Almost global solutions for the capillarity wave equation...
This is a modal window.
The media could not be loaded, either because the server or network failed or because the format is not supported.
Formal Metadata
| Title |
| |
| Title of Series | ||
| Part Number | 8 | |
| Number of Parts | 21 | |
| Author | ||
| License | CC Attribution 3.0 Unported: You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor. | |
| Identifiers | 10.5446/20765 (DOI) | |
| Publisher | ||
| Release Date | ||
| Language |
Content Metadata
| Subject Area | ||
| Genre | ||
| Abstract |
|
00:00
ExpressionOperator (mathematics)InfinityGreatest elementHausdorff dimensionCausalityMaxima and minimaInsertion lossDiscrete element methodExistenceSinePositional notationModulo (jargon)Normed vector spaceTheoremLogical constantSet theoryFunction (mathematics)Matrix (mathematics)Ext functorMereologyReal numberGamma functionPropositional formulaOrder (biology)Physical systemTranslation (relic)Energy levelIndependence (probability theory)Differential (mechanical device)Normal (geometry)ModulformChi-squared distributionDiagonalReliefTerm (mathematics)Divisor (algebraic geometry)Lemma (mathematics)Parameter (computer programming)Matrix (mathematics)RootFood energySquare numberEigenvalues and eigenvectorsPrime idealTheoryOrder (biology)Derivation (linguistics)ExistenceCohen's kappaLocal ring2 (number)EstimatorFree groupGenetic programmingVector potentialFunctional (mathematics)FluidGlattheit <Mathematik>Slide ruleMultiplication signNichtlineares GleichungssystemWater vaporNumerical analysisOperator (mathematics)Graph (mathematics)Well-formed formulaLine (geometry)GravitationCommutatorStatistical hypothesis testingEvoluteTerm (mathematics)VelocityPhysical systemGradientSpacetimeBeta functionNormal (geometry)ExpressionCategory of beingMultilaterationMultiplicationObservational studyCoefficientConjugacy classVariable (mathematics)Geometric quantizationTransformation (genetics)Right angleClosed setCondition numberParity (mathematics)Associative propertyPower (physics)Translation (relic)Natural numberAnalytic continuationArithmetic meanAveragePosition operatorPrice indexSummierbarkeitMeasurementDiagonalModulformReduction of orderDivision (mathematics)SequenceSinc functionMereologyLogarithmConnectivity (graph theory)ResultantIntegerLimit of a functionChi-squared distributionFrequencyTheoremSubsetLengthCapillary actionSurfaceInfinityDiagonal matrixOrdinary differential equationSet theoryHyperbolic functionSinguläres IntegralInequality (mathematics)Degree (graph theory)Equaliser (mathematics)Complex numberProof theoryBodenwelleLinear mapEnergy levelModulo (jargon)Differential operatorFree surfaceGreatest elementPositional notationPoint (geometry)Sobolev spaceNegative numberVector spaceProjective planeCircleLaplace-OperatorLogical constantSign (mathematics)Basis <Mathematik>Algebraic structureBoundary value problemWave equationPhysical lawPairwise comparisonNormal-form gameIdentical particlesLinear equationCorrespondence (mathematics)TangentResonatorRichtungsableitungSpherical capPotenz <Mathematik>Mathematical analysisDifferent (Kate Ryan album)Imaginary numberCoordinate systemMathematicsLAN partyCalculationBound statePolarization (waves)Group action1 (number)Hamiltonian (quantum mechanics)FinitismusSineTrigonometric functionsGaussian eliminationFinite differenceProcess (computing)Bootstrap aggregatingElementary arithmeticSelf-adjoint operatorAdditionExplosionLattice (group)State of matterConservation lawMusical ensembleBounded variationSolid geometryMaß <Mathematik>Prisoner's dilemmaMortality rateTheory of relativityAreaArithmetic progressionForcing (mathematics)Rule of inferenceIncidence algebraSubstitute goodFraction (mathematics)Many-sorted logicWeightLattice (order)Directed graphInjektivitätIndependence (probability theory)Metric systemVolume (thermodynamics)Complex (psychology)Inclusion mapGrothendieck topologyDatabase normalizationCharge carrierReal numberInsertion lossLink (knot theory)Similarity (geometry)ComputabilityHelmholtz decompositionSpecial unitary groupConsistencyModel theoryHill differential equationStandard deviationDressing (medical)Surjective functionArchaeological field surveyStress (mechanics)Resolvent formalismValuation (algebra)CausalityDecision theoryGoodness of fitQuadratic equationReflection (mathematics)Complete metric spaceVarianceSeries (mathematics)VotingGame theoryCentralizer and normalizerAnnihilator (ring theory)Regulator geneFreezingStudent's t-testComplex analysisZustandsgrößeRecurrence relationConnected spaceComputer animation
Transcript: English(auto-generated)
00:16
My talk will be about a joint work with Massimiliano Berti,
00:20
so concerning the most global solutions for the periodic gravity capillarity equation. So we consider a fluid, say water, which lies between a flat bottom given by this line and a free surface, which is a graph of some function y equals eta of t and of x.
00:43
So this function eta will be periodic, and we assume that the fluid is incompressible and irrotational. So that in the water the velocity of the fluid will be given by the gradient of some potential phi,
01:01
and phi will solve the Laplace equation, Laplacian phi equals zero. And we assume a boundary condition at the bottom, which is that the normal derivative of phi should vanish, and so if we are given the value of this potential on the free surface,
01:22
we may, solving this equation, deduce the value of phi and solve the velocity everywhere. So the motion of the fluid is determined by the function eta and by the restriction psi of capital phi to the free surface. And so if you write the early equations inside the water,
01:42
and if you write a convenient boundary condition at the free surface, you get an evolution equation for eta and psi, which is the so-called Craig-Sulam-Zakharov formulation of the gravity-capillarity wave equations that I describe now. So the first equation in that system is dt eta equals g of eta psi,
02:10
where g of eta is the Dirichlet-Nemann operator, which is defined in the following way.
02:20
So given a function psi, you solve the Dirichlet problem with boundary value psi, and so you find a potential capital phi, and then you compute a normal derivative of potential, dy phi minus dx eta dx phi.
02:40
At the free surface, y equals eta of Tn of x. And this defines g of eta psi, which is just the usual Dirichlet-Nemann operator. The second equation says that dt psi is given by some complicated expression,
03:02
say minus g eta minus one half of dx psi to the square, plus g of eta acting on psi plus dx psi dx eta to the square, divided by twice one plus dx eta to the square, plus kappa h of eta, where h of eta is some explicit function, that's the derivative
03:26
of eta prime over square root of one plus eta prime square, eta prime denoting just dx eta. And so we have two parameters in that equation. Namely, g, which is the gravity,
03:41
and kappa, which is the surface tension. So we shall consider these equations when x belongs to T1. That is, we consider these equations with functions which are periodic in x, and we want to solve that system for long times.
04:07
So let me give some non-results. So I shall not read all the names which are on this slide. I will just say that concerning local existence theory for the preceding system with smooth enough data,
04:24
the most basic work was due to Cjoueau, was proved that when x belongs to R or R2 instead of T1, when kappa, the surface tension, is zero,
04:41
and when the fluid has infinite depth, that local solutions exist over some time interval. So then there have been quite a lot of other works treating different other cases. The only one I shall mention explicitly is the one of Freuser, which obtained local existence in the periodic framework
05:03
for finite depth and for positive kappa. That is the problem we're interested in. And then there have been quite a lot of works devoted to long-time existence when you consider small and decaying data.
05:24
So in that case, you may exploit the dispersive properties of the equation so that the solutions of the linearized equation will decay when time goes to infinity. And this allows you to prove that if moreover the data are small and smooth, then you may get a solution defined over a long time interval
05:42
and even in some cases, a global solution. So I shall not read all these names of contributors. I will just say that there are nevertheless also some results which have been obtained
06:01
when you consider periodic, not necessarily periodic, but data which do not decay at infinity. By Ifrim and Tataru, and also by Alazar-Burk and Zulli.
06:23
And well, it has been shown in different cases that say for instance for the periodic Cauchy data, you have a so-called cubic lifespan. That is, if you take Cauchy data of size epsilon, you get solutions defined over an interval of time of length epsilon to the minus two,
06:44
while local existence theory provides a solution defined over an interval of time of length epsilon to the minus one. And so what we want to do is to try under convenient assumption to do better than getting this cubic lifespan.
07:03
Well, you should ask this question to Daniel and Mia and Ifrim. I guess that this is because quadratic corresponds to a quadratic non-linearity. And then when you have local existence theory, you get one over epsilon. If you have a cubic non-linearity, you get one over epsilon to the square.
07:23
But of course, here the non-linearity is quadratic and nevertheless, you get this cubic lifespan. That's why the origin, I guess, of the terminology. So, let me give the assumptions we shall be working with.
07:45
Let me introduce some operator, just this matrix S, which is a matrix 1, 0, 0, minus 1. And let me say that a solution, eta psi of our capillarity gravity system,
08:03
is reversible, if and only if, when we compute the value of the solution at time minus t, it is equal to the matrix S acting on the solution computed at time t.
08:22
So, of course, it follows immediately from this equality that if this property holds, then the second component, psi, has to vanish at time t equals 0. And conversely, if you start from a Cauchy data whose second component vanishes at time 0,
08:43
then your solution will be reversible that you will satisfy this equality. And this just follows from the fact that your equation may be written as, say, eta dot psi dot equals some function f of eta psi,
09:02
where f has the property that making at this matrix S, at the left, on f of eta psi, you get something which is equal to minus f of S of eta psi. And so, you deduce immediately from that property that under that assumption,
09:22
the solution is reversible. And so, to state the main theorem, I will use this property of reversibility and I will use also a couple of notation that I introduced now.
09:43
So, we shall look for the first component, eta, of our solution in a Sobolev space, h s plus one over four of t one. So, periodic functions which are Sobolev on each period,
10:01
we shall assume, moreover, that these functions are even and we shall assume that the average of eta is zero. So, if these properties are satisfied at the initial time, then they hold for any positive time as a consequence of the structure of the equation.
10:25
So, concerning the second component of eta psi, we shall take it in a Sobolev space, h s minus one over four, of even functions on the torus, on t one.
10:41
And actually, I'm taking an homogeneous Sobolev space, that is a Sobolev space modulo constants. So, we do that because the system is actually well-defined on functions, modulo constants. So, we may consider the second equation of the system, the one on psi as projected on functions and modulo constants.
11:04
And it's natural to do so because actually only the gradient of psi if you want to add the physical meaning, not psi itself. And now that you've introduced these notations and definitions, I may state the main theorem.
11:22
So, the theorem says that you may find a zero measure subset, capital N, of zero plus infinity to the square. And for any couple, G kappa, of gravity and surface tension
11:45
taken outside this subset of zero measure, for any given integer, capital N, there is some index of regularity, zero, large enough,
12:03
such that if you take any f larger than zero, well, you may find some positive constants, c, epsilon zero, such that for any epsilon smaller than epsilon zero,
12:25
for any function eta zero in the Sobolev space that I introduced above with zero average and even, with norm in that space smaller than epsilon, if you solve your capital gravity system
12:45
with Cauchy data at t equals zero given by eta zero and zero, then you get a unique solution, continuous on some interval with values in the natural Sobolev space
13:02
where the length of the interval of existence is bounded from below by C epsilon to the minus N. So, in other words, the theorem says that you may get almost global solutions that these solutions define on an interval of length epsilon to the minus N
13:21
for any given a priori N. If you take Cauchy data which are small enough, which are smooth enough, which are reversible in that sense, in the sense that the second component should be zero at t equals zero
13:40
and if, moreover, you have taken your parameters outside some exceptional subset of zero measure. So, before giving some elements of proof, let me make a remark
14:03
which is that actually you may reduce to the case when G equals one just using the homogeneity of the equation and then the parameter that you take outside a subset of zero measure is just the parameter kappa.
14:22
And let me say also that this theorem provides almost global solutions for the Cauchy problem under the assumptions that I indicated. We do not know if one can obtain global solutions for the Cauchy problem.
14:41
Or even e to the one over epsilon, we do not know that. But on the other hand, there are non-trivial examples of global solutions that is quasi-time-prosperiodic solutions which have been constructed by Berti and Montalto and which will be the subject of the talk of Massimiliano tomorrow.
15:02
Is it with zero? So, your initial data is special. The initial data is special because we want to have these reversible solutions so this corresponds to functions which satisfy the properties that I wrote before eta psi times t equals... No, we need these assumptions and I shall explain why later.
15:25
Is it true or not do you think? Well, you know I am not the kind of person who is making conjectures on the air so I will not...
15:41
No, psi physically is the restriction of the potential to the free surface. So if it's zero, then there's zero initial velocities. You raise the surface and then you let it go but with no initial velocities.
16:04
Ok, so now we shall in a first step try to forget the explicit expression of the equation and reduce ourselves to some parallel differential formulation. So, first of all let me fix some notations.
16:23
So, if a is a symbol, so say a function of x and psi such that d psi beta a is bounded from above by psi power m minus beta for a given m and any beta one associates to this a an operator, a parallel differential operator
16:44
in the following way. One takes chi, some cut off function with more enough support equal to one close to zero and one defines from the function a, a new function a chi
17:02
in the following way. One takes the free transform of a relatively to the variable this gives a hat of x hat psi one multiplies by the cut off chi of x hat of psi and takes inverse free transform.
17:22
And then in that way you define the Bonneville quantization of the function a as the operator associating to some test function u a new function given by this integral one over two pi integral e to the i x minus y xi
17:44
a chi of x plus y over two n of xi u of y dy xi. So in that way one gets an operator which is bounded from h s to h s minus m for n s.
18:01
If you assume moreover that your symbol a is periodic in it your operator will send periodic functions of h s of t one to functions of h s minus m of t one. And then using the celebrated Bonneville parallelization formula
18:21
you may rewrite the equation we started from as a partial equation. That is, you may write the initial equation under the following form. You have capital Dt, where capital Dt will be one over i d over dt
18:45
minus the operator associated to a matrix of symbols capital A. These symbols depend on eta psi because we treat a non-linear equation and all that operator acting on the unknown eta psi
19:04
should be equal to some smoothing operator that is an operator that will gain rho derivatives for rho a given large number that we may choose if we work in spaces of smoothing functions acting on eta psi.
19:23
So usually when you do that from a nice quasi-linear public equation you end up with a system for which you can get easily some energy estimates and then eventually local existence theory.
19:41
But in the case of the water waves equation it is well known that when you do that you get into trouble because the matrix of symbols A that you get are the eigenvalues whose imaginary part goes to infinity when psi goes to infinity.
20:05
In other words, you have instabilities which prevent you from getting energy estimates without derivative velocities from such a formulation when you apply this polarization method
20:23
to a water waves equation. So this problem has been solved years ago first of all by Sijui Wu who designed some Lagrangian formulation of the equations
20:41
that allow one to overcome the difficulty. Can I say something about that? Nadimov invented that. Yeah, but it was for small data. It's the same formulation. The same calculation. So then there has been another approach
21:02
that has been used relying on the so-called good and known of Allin-Harg by Alazar Metivier, Alazar Bergvili, Lan and this is the method we shall adopt and then more recently, Antari-Freeman-Tataro
21:21
proposed a third approach using some holomorphic coordinates. And so we shall use, as I said, this good and known approach which consists in saying that this problem that I was mentioning
21:40
comes from the fact that we didn't use the good and known to express a problem and that instead of writing the equation on eta psi as I did one should introduce instead omega which is psi
22:02
minus some operator acting on eta where b, the symbol of this operator is just a function, an explicit function g of eta psi plus dx psi dx eta divided by 1 plus dx eta to the square. And it turns out that if you write the system
22:24
in these unknowns then things are much better. So more precisely if we start from eta psi the solution of our equation defined on some interval let me introduce
22:42
a complex unknown the following way. Call lambda kappa of d the operator d hyperbolic tangent of d divided by 1 plus kappa d square to the 1 over 4 this is the operator of order minus 1 over 4 and defined from
23:01
the good and known omega and from eta a complex valued function u which is just lambda kappa of d omega plus i lambda kappa of d minus 1 eta. Next, introduce capital U
23:20
the vector u u bar and write your equation under a parameterized form on capital U. So when you do that you obtain the following well capital Dt
23:41
minus the operator with a new matrix of symbols capital A of u dx psi acting on u equals some smoothing operator for our view acting on u. And now what is important is the information we have
24:02
on the structure of that matrix A. So A is a matrix of symbols of order 3 halves so the main contribution in terms of the order
24:22
is given by some Fourier multiplier m kappa of xi where m kappa of xi is given explicitly by xi hyperbolic tangent of xi power one half times 1 plus kappa xi square to the one half so this is a symbol of order 3 halves
24:41
which is multiplied by 1 plus zeta of u t x zeta being some real valued function multiplied by this matrix 1 0 0 minus 1 and you have also another contribution of order 3 halves m kappa zeta
25:01
multiplied by a matrix 0 minus 1 1 0 then you have also contributions of positive but lower order namely lambda one half of u times so matrix and lambda one of u
25:21
times the identity matrix. These two symbols are of order one half and 1 respectively so they are of positive order but it turns out that their imaginary part is of order j minus 1 so of
25:42
negative order or at least non-positive order for lambda one. And then you have contributions of non-positive order given by symbols lambda minus one half of order minus one half lambda zero and what is important here
26:03
is that when you compute the eigenvalues of that new matrix you get that the imaginary part of these eigenvalues is bounded when xi goes to infinity which was a property which was not true in the initial formulation
26:20
and this property comes from the fact that using the good and known you make appear a symbol lambda one half here which has the imaginary part of negative order while you are doing that starting from the initial unknowns psi and eta instead of omega and eta
26:41
you would have got a similar expression but the imaginary part of this symbol would have been of order still one half and this is what would be responsible of the fact that the eigenvalues of the imaginary part of order one half that may go to infinity when xi goes to infinity and this is
27:03
the point that creates instability in the initial formulation so now that we have this form of the equation let me also introduce three properties satisfied by the matrix of symbols a
27:21
that will play an essential role so recall that I define reversibility using some linear operator S and now I will call S the matrix minus zero one one zero that's just a translation
27:41
of the same operator but on the new basis we are using in the complex formulation of our equation and the matrix of symbols that you get for the expression of your system satisfies three properties
28:01
the first one says that if you compute this matrix of symbol a at u t x minus xi and conjugate it you get minus S a of u t xi S so this property just means that when you consider
28:20
the associated operator make it act on v and take the conjugate what you obtain is minus S the operator with symbol a and this property is a property that just reflects the fact that we are working with a system
28:40
on capital U which is u bar whereas the second equation of the system is obtained from the first one by conjugation so this is just a property coming from the fact that we started from an equation from a system with real valued functions the second property satisfied
29:06
minus xi is equal to a of u t xi and this means that the associated operator in particular preserves even functions which is a property we need since we shall be working with even functions
29:20
and finally the third property is the so called reversibility which says that if you compute the matrix a at SU t xi it is equal to
29:40
minus S a of u t xi S so the meaning of this property is as follows if you consider the right hand side of our system which was op a of u acting on u
30:00
and if you apply to it S this is equal to minus the operator with symbol a of SU acting on SU and this just reflects the fact that the system is reversible the properties that I used before when I wrote that
30:21
S of some function f of eta psi was equal to minus f of S of eta psi so that's the third property we shall be using to obtain the result of log time existence and now let me give some ideas on the proof
30:44
so the first step is a sequence of reductions that have been used in other contexts by Alsabagli by Betty Montart so I will not give any detail about these reductions I will just say that
31:03
making a diagonalization of the symbol of your operator and then making some change of variable or actually some para composition and making conjugations by convenient free integral operators
31:20
you may reduce the system that I wrote before to a system which has constant coefficients so in other words you may introduce some new unknown capital V that may be expressed explicitly from u
31:42
that will be of the same size of u in Hs when both terms are small enough and such that these new unknowns will solve new equations that may be written as follows so you have capital Dt
32:04
minus m kappa of dx the Fourier multiplier that I introduced before which is dx hyperbolic tangent of dx to the one half 1 plus kappa square dx square to the one half
32:20
times some function 1 plus zeta underline of u and of t which is only a function of t so this operator is a constant coefficient operator at fixed time so this multiplies the matrix diagonal matrix 1 0 0 minus 1
32:42
and then you have another operator capital H of u t dx is also a diagonal matrix of constant coefficients through different operators which are of order 1
33:01
and what is important is that the imaginary part of the symbol H is of order 0 while H itself is of order 1 and finally this operator H satisfies the properties
33:24
the reality, parity preservation and reversibility conditions and in the right hand side of the equation you have r of u acting on v where r is again some smoothing operator that satisfies also these three properties
33:44
so in other words we have reduced ourselves to a new equation where in the left hand side you have just constant coefficient operators and so it's very easy now
34:01
to make energy estimates on that equation well actually when you make an energy estimate on such an equation the real part of H gives a self-adjoint operator which disappears in energy estimates
34:22
and so what you are left with is just contributions coming from the imaginary part of H which is an operator of order 0 which is bounded on v2 and moreover since we have constant coefficients we may commute
34:40
as many derivatives as we want and so we can write as well L2 or Sobolev energy estimates so in other words if you write on the pre-symmetric equation a very basic energy estimate you get that
35:00
the Sobolev norm of the solution of time t is bounded by the Sobolev norm at time 0 plus constant the integral from 0 to t of the operator imaginary part of H of u tau dx acting on v at time tau in Hs d tau
35:23
so if we knew in addition that this symbol imaginary part of H not only is over the 0 but that it vanishes like norm of u in Hs to the n when u goes to 0
35:40
then the proof will be finished because in this case we could write the preceding inequality saying that the Sobolev norm of v at time t is bounded by the Sobolev norm of v at time 0 plus an integral from 0 to t
36:01
of the norm of u at time tau in Hs power n norm of v at time tau in Hs d tau and since we work with Cauchy data of size epsilon and since we aim at propagating the fact that the solution
36:21
stays of such a size over some time interval morally in this integral this norm of u to the n is of size epsilon to the n so that some elementary bootstrap argument will show you that if you consider this inequality
36:40
for times small than c divided by epsilon to the n for small positive c then you will be able to prove an a priori rebound saying that the left hand side is more than k times epsilon where k is some fixed constant
37:01
and so this a priori rebound will tell you that the solution may be extended up to such a time but of course to do so we would need to have this property that the imaginary part of H vanishes like a large power of u when u goes to zero
37:21
and this is not true in H vanishes like u power one when u goes to zero so we have to do a first step in order to arrive to such a property and so this step
37:46
is let me call b of capital U dxi some diagonal matrix of symbols over zero to be determined and let me look for
38:00
again a new variable a new unknown v tilde given by the exponential of b of ut dx acting on v so in other words what was on the preceding slide
38:21
namely dt minus m cap of dx times one plus zeta underline one zero zero minus one plus h of ut dx acting on v equals some smoothing operator acting on v and what I'm doing is just that I intertwine this operator
38:41
by this exponential so since I have constant coefficients operators this exponential of b of ut dx commutes with all these operators and so when I'm making the conjugation of this equation
39:00
by this exponential the only new contribution that will appear will come from the conjugation of dt with xb that is I will get another term the new unknown v tilde
39:20
satisfies an equation which is capital dt minus dt b of ut dx minus m cap of dx one plus zeta underline one zero zero minus one plus h of ut dx v tilde equals r of u v tilde
39:41
and now what we have to do since I recall that what we want to do is to cancel the contributions to the imaginary port of h that are vanishing when u goes to zero at order lower than
40:02
some u to the n for a given n so in other words what we want to do is to choose capital b such that this term will cancel the contributions
40:21
so why do you worry only about cancelling the contribution from h and not r oh no of course I have also to do the same thing for r later of course of course but here I'm you know the method is pretty much the same in both cases and so I explain it
40:54
so to construct b I'm looking for it as a sum
41:01
of expressions bp of u u u which are homogeneous of degree p p going from one up to n minus one the last degree that I want to cancel out each bp being so p linear map
41:21
that are restricted to the diagonal u equals u u1 equals u2 et cetera so now let me compute dtb when I compute dtb the t derivative will act successively on each argument of bp and I will replace
41:43
the corresponding dtu using the equation which was that dtu is some free multiplier m kappa of dx times some matrix k which was one zero zero minus one
42:01
acting on capital u plus some non-linear terms which will generate contributions of higher degree of homogeneity so if I write what I get as a level of the contributions of homogeneous of degree p
42:20
so I will get bp in which I have replaced one of the arguments by the linear part of the equation this j corresponding to the place of this argument going from one to p the non-linearities will give terms of higher degree of homogeneity
42:41
that I may forget at that level and I want to choose bp in order to get rid of the contribution to in h homogeneous of degree p that I may write as some imaginary part of hp of uu dxi
43:02
and now this is essentially the equation to which I reduce myself to finish the proof so I wrote again that equation at the top of this slide and to solve it
43:21
let me decompose each argument capital u as a sum of pi as a sum in n of pi n plus capital u plus pi n minus capital u where what I denoted by pi n is just the spectral projector
43:41
associated to the nth eigenmode of the Laplacian on the circle and where since capital u is a two vector the plus and the minus denote projection on the first or the second component on that vector
44:01
so if I replace each argument u by such a decomposition and if I look at what happens when I make act this operator on for instance pi n plus capital u one sees immediately that
44:21
the result of this operation acting on such a localized is given by multiplication of the function by the symbol m kappa computed at n so in other words when we write the preceding equation
44:40
replacing the arguments by pi n one plus u pi n l plus u pi n l plus one minus u up to pi n p minus u when you make act this operator on one of the first l terms you get multiplication by m kappa of nj
45:01
with a plus sign and of course this multiplication by a function may move out of vp by denarity in the same way for the last terms when you make act such an operator on the last terms you get multiplication
45:28
where you replace each argument by a spectrumly localized one what you obtain is just bp computed at these arguments multiplied by some function
45:42
dl which is just given by this sum m kappa of nj from one to l minus the same sum from l plus one to p so our problem is to determine bp and so we have just to divide
46:02
by dl and so we have to know that dl is not zero so clearly there is a case when dl will be zero whatever you do which is a case when you have the same number of terms in both sums
46:21
and when you have two by two cancellations between one term in the first term and one term in the second one this may happen and so I have to distinguish two cases
46:40
the first one being the case when such a scenario does not happen so let me assume that I am not in the preceding case that is in the case when p is even l is p over two
47:01
and the set n1 and l coincide with nl plus one and p since in this case the function dl vanishes identically so the lemma says that if you are not in this case then if the parameter kappa surface tension
47:21
is taken outside a convenient subset of zero measure then you may ensure a small device is estimated saying that not only dl does not vanish moreover dl is bounded from below
47:40
by the constant times n1 plus d3 represent p power minus some integer n0 so in other words when we are not in the forbidden case we shall be able to divide by dl that is we shall be able to solve the equation
48:01
we were interested in dl times bl equals i in the front part of hl since dl does not vanish so of course when we make such a division we lose some large power over n1 plus np which means that we lose some derivatives
48:21
on the coefficients of our equation but this does not matter because the coefficients of the equation are low frequencies that is we may afford to lose a large number of derivatives of these coefficients if of course we work
48:41
okay so this solves the equation we wanted to study in the case which except in the case p even l equals p over 2 and n1 l and l equals n l plus 1 n as we have seen in that case this dl will vanish identically
49:00
whatever the value of the parameter kappa and so to solve the equation in that last case we use the three properties that I was mentioning reality, parity preservation and reversibility because combining these three properties one may check that
49:22
the right hand side of the equation in hl or in hp in this case computed at pi 1 plus u up to pi np minus w with n1 and p satisfying this equality vanishes identically so that in the remaining case
49:41
the equation to be solved was just 0 equals 0 and we know how to solve such an equation and consequently we have been able to solve in all cases this equation and we have seen before that solving this equation was allowing one to eliminate the contributions to
50:01
in hl which have a low degree of homogeneity and that doing that by energy inequality our long time existing result over an interval of time of length epsilon minus n and being the level
50:20
at which we stop this process of elimination so this concludes the proof of the theorem and this concludes also my talk which is fortunate because my time is essentially over
50:43
Are there some questions? There is a very interesting cancellation in this reversible and parity setting so in the general case that is allowing cosines and sines
51:01
of x I mean because those somehow from my Hamiltonian point of view those terms are very important and those are ones which really are important to the dynamics and so I imagine what caused the action in terms of frequency
51:21
non-degeneracy so I'm just wondering what happens in the general case So in the general case you know what one is able to do in other settings and what one would like to be able to do here would be to use the fact that the equation is Hamiltonian and then the terms
51:40
that you cannot eliminate are dependent only on the actions so these terms and so these terms they cannot grow their Sobolev norms do not grow because they depend just on the actions and so these are terms
52:01
that you need to eliminate in a Birkhoff normal form using the Hamiltonian character of the equation and so why didn't we use this approach well this is because in the Hamiltonian
52:20
the reversibility condition is trivially preserved but the point is that the passage from the old unknown to the new unknown is not something that is Hamiltonian or at least we do not know to do that in an Hamiltonian way if we could
52:40
design some Hamiltonian way of doing that step then I'm pretty confident that we could write this framework and so get rid of these conditions of parity and reversibility sorry? I think I answered the question
53:00
I think some other questions but that doesn't work in this case in any case that would not work in this case you have to go to another point of view maybe I have a short question concerning this poverty the one you spoke
53:21
so usually the papers are in the zero order term of the equation of the symbol so here your kappa is in the main part of the symbol so it's something which I never saw it makes really a difference in the analysis
53:40
you did not mention much about this minoration dL how you prove it so this follows from essentially from some old result with Jeremy to say to some general functions
54:00
satisfying some conditions so the fact that it depends this m kappa which was written here so the fact that the kappa here
54:21
is in front of a derivative of i order does not matter at all of this lower bound what is important is the fact that when you consider this function as a function of xi the dependence on kappa is essentially a non-trivial dependence
54:41
and this is what allows you to obtain this small divisor estimate to say that when you evaluate this or what I call dL only at integers then moving a little bit kappa you may arrange so that this lower bounds are satisfied
55:02
so the fact that kappa is here or here doesn't matter at all actually you could factor out kappa and get a parameter here and you wouldn't change anything In terms of the method of proof can you
55:21
make a comparison with the result of of tataru and fin so the law of epsilon so the basic point the basic stopping point is the same you want to get rid of terms of lower degree
55:41
in the non-linearity so to get their cubic result you have to get rid of the quadratic term of the non-linearity oh no no this is not a parameter but this is because the point is that they don't have parameter because it's at the level
56:01
of the quadratic non-linearity for which you will have essentially no resonance and they don't need disparity condition also they don't need disparity condition also they don't need disparity condition but disparity condition actually we need it to deal with the bad terms
56:22
for which you cannot make non-zero your function dl and so you need it only to get rid of terms of odd degree in your non-linearity you need it to get rid of a cubic term you do not need it to get rid of a quadratic term or a quartic term so up to those cubic terms
56:42
that proof will be the same yeah except that you have kappa except zero that's important yeah so you would you be able to recover
57:02
in other words without the generic condition well I suppose we could but we didn't try to write it down we wrote the general thing