Well-posedness results for dispersive equations with derivative nonlinearities [Elektronische Ressource] / vorgelegt von Sebastian Herr

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• Mathematics

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• Publié par: technischen_universitat_dortmund
• Publié le: 01 janvier 2006
• Nombre de lectures: 45
• Langue: English

Extrait

Well-posedness results for dispersive
equations with derivative nonlinearities
Dissertation
zur Erlangung des Grades
eines Doktors der Naturwissenschaften
Dem Fachbereich Mathematik der Universit at Dortmund
vorgelegt von
Sebastian Herr
am 2. Mai 2006Gutachter der Dissertation:
Prof. Dr. Herbert Koch (Universit at Dortmund)
Prof. Carlos E. Kenig, Ph.D. (University of Chicago)
Tag der mund lichen Pruf ung: 8. August 2006Contents
Introduction v
1 Cauchy problems and well-posedness 1
1.1 Basic function spaces . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 The Fourier transformation and Sobolev spaces . . . . 2
1.1.2 The periodic case . . . . . . . . . . . . . . . . . . . . . 3
1.1.3 Conventions . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Linear equations . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.1 Linear homogeneous equations . . . . . . . . . . . . . 6
1.2.2 Linear inhomogeneous equations . . . . . . . . . . . . 7
1.3 The nonlinear Cauchy problem and well-posedness . . . . . . 7
1.4 Analytic maps between Banach spaces . . . . . . . . . . . . . 12
1.5 Notes and References . . . . . . . . . . . . . . . . . . . . . . . 13
2 Dispersive estimates and Bourgain spaces 15
2.1 Dispersive estimates . . . . . . . . . . . . . . . . . . . . . . . 15
2.1.1 The periodic case: The Schodr inger equation . . . . . 15
2.1.2 The non-periodic case: Generalized dispersion . . . . . 16
2.2 Fourier restriction norm spaces . . . . . . . . . . . . . . . . . 20
2.2.1 The periodic case: The Schodr inger equation . . . . . 20
2.2.2 The non-periodic case: Benjamin-Ono type equations 27
2.3 Notes and References . . . . . . . . . . . . . . . . . . . . . . . 33
3 Derivative nonlinear Schr odinger equations 35
3.1 Motivation and main results . . . . . . . . . . . . . . . . . . . 35
3.2 The gauge transformation . . . . . . . . . . . . . . . . . . . . 37
3.3 Multi-linear estimates . . . . . . . . . . . . . . . . . . . . . . 44
3.4 Proof of the well-posedness results . . . . . . . . . . . . . . . 52
3.4.1 The gauge equivalent Cauchy problem . . . . . . . . . 53iv Contents
3.4.2 Proof of the main results . . . . . . . . . . . . . . . . 57
3.4.3 A Counterexample to tri-linear estimates . . . . . . . 61
3.5 Notes and References . . . . . . . . . . . . . . . . . . . . . . . 64
4 Benjamin-Ono type equations 67
4.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.2 Equations of Benjamin-Ono type . . . . . . . . . . . . . . . . 68
4.2.1 A bilinear estimate . . . . . . . . . . . . . . . . . . . . 69
4.2.2 Proof of well-posedness . . . . . . . . . . . . . . . . . 81
4.2.3 Sharpness of the low frequency condition . . . . . . . 88
4.3 The Benjamin-Ono equation in the periodic case . . . . . . . 89
4.3.1 A Counterexample to bilinear estimates . . . . . . . . 91
4.4 Equations with weak dispersion . . . . . . . . . . . . . . . . . 95
4.4.1 Review of a re ned energy method . . . . . . . . . . . 96
4.5 Notes and References . . . . . . . . . . . . . . . . . . . . . . . 99
A Auxiliary estimates 101
A.1 A Gagliardo-Nirenberg estimate in the periodic case . . . . . 101
A.2 An estimate involving exponentials . . . . . . . . . . . . . . . 102
A.3 An estimate for . . . . . . . . . . . . . . . . . . . . . . . . . 103
A.4 Energy estimate for the DNLS . . . . . . . . . . . . . . . . . 104
A.5 Conservation laws for the DNLS . . . . . . . . . . . . . . . . 105
Bibliography 116Introduction
The present work is devoted to the study of Cauchy problems for nonlinear
evolutionequationswithinitialdatainSobolevspacesoflowregularitywhich
describe the propagation of nonlinear dispersive waves.
We are interested in a well-posedness theory for these problems, i.e. for
given initial data we try to nd
(i) unique
(ii) solutions
(iii) whose initial regularity persists
(iv) and which depend continuously on the initial data.
We are challenged to prove results with regularity assumptions on the initial
1data which are as weak as possible . It is part of the problem to nd an
adequate way to express all these four aims precisely and consistently in a
low regularity context.
The examples discussed here arise as one-dimensional model equations
for nonlinear wave propagation in water wave theory (Benjamin-Ono type
equations) or plasma physics (derivative nonlinear Schodir nger equation).
2Inordertointroducetheprincipleofdispersion letusconsiderthelinear
equation
3∂ u+∂ u = 0 (Airy)t x
We may calculate explicit solutions u : [ T,T]R →R with the help
of Fourier analysis: Let the periodic initial datum be given by u (x) =0P
ikxc e , then the periodic solution isk
X
3i(kx+tk )u(t,x) = c ek
1 sOn the Sobolev scaleH this means that we try to chooses ∈R as small as possible.
2In the present work we focus on the analytical eects of special dispersion relations
and do not give a formal de nition of dispersive waves in general cp. [Whi74], pp.363–369.vi Introduction
which shows that the k-th Fourier mode of the initial datum propagates
2with group velocity 3k . The di erent speed of Fourier modes has certain
regularizing e ects such as higher integrability and in the nonperiodic case
2the gain of fractional derivatives in L . On the other hand, for each t > 0loc
we observeku(t)k s =kuk s and the solution operator is unitary in theH 0 H
sSobolev spaces H such that the solution has exactly the same regularity as
sthe initial datum in the H sense.
Let us consider a nonlinear version of this equation, the Korteweg-de
Vries equation
3 2∂ u+∂ u =∂ u (KdV)t xx
One idea to establish well-posedness for nonlinear equations is to apply the
Picard iteration scheme to a related integral equation as in the case of or-
dinary di erential equations. The nonlinearity has to be small in a suitable
sense such that the Duhamel term is a strict contraction and its in uence
on the linear solution is not too strong. To control all possible nonlinear
interactions in the quadratic term and to gain the derivative one may ex-
ploit the above mentioned dispersive properties of the linear equation. This
strategywasusedbyC.E.Kenig-G.Ponce-L.Vega[KPV93c]toprovewell-
posedness results for (generalized) KdV equations in the non-periodic case.
In [Bou93] J. Bourgain developed a general approach to the well-posedness
of nonlinear dispersive equations which reduces the problem to multi-linear
estimates in spaces which are de ned according to the symbol of the linear
equation and inherit the dispersive properties of the solutions.
These harmonic analysis techniques in combination with the Picard it-
eration found applications in many di erent situations and lead to strong
well-posedness results in spaces of low regularity. All of them share the
3property that the ow map (data upon solution) is necessarily analytic .
Later, it was observed that there are many interesting equations where a
smooth dependence on the data or at least multi-linear estimates fail to hold
[MST01, KT05b] even for regular data, although there are well-posedness
results which include the continuous dependence on the data and which are
based on energy type arguments.
Another approach to many nonlinear dispersive equations, such as the
Korteweg-deVries,theBenjamin-OnoorthederivativenonlinearSchodir nger
equation is provided by the inverse scattering theory, cp. [AC91]. However,
the results in the present work do not rely on inverse scattering techniques.
Here, we are particularly interested in situations where standard multi-
linear estimates for the nonlinear term cannot be true and a direct approach
3This holds true if the nonlinear terms are analytic.Introduction vii
via the Picard iteration is not applicable. Our aim is to overcome these dif-
culties by identifying the strongest interactions and modifying the method
accordingly.
In Chapter 1 the basic notation and a notion of well-posedness is in-
troduced. In Chapter 2 the dispersive properties of solutions to the linear
equations
(∂ | D| ∂ )u =0t x
in the non-periodic and
2
(∂ i∂ )u =0t x
in the periodic setting as well as related function spaces are discussed.
In Chapter 3 derivative nonlinear Schr odinger equations in the periodic
setting are considered, in particular
2 2∂ u(t) i∂ u(t) =∂ (|u| u)(t) for t∈( T,T)t xx
u(0) =u0
1sA local well-posedness result for initial data in H (T) for all s is proved
2
which extends to global well-posedness for s 1 and data which satis es
2a L smallness condition, cp. [Her05a]. A detailed uniqueness statement is
given and it is shown that the ow map is not uniformly continuous on balls
s 1 2in H (T) for s , but locally Lipschitz on subsets of data with xed L2
norm. Similarly, for a version of this equation with a regularized nonlinear
term well-posedness follows with real analytic dependence on the intial data.
The results are shown to be sharp in certain directions.
4In Chapter 4 equations of Benjamin-Ono type
1 2∂ u(t)| D| ∂ u(t)+ ∂ u (t) =0 for t∈( T,T)t x x
2
u(0) =u0
are studied. Section 4.2 deals with the cases 1 < < 2 in the non-periodic
(s,ω)setting. Local well-posedness for initial data in spaces H (R) for s >
3 1 1( 1)andω = andglobalwell-posednessforrealvalueddatainthe
4 2
1 1range s 0 and ω = is shown, cp. [Her05b]. These space