exceptional 2 solitons Conclusion Other considered equations: ∂ t u + ∂ 1 (∆ u + u 2 ) = 0 , with x ∈ R 2 , (ZK 2 D ) and ∂ t u + ∂ 1 (∆ u + u 3 ) = 0 , with x ∈ R 2 . (mZK 2 D ) Cauchy problems in Sobolev spaces H s : 6 / 27
Introduction Growth of Sobolev Norms What kind of plasma? Multi-solitons The Zakharov-Kuznetsov equations Exceptional 2 solitons Conclusion Other considered equations: ∂ t u + ∂ 1 (∆ u + u 2 ) = 0 , with x ∈ R 2 , (ZK 2 D ) and ∂ t u + ∂ 1 (∆ u + u 3 ) = 0 , with x ∈ R 2 . (mZK 2 D ) Cauchy problems in Sobolev spaces H s : (ZK 3 D ): LWP for s > − 1 2 (Herr and Kinoshita 2020); GWP s ≥ 0 (Herr and Kinoshita 2020). 6 / 27
Introduction Growth of Sobolev Norms What kind of plasma? Multi-solitons The Zakharov-Kuznetsov equations Exceptional 2 solitons Conclusion Other considered equations: ∂ t u + ∂ 1 (∆ u + u 2 ) = 0 , with x ∈ R 2 , (ZK 2 D ) and ∂ t u + ∂ 1 (∆ u + u 3 ) = 0 , with x ∈ R 2 . (mZK 2 D ) Cauchy problems in Sobolev spaces H s : (ZK 3 D ): LWP for s > − 1 2 (Herr and Kinoshita 2020); GWP s ≥ 0 (Herr and Kinoshita 2020). (ZK 2 D ): LWP for s > − 1 4 (Kinoshita 2019a); GWP s > − 1 13 (Shan, Wang, and Zhang 2020). 6 / 27
Introduction Growth of Sobolev Norms What kind of plasma? Multi-solitons The Zakharov-Kuznetsov equations Exceptional 2 solitons Conclusion Other considered equations: ∂ t u + ∂ 1 (∆ u + u 2 ) = 0 , with x ∈ R 2 , (ZK 2 D ) and ∂ t u + ∂ 1 (∆ u + u 3 ) = 0 , with x ∈ R 2 . (mZK 2 D ) Cauchy problems in Sobolev spaces H s : (ZK 3 D ): LWP for s > − 1 2 (Herr and Kinoshita 2020); GWP s ≥ 0 (Herr and Kinoshita 2020). (ZK 2 D ): LWP for s > − 1 4 (Kinoshita 2019a); GWP s > − 1 13 (Shan, Wang, and Zhang 2020). (mZK 2 D ): LWP for s ≥ 1 4 (Kinoshita 2019b); blow up possible (Farah, Holmer, Roudenko, and Yang 2018). 6 / 27
Introduction Growth of Sobolev Norms The cascade phenomenon Multi-solitons Theorem Exceptional 2 solitons Perspectives on the growth of Sobolev norms Conclusion Dispersive equations: each frequency moves at a different velocity. (ZK 2 D ) 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 The cascade phenomenon Multi-solitons Theorem Exceptional 2 solitons Perspectives on the growth of Sobolev norms Conclusion Dispersive equations: each frequency moves at a different velocity. (ZK 2 D ) 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): ∂ 2 x u + u 2 � � ∂ t u + ∂ x = 0 , x ∈ R , u ( t , x ) ∈ R , (KdV) and i ∂ t u + ∆ u + α u | u | 2 = 0 , x ∈ R d , u ( t , x) ∈ C , d = 1 , 2 , 3 . (NLS) A way to study this velocity is by studying the H s norm of the solution. Here, we study (ZK 2 D ). 7 / 27
Introduction Growth of Sobolev Norms The cascade phenomenon Multi-solitons Theorem Exceptional 2 solitons Perspectives on the growth of Sobolev norms Conclusion Theorem (Cˆ ote,V. ;19) Let an integer s ≥ 2 , and an initial condition u 0 ∈ H s ( R 2 ) . Let u the solution of (ZK 2 D ) with the initial condition u 0 , and A := sup � u ( t ) � H 1 . t ≥ 0 Then u ∈ C ( R , H s ) , and for any β > s − 1 2 , there exists a constant C = C ( s , β, A ) such that: � u ( t ) � H s ≤ C (1 + | t | ) β (1 + � u 0 � H s ) . ∀ t ∈ R , 8 / 27
Introduction Growth of Sobolev Norms The cascade phenomenon Multi-solitons Theorem Exceptional 2 solitons Perspectives on the growth of Sobolev norms Conclusion Theorem (Cˆ ote,V. ;19) Let an integer s ≥ 2 , and an initial condition u 0 ∈ H s ( R 2 ) . Let u the solution of (ZK 2 D ) with the initial condition u 0 , and A := sup � u ( t ) � H 1 . t ≥ 0 Then u ∈ C ( R , H s ) , and for any β > s − 1 2 , there exists a constant C = C ( s , β, A ) such that: � u ( t ) � H s ≤ C (1 + | t | ) β (1 + � u 0 � H s ) . ∀ t ∈ R , Remarks: Concerning the H s -norm: only the H s -norm of the IC � u 0 � H s . Dependency of the parameters: the H 1 norm appears, but bounded by the energy and the mass (conserved quantities). Lower bound of any solution? 8 / 27
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