SLIDE 1
On Zakharov-Kuznetsov Equation VIII Workhsop on Partial Differential - - PDF document
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 ,
SLIDE 2
SLIDE 3
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
SLIDE 4
The equation under consideration is a 2D gen- eralization of the Zakharov-Kuznetsov equa- tion, that is, ut + u ∂xu + ∂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
SLIDE 5
Even though the Zakharov-Kuznetsov equa- tion seems a natural generalization of the Korteweg- de Vries equation, ∂tv + v∂xv + ∂3
xv = 0
The ZK equation is derived from the Euler- Poisson system for nonlinear ion-acoustic waves in a magnetized plasma.
nt + div(nv) = 0 vt + (v · ∇)v + ∇ϕ + a ex × v = 0 ex =
- 1
T
∆ϕ − eϕ + n = 0 where n =ion density v =ion velocity ϕ = electrostatic potential a ≥ 0 measures the applied magnetic field
5
SLIDE 6
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., ut + up ∂xu + ∂x∆u = 0, u(x, y, 0) = u0(x, y). We can see that if u is a solution with data u0, then uλ(x, y, t) = λu(λx, λy, λ3t) is also a solution with data uλ(x, y, 0) = λu0(λx, λy). In particular, we have that uλ(0) ˙
Hs(R2) = λs−1+2
pu0 ˙
Hs(R2),
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 L2(R2)
as the possible larger space where local well- posedness can be obtained.
6
SLIDE 7
– Global well-posedness We note that the modified Zakharov-Kuznetsov equation has two conserved quantities, namely, I1(u(t)) =
- R2 u2(t) dxdy =
- R2 u2
0 dxdy,
I2(u(t)) =
- R2(u2
x + u2 y − 1
6u4)(t) dxdy =
- R2(u2
0x + u2 0y − 1
6u4
0) dxdy.
One can establish a H1(R2) an a priori esti- mate combining I1 and I2. Indeed, u(t)2
H1 = u(t)2 L2 + ∂xu(t)2 L2 + ∂yu(t)2 L2
= u02
L2 + I2(u(0)) + 1
6
- u4(t) dxdy
Using Gagliardo-Nirenberg interpolation esti- mate we see that the last term is bounded by cu(t)2
L2
- ∂xu2
L2 + ∂yu2 L2
- = cu02
L2
- ∂xu2
L2 + ∂yu2 L2
- .
7
SLIDE 8
Thus to obtain an a priori estimate we require cu02
L2 < 1. In fact,
u(t)H1 ≤ u0L2 + (1 − cu02
L2)−1 I2(u0).
One can be more precise regarding the size of the L2-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 H1 can be established for data in H1 with H1-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
SLIDE 9
– Stability / Instability of solitary wave solu- tions The existence of solitary wave solutions of the form ϕ(x, y) = ϕ(r), r =
- x2 + y2 for the
generalized ZK equation was established by de Bouard.
- p = 1 stable
- p ≥ 3 unstable
9
SLIDE 10
Previous Results
- Faminskii (1995) Local and Global well-
posedness for ZK equation for data in H1(R2)
- Biagioni-L (2003) Local and Global well-
posedness for modified ZK equation for data in H1(R2)
- L-Saut (2008) Local well-posedness for ZK
equation in 3D for data H1+(R3) The notion of well-posedness we use is the
- ne given by Kato, that is, existence, unique-
ness, persistence property and continuous de- pendence upon the data.
10
SLIDE 11
Main Results Theorem 1. For any u0 ∈ Hs(R2), s > 3/4, there exist T = T(u0Hs) > 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]; Hs(R2)), (3) Ds
xuxL∞
x L2 yT + Ds
yuxL∞
x L2 yT < ∞,
(4) uL3
T L∞ xy + uxL9/4 T
L∞
xy
< ∞, (5) and uL2
xL∞ yT < ∞.
(6) Moreover, for any T ′ ∈ (0, T) there exists a neighborhood W of u0 in Hs(R2) such that the map u0 → u(t) from W into the class defined by (3)–(6) is smooth.
11
SLIDE 12
Consider ϕ the unique (up to translation) pos- itive radial solution of the equation −∆ϕ + ϕ − ϕ3 = 0. (7) Then we have the next global well-posedness result: Theorem 2. Let u0 ∈ H1(R2). If u0L2 < √ 3 ϕL2, 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 u0 belongs to Hs(R2), s > 19/21, and sat- isfies u0L2 < √ 3 ϕL2, 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, vt + v4vx + vxxx = 0.
12
SLIDE 13
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 Hs(R2), s ≤ 0 in the sense that the map data-solution, u0 → 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 Hs(R2), s ≤ 0.
13
SLIDE 14
Ingredients – Local well-posedness we consider the linear initial value problem
ut + ∂x∆u = 0, (x, y) ∈ R2, t ∈ R, u(x, y, 0) = u0(x, y). (8) The solution of (8) is given by the unitary group {U(t)}∞
t=−∞ such that
u(t) = U(t)u0(x, y) =
- R2 ei(t(ξ3+ξη2)+xξ+yη)
u0(ξ, η)dξdη.
14
SLIDE 15
Strichartz Estimates Proposition 1. Let 0 ≤ ε < 1/2 and 0 ≤ θ ≤ 1. Then the group {U(t)}∞
t=−∞ satisfies
Dθε/2
x
U(t)fLq
tLp xy ≤ cfL2 xy,
Dθε
x
∞
−∞ U(t − t′)g(·, t′)dt′Lq
tLp xy ≤ cgLq′ t Lp′ xy
, Dθε
x
∞
−∞ U(t)g(·, t)dtL2
xy ≤ cgLq′ t Lp′ xy
, where 1
p + 1 p′ = 1 q + 1 q′ = 1 with
p = 2 1 − θ and 2 q = θ(2 + ε) 3 .
15
SLIDE 16
As a consequence of Proposition 1 we have Let 0 ≤ ε < 1/2. Then the group {U(t)}∞
t=−∞
satisfies U(t)fL2
T L∞ xy ≤ cT γD−ε/2
x
fL2
xy
(9) and U(t)fL9/4
T
L∞
xy
≤ cT δD−ε/2
x
fL2
xy,
(10) where γ = (1 − ε)/6 and δ = (2 − 3ε)/18.
16
SLIDE 17
Smoothing Effect Lemma 1. Let u0 ∈ L2(R2). Then, ∂xU(t)u0L∞
x L2 yT ≤ cu0L2 xy.
Maximal Function Estimate Lemma 2. Let u0 ∈ Hs(R2), s > 3/4. Then, U(t)u0L2
xL∞ yT ≤ c(s, T)u0Hs xy,
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, Dα(fg)−fDαg−gDαfLp(R) ≤ cgL∞(R)DαfLp(R), where Dα denotes either Dα
x or Dα y .
17
SLIDE 18
Proof of Theorem 1 Consider the integral operator Ψ(u)(t) = Ψu0(u)(t) = U(t)u0 +
t
0 U(t − t′)(u2ux)(t′)dt′
and define the metric spaces
YT = {u ∈ C([0, T]; Hs(R2));
| | |u| | | < ∞} and
Ya
T = {u ∈ XT;
| | |u| | | ≤ a}, with | | |u| | | : = uL∞
T Hs xy + uL3 T L∞ xy + uxL9/4 T
L∞
xy
+ Ds
xuxL∞
x L2 yT + Ds
yuxL∞
x L2 yT + uL2 xL∞ yT ,
where a, T > 0 will be chosen later. We assume that 3/4 < s < 1 and T ≤ 1.
18
SLIDE 19
We only sketch the estimate of the Hs-norm
- f Ψ(u). Let u ∈ YT. By using Minkowski’s in-
equality, group properties and H¨
- lder inequal-
ity, we have Ψ(u)(t)L2
xy
≤ cu0Hs + c
T
0 uL2
xyuuxL∞ xydt′
≤ cu0Hs + cT 2/9uL∞
T L2 xyuL3 T L∞ xyuxL9/4 T
L∞
xy
.
19
SLIDE 20
Using group properties, Minkowski’s inequal- ity, Leibniz rule for fractional derivatives and H¨
- lder’s inequality, we have
Ds
xΨ(u)(t)L2
xy
≤ Ds
xu0L2
xy +
T
0 Ds x(u2ux)(t′)L2
xydt′
≤ c u0Hs + c
T
0 uxL∞
xyuL∞ xyDs
xuL2
xydt′
+
T
0 u2Ds xuxL2
xydt′
≤ cu0Hs + c T 2/9uL∞
T Hs xyuL3 T L∞ xyuxL9/4 T
L∞
xy
+
T
0 u2Ds xuxL2
xydt′. 20
SLIDE 21
From H¨
- lder’s inequality we get
T
0 u2Ds xuxL2
xydt′ ≤
T
0 uL∞
xyuDs
xuxL2
xydt′
≤ u2
L2
T L∞ xyuDs
xuxL2
xyT
≤ cT 1/6uL3
T L∞ xyuL2 xL∞ yT Ds
xuxL2
xL∞ yT .
Thus, Ds
xΨ(u)(t)L2
xy
≤ cu0Hs + cT 2/9uL∞
T Hs xyuL3 T L∞ xyuxL9/4 T
L∞
xy
+ cT 1/6uL3
T L∞ xyuL2 xL∞ yT Ds
xuxL2
xL∞ yT . 21
SLIDE 22
Similarly, Ds
yΨ(u)(t)L2
xy
≤cu0Hs + cT 2/9uL∞
T Hs xyuL3 T L∞ xyuxL9/4 T
L∞
xy
+ cT 1/6uL3
T L∞ xyuL2 xL∞ yT Ds
yuxL2
xL∞ yT .
Therefore, Ψ(u)L∞
T Hs ≤ cu0Hs + cT 1/6|
| |u| | |3.
22
SLIDE 23
Proof of Theorem 2 The main ingredients are the conserved quanti- ties I1, I2 and the following Gagliardo-Nirenberg interpolation inequality 1 6u(t)4
L4 ≤ 1
3
u(t)L2
ϕL2
2
∇u(t)2
L2,
where ϕ is the solution of (7). Thus we can estimate u(t)2
H1 ≤ u(t)2 L2 + I2(u(t)) + 1
6u(t)4
L4
=u02
L2 + I(u0) + 1
6u(t)4
L4
≤u02
L2 + c u02 H1 + 1
3
u0L2
ϕL2
2
∇u(t)2
L2.
Hence using the hypothesis we obtain u(t)H1 ≤ cu0H1. This a priori estimate and a standard argument yield the desired result.
23
SLIDE 24
Proof of Theorem 3 As we have already pointed out, a scaling argu- ment suggests the IVP being locally well-posed for data in Hs(R2), s ≥ 0. We will show us- ing an example that the IVP () is ill-posed in Hs(R2), s ≤ 0, in the sense that the map data- solution is not uniformly continuous. Let f(x, y) be the positive radial solution in H1(R2) of ∆f − f + f3 = 0. So, f(x, y) = f(r), r2 = x2+y2. Now we define g = gc(r) = √cf(√cr). It is easy to see that g satisfies ∆g − cg + g3 = 0.
24
SLIDE 25
Next we define u = uc(x, y, t) = gc( r), where
- r2 = ξ2 + y2 with ξ = x − ct. Then it is easy
to check that u is a solution of () with initial datum u0(x, y) = √cf(√cx, √cy). Moreover,
- uc(0)(ξ, η) =
uc(ξ, η, 0) = 1 √c
- f
- ξ
√c, η √c
- .
Let c1, c2 > 0. Let us evaluate the L2-norm
- f the difference uc1(0) − uc2(0).
First, we note that if ·, ·0 denotes the inner product in L2(R2) then uc1(0), uc2(0)0 = 1 √c1c2
- R2
- f
- ξ
√c1 , η √c1
- f
- ξ
√c2 , η √c2
- dξdη
= √c1 √c2
- R2
- f (ξ1, η1)
f
√c1
√c2 ξ1, √c1 √c2 η1
- dξ1dη1.
Hence, when θ := c1
c2 → 1, we have
lim
θ→1 uc1(0), uc2(0)0 = f2 0.
25
SLIDE 26
Also, uc(0)2
0 =
- R2
- √cf(√cx, √cy)
- 2 dxdy
=
- R2 |f(x, y)|2dxdy = f2
0.
Therefore, we get lim
θ→1 uc1(0) − uc2(0)2 0 = 0.
Now for t > 0, uc(x, y, t) = √cf
√c(x − ct, y)
- = √cf
- (√cx, √cy) − (c3/2t, 0)
- = √c δ√c τhf(x, y),
where h = (c3/2t, 0), τ(a,b)f(x, y) = f((x, y) − (a, b)) δaf(x, y) = f(ax, ay).
26
SLIDE 27
Thus,
- uc(t)(ξ, η) =
uc(ξ, η, t) = 1 √c
- τhf
- ξ
√c, η √c
- = 1
√ce−2πi(c3/2t,0)·(c−1/2ξ,c−1/2η) f
- ξ
√c, η √c
- = 1
√ce−2πictξ f
- ξ
√c, η √c
- .
Next, we evaluate the L2-norm of uc1(t)−uc2(t): uc1(t), uc2(t)0 =
- R2
uc1(ξ, η, t) uc2(ξ, η, t)dξdη = 1 √c1c2
- e2πitξ(c2−c1)
f
- ξ
√c1 , η √c1
- f
- ξ
√c2 , η √c2
- =
√c1 √c2
- e2πitξ1
√c1(c2−c1)
f (ξ1, η1) f
√c1
√c2 ξ1, √c1 √c2 η1
- 27
SLIDE 28
Taking c1 = n + 1 and c2 = n we have, by the Riemann-Lebesgue lemma, lim
n→∞un+1(t), un(t)0 = 0.
But, since un+1(t)0 = un0 = f0 we have lim
θ→1 uc1(t) − uc2(t)0 = lim n→∞ un+1(t) − un(t)0
= √ 2 f0 = 0. This finishes the proof of Theorem 3.
28
SLIDE 29
Final Remarks We observe that Bourgain’s approach to deal with the KdV equation, does not seem to work in our case. Indeed, it is well-known that to
- btain “good bounds” by using the Fourier re-
striction method we need to know very well the behavior of the resonant function, or equiva- lently, the geometry of the resonant set, which is the zero set of the resonant function. In gen- eral, if the geometry of the resonant set is too “complicated” then it is not clear how to per- form dyadic decompositions to get the needed estimates. This is the situation in our case where the resonant function is given by h(ξ, ξ1, η, η1) = (ξ − ξ1)(3ξξ1 + ηη1) + (η − η1)(ξη1 + ξ1η).
29
SLIDE 30
In trying to improve the local well-posedness result in Theorem 1 we note that for the mKdV equation, by using a L4-maximal function esti- mate, Kenig, Ponce and Vega obtained sharp well-posedness results for that equation. So, in lights of the mKdV we may ask if, taking into account a L4
x-maximal function estimate,
we can improve the result in Theorem 1. We establish the following sharp maximal function estimate for solutions of the linear problem as- sociated to (): Proposition 2. For any s > 1/4, r > 1/2 and 0 ≤ T ≤ 1, we have U(t)u0L4
xL∞ yT ≤ C(1 + Dx)s(1 + Dy)ru0L2 xy.
A similar estimate was proved by Kenig and Ziesler for solutions of the linear problem as- sociated to the modified KP equation.
30
SLIDE 31
The Zakharov-Kuznetsov (ZK) equation ut + u ∂xu + ∂x∆u = 0, (11) admits as a solution the well-known KdV soli- tary wave φω(x, t) = φω(x − ωt), where φω(ξ) = 3ωsech2
√ω
2 ξ
- ,
ω > 0. More generally, the N-soliton φN of the KdV equation is also a particular solution of the ZK equation which is smooth and bounded together with its time and space derivatives and behaves as a sum of solitons of velocities 4n2, 1 ≤ n ≤ N when t → ∞. For instance, the 2-soliton φ2 is given by φ2(x, t) = 723 + 4 cosh(2x − 8t) + cosh(4x − 64t) {3 cosh(x − 28t) + cosh(3x − 36t)}2
31
SLIDE 32
A fundamental issue is that of the transverse stability/instability of those one-dimensional “lo- calized” solutions of the KdV equation (such as the solitary wave) with respect to trans- verse perturbations governed by the ZK equa- tion. This question was rigorously addressed recently by Rousset and Tzvetkov who devel-
- ped a general theory which applies in particu-
lar to one-dimensional transverse perturbations
- f the KdV solitary wave.
Functional framework for the Cauchy problem which should be suitable to describe the afore- mentioned transverse perturbations. This framework cannot be the classical Sobolev spaces Hs(Rd) since the KdV soliton or multi- soliton do not belong to this class of spaces. A natural space to study the transverse stability
- f localized one-dimensional solutions should
contain those solutions.
32
SLIDE 33
A first possibility consists in functions which are “localized” in x and periodic in y. This leads to our study of the Cauchy problem for the ZK equation in Hs(R × T). Let T = R/2πZ be the one-dimensional torus. We will thus consider the IVP associated to the ZK equation in a cylinder
∂tu + ∂x∆u + u∂xu = 0, (x, y) ∈ R × T, t > 0, u(x, y, 0) = u0(x, y)
SLIDE 34
A second possibility is to consider two-dimensional “localized” perturbations of the one-dimensional solution φ. This motivates the study of the Cauchy problem,
- ut + ∂x∆u + u∂xu + ∂x(φu) = 0,
(x, y) ∈ R2, t > u(x, y, 0) = u0(x, y). where ∆ = ∂2
x + ∂2 y is the Laplace operator
and φ is the KdV solitary wave solitary wave
- r more generally any N-soliton of the KdV
- equation. Actually, we will only use that φ =
φ(x, t) is a solution of the KdV equation which is smooth and bounded together with its time and space derivatives, and furthemore belongs to the space L2
xL∞ T
. Those assumptions are
- bviously satisfied by the N-soliton solution of