On Zakharov-Kuznetsov Equation VIII Workhsop on Partial Differential - PDF document

On Zakharov-Kuznetsov Equation VIII Workhsop on Partial Differential Equations Felipe Linares IMPA 1

In this talk we will consider the initial value problem associated to the nonlinear equation  u t + u 2 ∂ x u + ∂ x ∆ u = 0 ,  (1)  u ( x, y, 0) = u 0 ( x, y ) called the modified Zakharov-Kuznetsov equa- tion, where u is a real function defined in R 2 × R . 2

Outline • Model • Motivation and Previous Results • Main Results • Ingredients • Ideas of the Proofs • Final Remarks Joint work with Ademir Pastor (IMPA), Jean- Claude Saut (Paris-Sud) 3

The equation under consideration is a 2D gen- eralization of the Zakharov-Kuznetsov equa- tion, that is, u t + u ∂ x u + ∂ x ∆ u = 0 , (2) This equation was first derived by Zakharov and Kuznetsov (1974) in three-dimensional form to describe nonlinear ion–acoustic waves in a magnetized plasma. A variety of physical phe- nomena, are governed by this type of equation; for example, the long waves on a thin liquid film, the Rosby waves in rotating atmosphere, and the isolated vortex of the drift waves in three-dimensional plasma. 4

Even though the Zakharov-Kuznetsov equa- tion seems a natural generalization of the Korteweg- de Vries equation, ∂ t v + v∂ x v + ∂ 3 x v = 0 The ZK equation is derived from the Euler- Poisson system for nonlinear ion-acoustic waves in a magnetized plasma.   n t + div( nv ) = 0    � � T v t + ( v · ∇ ) v + ∇ ϕ + a e x × v = 0 e x = 1 0 0    ∆ ϕ − e ϕ + n = 0  where n =ion density v =ion velocity ϕ = electrostatic potential a ≥ 0 measures the applied magnetic field 5

Critical character of the modified ZK equation. – Local well-posedness If we consider the IVP associated to the gen- eralized Zakharov-Kuznetsov equation, i.e., u t + u p ∂ x u + ∂ x ∆ u = 0 , u ( x, y, 0) = u 0 ( x, y ) . We can see that if u is a solution with data u 0 , then u λ ( x, y, t ) = λu ( λx, λy, λ 3 t ) is also a solution with data u λ ( x, y, 0) = λu 0 ( λx, λy ). In particular, we have that H s ( R 2 ) = λ s − 1+ 2 � u λ (0) � ˙ p � u 0 � ˙ H s ( R 2 ) , This means that derivatives of the solutions remain invariant only if s = 1 − 2 p This scaling argument suggests local well-posedness for s ≥ 1 − 2 p . In case p = 2, we have L 2 ( R 2 ) as the possible larger space where local well- posedness can be obtained. 6

– Global well-posedness We note that the modified Zakharov-Kuznetsov equation has two conserved quantities, namely, � � R 2 u 2 ( t ) dxdy = R 2 u 2 I 1 ( u ( t )) = 0 dxdy, � y − 1 R 2 ( u 2 x + u 2 6 u 4 )( t ) dxdy I 2 ( u ( t )) = � 0 y − 1 R 2 ( u 2 0 x + u 2 6 u 4 = 0 ) dxdy. One can establish a H 1 ( R 2 ) an a priori esti- mate combining I 1 and I 2 . Indeed, � u ( t ) � 2 H 1 = � u ( t ) � 2 L 2 + � ∂ x u ( t ) � 2 L 2 + � ∂ y u ( t ) � 2 L 2 � L 2 + I 2 ( u (0)) + 1 = � u 0 � 2 u 4 ( t ) dxdy 6 Using Gagliardo-Nirenberg interpolation esti- mate we see that the last term is bounded by � � c � u ( t ) � 2 � ∂ x u � 2 L 2 + � ∂ y u � 2 L 2 L 2 � � = c � u 0 � 2 � ∂ x u � 2 L 2 + � ∂ y u � 2 . L 2 L 2 7

Thus to obtain an a priori estimate we require c � u 0 � 2 L 2 < 1. In fact, L 2 ) − 1 I 2 ( u 0 ) . � u ( t ) � H 1 ≤ � u 0 � L 2 + (1 − c � u 0 � 2 One can be more precise regarding the size of the L 2 -norm of the data. Observe that a similar analysis can be done for the generalized ZK equation. In particular, so- lutions of the generalized ZK equation satisfy two conserved quantities as above and a priori estimate in H 1 can be established for data in H 1 with H 1 -norm small and p ≥ 3. It is an open problem to show global well- posedness for the modified ZK equation (and generalized ZK equation) for any data. Nu- merical evidence suggests blow-up of solutions in finite time. 8

– Stability / Instability of solitary wave solu- tions The existence of solitary wave solutions of the � x 2 + y 2 for the form ϕ ( x, y ) = ϕ ( r ), r = generalized ZK equation was established by de Bouard. • p = 1 stable • p ≥ 3 unstable 9

Previous Results • Faminskii (1995) Local and Global well- posedness for ZK equation for data in H 1 ( R 2 ) • Biagioni-L (2003) Local and Global well- posedness for modified ZK equation for data in H 1 ( R 2 ) • L-Saut (2008) Local well-posedness for ZK equation in 3D for data H 1+ ( R 3 ) The notion of well-posedness we use is the one given by Kato, that is, existence, unique- ness, persistence property and continuous de- pendence upon the data. 10

Main Results Theorem 1. For any u 0 ∈ H s ( R 2 ) , s > 3 / 4 , there exist T = T ( � u 0 � H s ) > 0 and a unique solution of the IVP associated to the modified ZK equation, defined in the interval [0 , T ] , such that u ∈ C ([0 , T ]; H s ( R 2 )) , (3) � D s yT + � D s x u x � L ∞ y u x � L ∞ yT < ∞ , (4) x L 2 x L 2 xy + � u x � L 9 / 4 (5) � u � L 3 < ∞ , T L ∞ L ∞ xy T and (6) � u � L 2 yT < ∞ . x L ∞ Moreover, for any T ′ ∈ (0 , T ) there exists a neighborhood W of u 0 in H s ( R 2 ) such that the u ( t ) from W into the class defined map � u 0 �→ � by (3) – (6) is smooth. 11

Consider ϕ the unique (up to translation) pos- itive radial solution of the equation − ∆ ϕ + ϕ − ϕ 3 = 0 . (7) Then we have the next global well-posedness result: Theorem 2. Let u 0 ∈ H 1 ( R 2 ) . If � u 0 � L 2 < √ 3 � ϕ � L 2 , ϕ as in (7) , then the local solution given in Theorem 1 can be extended to any time interval [0 , T ] . Remark 1. One can prove that if the initial data u 0 belongs to H s ( R 2 ) , s > 19 / 21 , and sat- √ isfies � u 0 � L 2 < 3 � ϕ � L 2 , then the local solu- tion given in Theorem 1 can also be extended globally in time. To prove this one can follow the argument used by Fonseca, L- and Ponce, following the ideas introduced in Bourgain, to established a global result for the critical KdV equation, v t + v 4 v x + v xxx = 0 . 12

We show that the minimum index of local well- posedness cannot be achieved. Actually, we will establish that we cannot have local well- posedness for data in H s ( R 2 ), s ≤ 0 in the sense that the map data-solution, u 0 �→ u ( t ), where u ( t ) solves the IVP (), is not uniformly continuous. In other words, we prove the fol- lowing result: Theorem 3. The IVP () is ill-posed for data in H s ( R 2 ) , s ≤ 0 . 13

Ingredients – Local well-posedness we consider the linear initial value problem  ( x, y ) ∈ R 2 ,  t ∈ R , u t + ∂ x ∆ u = 0 ,  (8)   u ( x, y, 0) = u 0 ( x, y ) . The solution of (8) is given by the unitary group { U ( t ) } ∞ t = −∞ such that u ( t ) = U ( t ) u 0 ( x, y ) � R 2 e i ( t ( ξ 3 + ξη 2 )+ xξ + yη ) � = u 0 ( ξ, η ) dξdη. 14

Strichartz Estimates Proposition 1. Let 0 ≤ ε < 1 / 2 and 0 ≤ θ ≤ 1 . Then the group { U ( t ) } ∞ t = −∞ satisfies � D θε/ 2 U ( t ) f � L q xy ≤ c � f � L 2 xy , t L p x � ∞ � D θε −∞ U ( t − t ′ ) g ( · , t ′ ) dt ′ � L q xy ≤ c � g � L q ′ , t L p t L p ′ x xy � ∞ � D θε −∞ U ( t ) g ( · , t ) dt � L 2 xy ≤ c � g � L q ′ , t L p ′ x xy where 1 p + 1 p ′ = 1 q + 1 q ′ = 1 with 1 − θ and 2 2 q = θ (2 + ε ) p = . 3 15

As a consequence of Proposition 1 we have Let 0 ≤ ε < 1 / 2. Then the group { U ( t ) } ∞ t = −∞ satisfies xy ≤ cT γ � D − ε/ 2 � U ( t ) f � L 2 f � L 2 (9) x T L ∞ xy and ≤ cT δ � D − ε/ 2 � U ( t ) f � L 9 / 4 (10) f � L 2 xy , x L ∞ xy T where γ = (1 − ε ) / 6 and δ = (2 − 3 ε ) / 18. 16

Smoothing Effect Lemma 1. Let u 0 ∈ L 2 ( R 2 ) . Then, � ∂ x U ( t ) u 0 � L ∞ yT ≤ c � u 0 � L 2 xy . x L 2 Maximal Function Estimate Lemma 2. Let u 0 ∈ H s ( R 2 ) , s > 3 / 4 . Then, � U ( t ) u 0 � L 2 yT ≤ c ( s, T ) � u 0 � H s xy , x L ∞ where c ( s, T ) is a constant depending on s and T . Leibniz rule for fractional derivatives: Lemma 3. Let 0 < α < 1 and 1 < p < ∞ . Then, � D α ( fg ) − fD α g − gD α f � L p ( R ) ≤ c � g � L ∞ ( R ) � D α f � L p ( R ) , where D α denotes either D α x or D α y . 17

Proof of Theorem 1 Consider the integral operator Ψ( u )( t ) = Ψ u 0 ( u )( t ) � t 0 U ( t − t ′ )( u 2 u x )( t ′ ) dt ′ = U ( t ) u 0 + and define the metric spaces Y T = { u ∈ C ([0 , T ]; H s ( R 2 )); | | | u | | | < ∞} and Y a T = { u ∈ X T ; | | | u | | | ≤ a } , with | : = � u � L ∞ xy + � u � L 3 xy + � u x � L 9 / 4 | | | u | | T H s T L ∞ L ∞ xy T + � D s yT + � D s x u x � L ∞ y u x � L ∞ yT + � u � L 2 yT , x L ∞ x L 2 x L 2 where a, T > 0 will be chosen later. We assume that 3 / 4 < s < 1 and T ≤ 1. 18

We only sketch the estimate of the H s -norm of Ψ( u ). Let u ∈ Y T . By using Minkowski’s in- equality, group properties and H¨ older inequal- ity, we have � Ψ( u )( t ) � L 2 xy � T xy dt ′ ≤ c � u 0 � H s + c 0 � u � L 2 xy � uu x � L ∞ ≤ c � u 0 � H s + cT 2 / 9 � u � L ∞ xy � u � L 3 xy � u x � L 9 / 4 . T L 2 T L ∞ L ∞ xy T 19