Dispersion dynamics for the defocusing generalized Korteweg-de Vries equation
aa r X i v : . [ m a t h . A P ] J un DISPERSION DYNAMICS FOR THE DEFOCUSING GENERALIZEDKORTEWEG-DE VRIES EQUATION
STEFAN STEINERBERGER
Abstract.
We study dispersion for the defocusing gKdV equation. It is expected that it isnot possible for the bulk of the L − mass to concentrate in a small interval for a long time. Westudy a variance-type functional exploiting Tao’s monotonicity formula in the spirit of earlierwork by Tao as well as Kwon & Shao and quantify its growth in terms of sublevel estimates. Introduction
We are interested in the dispersion properties of the defocusing generalized Korteweg de Vries(gKdV) equation. The gKdV equation is given by ∂ t u + ∂ xxx u = µ∂ x ( | u | p − u ) , where p > µ is real ( µ = 1 corresponds to defocusing) and u is real-valued. There are conser-vation laws for mass and energy M ( u ) = Z R u ( t, x ) dxE ( u ) = Z R u x ( t, x ) + µp + 1 | u ( t, x ) | p +1 dx as well as a scaling symmetry u ( t, x ) → λ − p − u (cid:18) tλ , xλ (cid:19) . The focusing case µ = 1 has been intensively studied: we refer to the influential work of Martel &Merle; in particular, their Liouville-type theorem [11] states that (under some conditions) a per-turbation of the soliton (away from the soliton manifold) cannot be L − compact. Based on theextensive theory developed for the focusing gKdV, there are also results by de Bouard & Martel[2] and Laurent & Martel [9].The defocusing case µ = +1 does not only not have solitons, it is also expected to not have’pseudosolitons’ (solutions whose L − mass exhibits spatial concentration over time) in a fairlygeneral sense. This has been fully resolved in the mass-critical case p = 5 by Dodson [4] (buildingon earlier work by Killip, Kwon, Shao & Visan [7] and an older result [3] of his). We emphasizethat we are interested in the long-time dynamical behavior of smooth initial data u ∈ S ( R ): thedefocusing nature of the equation guarantees global well-posedness, including uniform bounds on k u k L ∞ t,x and k u k L ∞ t H x and regularity never becomes an issue. A similar problem occurs for thedefocusing, one-dimensional nonlinear wave equation − u tt + u xx = | u | p − u on R with k u (0) k H ( R ) + k ∂ t u (0) k L ( R ) < ∞ . Here, again, k u k L ∞ t,x bounds are easy and the questionis whether there is an actual decay of k u ( t ) k L ∞ x . In a certain averaged sense, this was proven byLindblad & Tao [10].Using weighted integrals of the conservation laws (an idea dating back at least to Friedrichsand famously used in the work of Morawetz [12]), the first result for the defocusing gKdV is due to Tao [15]. He introduced the normalized centers of mass and energy via h x i M := 1 M Z R xu dx and h x i E := 1 E Z R x (cid:18) u x + 1 p + 1 | u | p +1 (cid:19) dx, where M and E denote mass and energy of the solution, respectively. If we assume dynamics fromthe linear part to play the dominant role in the nonlinear behavior, then high frequencies of thesolutions should move to −∞ much quicker than slow frequencies. However, since energy is moreweighted toward high frequencies, we could hope for some connection between the centers of massand energy. Theorem (Tao) . Let p ≥ √ and let u be a global-in-time Schwartz solution. Then ∂ t h x i E < ∂ t h x i M . The condition p ≥ √ Theorem (Tao) . Let p ≥ √ and let u be a global-in-time Schwartz solution. Then, for anyfunction x : R → R , sup t ∈ R Z R | x − x ( t ) | (cid:18) u ( t, x ) + u x ( t, x ) | u ( t, x ) | p +1 p + 1 (cid:19) dx = ∞ . This has no implications on spatial decay of the solution because some energy might move faraway very quickly. Kwon & Shao [8] noticed that the monotonicity formula could be employed inanother way to exclude some type of spatial decay.
Theorem (Kwon & Shao) . Let p ≥ √ and let u be a global-in-time Schwartz solution. Then,for any function x : R → R , sup t ∈ R Z R ( x − x ( t )) u ( t, x ) dx = ∞ . The key observation is the simple fact that Z R ( x − x ( t )) u ( t, x ) dx ≥ Z R ( x − h x i M ) u ( t, x ) dx, where the right hand side allows for an explicit computation. In particular, if a solution satisfies | u ( t, x ) | . | x − x ( t ) | / ε for some x ( t ) : R → R , then u ≡ . Inspired by these results, we study the interaction functional I : L ( R ) × L ( R ) → R ∪ {∞} given by I ( f ) := Z R Z R f ( x ) ( x − y ) f ( y ) dxdy. Formal calculations with a translation invariant interaction term η ( x − y ) yields complicated ex-pressions of the form η ′′′ , which naturally suggests η ( x − y ) = ( x − y ) . Interaction estimates ofthis type have been used very effectively for NLS: we refer to the work of Colliander & Grillakis& Tzirakis [1] who use a weight that is essentially quadratic for small distances and results byPlanchon & Vega [14] (where a similar computation is mentioned but they ultimately manage touse a lower order expression η ( x − y ) = | ( x − y ) · ω | , where ω is a fixed unit vector). QuadraticMorawetz estimates for NLS and Hartree equations are also studied by Ginibre & Velo [6].Our functional incorporates structure coming from Tao’s theorem as follows: for any solution u ( t ) of the gKdV, the variation of the functional in time can be written as ∂ t I ( u ( t )) = 12 EM ( h x i M − h x i E ) + 4 p − p + 1 (cid:18) h x i M M Z R | u | p +1 dx − M Z R | u | p +1 xdx (cid:19) . ISPERSION DYNAMICS FOR THE DEFOCUSING GENERALIZED KORTEWEG-DE VRIES EQUATION 3
So far, this is very much in spirit of the earlier results by Tao and Kwon & Shao and unboundednessof the functional follows from their approach. A key novelty in our approach is a slight refinementof Tao’s monotonicity which yields additional control on ∂ t I ( u ( t )) in terms of mass, energy and I ( u ( t )), which allows for bootstrapping and the derivation of some additional information such assublevel estimates on I ( u ( t )). 2. Statement of Results
Our result relies on Tao’s monotonicity and inherits all its requirements. We believe the con-dition on p to be an artifact and that all statements hold true for much rougher solutions u aswell. Theorem.
Let p ≥ √ and let u be a global-in-time Schwartz solution. Then |{ t > I ( u ( t )) ≤ z }| . u (0) z p . Our emphasis is on the following: the functional I ( u ( t )) is neither monotonically increasing norconvex (except for p = 3) but can nonetheless only be small for a bounded amount of time. Theproof is based on showing that I ( u ( t )) satisfies a certain integrodifferential inequality, where theinfluence of nonlinear dynamics acting on regions of mass concentration can be clearly observed.The precise dependence of the implicit constant on the initial data follows from the proof and isrelated to mass, energy and the centers of mass and energy.The following minor improvement of the Kwon-Shao nonexistence result is a trivial consequence. Corollary 1.
Let ε > be fixed. Under the assumptions of Theorem 1, if | u ( t, x ) | . (1 + t ) p − ε | x − x ( t ) | + ε for some x : R → R , then u ≡ . For the particular case of the mKdV ( p = 3), some algebraic simplifications immediately implythat the functional I ( u ( t )) is convex. However, in this case the connection to the focusing KdVvia the Miura map should give a wealth of additional information anyway – the problem can beexpected to be much simpler in this special case.We consider the following corollary to be a much more interesting consequence: its conditionson center and mass are immediately seen to be necessary for the argument to work, however,heuristic arguments (to be described below) make them also seem necessary. Corollary 2.
Let p ≥ √ and let u be a global-in-time Schwartz solution. There exists a constant c > depending only on p such that if h x i E (cid:12)(cid:12) t =0 ≤ h x i M (cid:12)(cid:12) t =0 , and I ( u (0)) ≤ cM − p +3 p − E − then inf t> I ( u ( t )) ≥ I ( u (0)) . Remark.
As we will show below, the assumption I ( u (0)) ≤ cM − p +3 p − E − does imply M < c ′ for some constant c ′ = c ′ ( c, p ). The entire statement is thus only applicable to initial data withsmall mass. Interpretation.
If the function is sufficiently localized and very smooth, then at least some partof it needs to break off and go away quickly never to return. The assumption on the centers is notat all unreasonable: assume u (0) is some L − normalized bump function localized in space x ∼ STEFAN STEINERBERGER and Fourier space ξ ∼ w localized around x ∼ x ≫ ξ ∼ N .Then, for x ≫ k w k − L , I ( u (0)) ∼ x k w k L . Assuming w to be small in L ∞ , we expect linear dynamics to be dominating. This means that theperturbation w moves with speed − N and noticeably decreases the functional (if N ≫ √ x )while the big bump function u barely moves at all during that time. We need to exclude thisscenario and indeed, for centers of mass and energy, we have h x i M (cid:12)(cid:12) t =0 ∼ x k w k L h x i E (cid:12)(cid:12) t =0 ∼ x k w x k L ∼ x N k w k L . The condition h x i E (cid:12)(cid:12) t =0 ≤ h x i M (cid:12)(cid:12) t =0 now implies N . u (0) and the problem cannot occur.A small caveat: we need to make sure thatno inequality of the type I ( u (0)) ≥ cM − p +3 p − E − holds.Otherwise the statement would be a statement about the energy landscape of the functional I andnot about the dynamics of the equation. We give a quick classification of all ( α, β ) ∈ R for which I ( u ) & M ( u ) α E ( u ) β holds true. Proposition 1.
Let p > , ≤ α ≤ pp − and β = (4 − α ) p + 5 α − p + 3 . Then there exists a constant c p > such that for any function u ∈ H ( R ) (cid:18)Z R u x | u | p +1 p + 1 dx (cid:19) β (cid:20)Z R Z R u ( x ) ( x − y ) u ( y ) dxdy (cid:21) ≥ c p (cid:18)Z R u dx (cid:19) α . A simple scaling argument shows that these are the only ( α, β ) for which such an inequalitycan possibly hold. In particular, setting α = 3 gives β = − I (0) & M E − , which renders Corollary 2 nontrivial but also shows that it is only applicable in the case of smallmass. These inequalities seem fairly technical and of little intrinsic interest. That is why we weresurprised about the following: Proposition 1 is implied by Sobolev embedding and interpolationwith the following elementary inequality, which we couldn’t find in the literature. In an earlierversion of the manuscript, we proved existence of extremizers and gave some rough bounds on c p . However, these extremizers can be found explicitely via the Lagrange multiplier theorem (anobservation that was communicated to us by Soonsik Kwon). Proposition 2.
Let p > and c p = 12 π Γ (cid:16) p +1 p − (cid:17) Γ (cid:16) pp − (cid:17) Γ (cid:16) pp − (cid:17) Γ (cid:16) p − p − (cid:17) p +3 p − Γ (cid:16) p +12 p − (cid:17) Γ (cid:16) p +1 p − (cid:17) p +1 p − ≥ πe . Then, for every u ∈ L ( R ) , (cid:18)Z R u x dx (cid:19) (cid:18)Z R | u | p +1 dx (cid:19) p − ≥ c p (cid:18)Z R u dx (cid:19) p +1 p − . Additionally, all minimizers are given by translation, scaling and dilation of the compactly sup-ported function u ( x ) = ( (1 − x ) p − if | x | ≤ otherwise. ISPERSION DYNAMICS FOR THE DEFOCUSING GENERALIZED KORTEWEG-DE VRIES EQUATION 5
Remark.
Suppose u ∈ H ( R ). Then there is the classical uncertainty principle in the form k ux k L k u x k L ≥ k u k L . Combining our inequality with the Gagliardo-Nirenberg inequality k u k p +1 L p +1 ≤ G p k u k p +32 L k u x k p − L yields a version of the uncertainty principle with different constants and an additional term sand-wiched in the middle k ux k L k u x k L ≥ G p − p k ux k L k u k p +2 p − L p +1 k u k p +3 p − L ≥ √ c p G p − p k u k L . Proof of Theorem 1
We start by giving a refined version of Tao’s monotonicity formula, then derive a functionalinequality and conclude by showing the sublevel estimate for all solutions of the differential in-equality. Throughout the paper I ( u ( t )) := Z R Z R u ( t, x ) ( x − y ) u ( t, y ) dxdy. Refined monotonicity formula.
The statement is implicitely contained in Tao’s proof.
Lemma 1 (Refined monotonicity formula) . Let u ( t, x ) be a global-in-time Schwartz solution tothe defocusing gKdV for some p ≥ √ . Then ∂ t h x i M − ∂ t h x i E & p EM (cid:18)Z R | u ( x ) | p +1 dx (cid:19) . Proof.
Tao’s proof is based on introducing strictly positive quantities a, b, q, r, s by solving a M = Z R u xx dx b M = Z R | u | p dx aqM = Z R u x dxbrM = Z R | u | p +1 dx absM = p Z R | u | p − u x dx, where partial integration implies0 < q, r, s < − q − r − s + 2 qrs ≥ . Then the proof can be finished algebraically by showing that for p ≥ √ EM ( ∂ t h x i M − ∂ t h x i E ) = 32 (1 − q ) a + (cid:18) s − p + 3 p + 1 qr (cid:19) ab + 12 (cid:18) − p ( p + 1) r (cid:19) b > . However, he actually proves the stronger statement32 (1 − q ) a + (cid:18) s − p + 3 p + 1 qr (cid:19) ab + 12 (cid:0) − r (cid:1) b > . Hence EM ( ∂ t h x i M − ∂ t h x i E ) ≥ (cid:18) − p ( p + 1) (cid:19) b r & p M (cid:18)Z R | u ( x ) | p +1 dx (cid:19) . (cid:3) STEFAN STEINERBERGER
A differential inequality.
Repeated partial integration yields ∂ t Z R Z R u ( x ) u ( y ) ( x − y ) dxdy = 4 Z R Z R u ( y ) (cid:18) pp + 1 | u ( x ) | p +1 + 32 u x ( x ) (cid:19) ( y − x ) dxdy. By introducing the normalized centers of mass and energy, h x i M := 1 M Z R xu dx and h x i E := 1 E Z R x (cid:18) u x + 1 p + 1 | u | p +1 (cid:19) dx, we can rewrite the first derivative as( ♦ ) ∂ t Z R Z R u ( x ) u ( y ) ( x − y ) dxdy = 12 EM ( h x i M − h x i E )+ 4 p − p + 1 (cid:18) h x i M M Z R | u | p +1 dx − M Z R | u | p +1 xdx (cid:19) . In the special case p = 3, the monotonicity formula immediately implies convexity of I ( u ( t )).The following simple argument will be used also in later proofs. If I ( u ( t )) is small, then thereis a small interval containing a lot of the L − mass: for a fixed time t , let J be the unique intervalsuch that Z x< inf J u dx = Z x> sup J u dx = 14 Z R u dx, then M | J | ≤ Z x ∈ R \ J Z x ∈ R \ J u ( t, x ) u ( t, y ) ( x − y ) dxdy ≤ I ( u ( t )) , which implies | J | ≤ p I ( u ( t )) /M. Hence, with H¨older, M Z J u dx ≤ (cid:18)Z J | u | p +1 dx (cid:19) p +1 | J | p − p +1 and thus, as a consequence, (cid:18)Z R | u | p +1 dx (cid:19) ≥ (cid:18)Z J | u | p +1 dx (cid:19) & M p I ( u ( t )) / − p/ . The fundamental theorem of calculus and refined monotonicity (i.e. Lemma 1) give( h x i M − h x i E ) (cid:12)(cid:12) t − ( h x i M − h x i E ) (cid:12)(cid:12) t =0 & Z t EM (cid:18)Z R | u ( z, x ) | p +1 dx (cid:19) dz & M p − E Z t I ( u ( z )) − p dz. The remaining term on the right-hand side of ( ♦ ) can be easily controlled via (cid:12)(cid:12)(cid:12)(cid:12) h x i M M Z R | u | p +1 dx − M Z R | u | p +1 xdx (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z R Z R u ( x ) | u ( y ) | p +1 | x − y | dxdy, which, using H¨older and k u k L ∞ . (cid:18)Z R u dx (cid:19) (cid:18)Z R u x dx (cid:19) . M / E / can be bounded by Z R Z R u ( x ) | u ( y ) | p +1 | x − y | dxdy . M ( M E ) p − I ( u ( t )) / . Altogether, this yields that the real function f ( t ) := I ( u ( t )) satisfies the differential inequality f ′ ( t ) ≥ α Z t f ( z ) − p dz − β p f ( t ) − γ, ISPERSION DYNAMICS FOR THE DEFOCUSING GENERALIZED KORTEWEG-DE VRIES EQUATION 7 for positive constants α ∼ M p − , β ∼ M ( EM ) p − and a constant γ which encodes the initialdifference between the center of mass and the center of energy γ = EM ( h x i M − h x i E ) (cid:12)(cid:12) t =0 . Conclusion.
This section finishes the proof of Theorem 1 by showing that the derived dif-ferential inequality alone already implies the sublevel estimate.
Lemma 2.
Let f ∈ C ( R , R + ) , α, β > and γ ∈ R arbitrary. Assume that f ( x ) > δ > and f ′ ( x ) ≥ α Z x f ( y ) − p dy − β p f ( x ) − γ, then µ ( { x > f ( x ) ≤ z } ) . α,β,γ,δ z p . Proof.
Let us quickly describe the argument: the lower bound is comprised of one trivial compo-nent − β p f ( x ) − γ , which merely depends on the actual value of f ( x ) and one term with ’memory’:whether the integral is large compared to the trivial component depends on whether or not thefunction has been small in the past. In particular, if the function has been small in the past for along time, the integral will dominate the trivial component and force the function to grow. Fix a z > δ , let c to be a large positive constant, consider I = { x > f ( x ) ≤ z } and take K sufficientlylarge such that | [0 , K ] ∩ I | = cz p . It remains to show that taking c > x = K (trivially, K can be chosen such that K ∈ I ) f ′ ( K ) ≥ α Z K f ( y ) − p dy − β p f ( K ) − γ ≥ α Z I ∩ [0 ,K ] f ( y ) − p dy − β √ z − γ ≥ αc √ z − β √ z − γ Since f ( x ) > δ >
0, the statement we are trying to prove is trivially true for z ≤ δ and we mayassume z ≥ δ . Then, for any c suffiently large depending on α, β, γ , this implies that at x = K the derivative is of order f ′ ( x ) & √ z . By the same reasoning, the same holds true for all points in( K, ∞ ) ∩ I . This growth implies that ( R \ I ) ∩ ( K, K + C √ z ) = ∅ , where the constant C dependson α, β, γ but not z . We show now that sup I ≤ K + C √ z . Suppose, this were not the case. Since( R \ I ) ∩ ( K, K + C √ z ) = ∅ , there would then be a smallest point y ∗ with f ( y ∗ ) = z , where theprevious inequality implies f ′ ( y ∗ ) > f ( x ) > δ > | I | ≤ cz p + C √ z . δ ( c + C ) cz p , where c could be chosen depending only on α, β, γ and C was finite. (cid:3) Remark.
Note that in applying this result to our case, the constants behave as α ∼ M p − , β ∼ M ( EM ) p − and γ = EM ( h x i M − h x i E ) (cid:12)(cid:12) t =0 . For the constant δ >
0, we have δ & M E − (which is proven further below), hence δ = δ ( α, β ),which is why it is not mentioned in the formulation of the Theorem 1.4. Proof of Corollary 2
This section first points out in which way the previous argument can be strenghtened to yield(very minor) additional information in a special case. Its remainder is then devoted to the con-nections between the functional, energy and mass.
STEFAN STEINERBERGER
Proof of Corollary 2.
The argument is very easy and uses nothing but the differentialinequality.
Lemma 3.
Let f ∈ C ( R , R + ) and α, β > , γ ≤ . If f ′ ( x ) ≥ α Z x f ( y ) − p dy − β p f ( x ) − γ and f (0) < (cid:18) αβ (cid:19) p − , then inf x> f ( x ) ≥ f (0)4 . Proof.
It follows from the structure of the inequality that we can restrict ourselves to functionsbeing monotonically increasing until they reach a global minimum. The function g ( x ) = 14 (2 p f (0) − βx ) satisfies g (0) = f (0) and g ′ ( x ) = − β p g ( x ). At x = p f (0) /β , we have from monotonicity andthe initial bound on f (0) that f ′ ( x ) ≥ α Z x f ( y ) − p dy − β p f ( x ) ≥ αβ p f (0) f (0) − p − β p f (0) > x = p f (0) /β . However, g p f (0) β ! = 14 (2 p f (0) − β p f (0) β ) = f (0)4 . (cid:3) Reducing Proposition 1 to Proposition 2.
The product structure of the inequalitiesimplies that it is sufficient to prove the two endpoints α = 3 and α = 4 p/ ( p − α = 3 follows quickly from Sobolev embedding. All further considerations are for general functions u ∈ H ( R ), where we use M and E to denote their mass and energy, respectively. There are notime-dependent elements in the arguments nor does the gKdV equation play any role. Lemma 4.
For any u ∈ H ( R ) I ( u ) & M E . Proof.
By Sobolev embedding k u k L ∞ ≤ (cid:18)Z R u dx (cid:19) (cid:18)Z R u x dx (cid:19) . M / E / . At the same time, reusing an argument employed earlier, there is an interval J of length | J | . p I ( u ) /M such that J contains half of the L − mass of u . Therefore k u k L ∞ ≥ k u k L ∞ ( J ) ≥ | J | Z J u ( x ) dx & M | J | & M p I ( u ) . Combining these two inequalities gives the result. (cid:3)
The proof of the second endpoint uses symmetric decreasing rearrangement to gain an additionalsymmetry, which then yields an algebraic simplification of the functional. The following statementwill come as no surprise at all, it can certainly be founded in the literature in more general form.
Lemma 5.
The functional I is decreasing under symmetrically decreasing rearrangement. ISPERSION DYNAMICS FOR THE DEFOCUSING GENERALIZED KORTEWEG-DE VRIES EQUATION 9
Proof.
We use the familiar layer-cake decomposition Z R Z R u ( t, x ) u ( t, y ) ( x − y ) dxdy = Z ∞ Z ∞ rs Z { u ( x ) = r }×{ u ( y ) = s } ( x − y ) d H drds, where H is the 2 − dimensional Hausdorff measure. The statement would then follow if it werethe case that for fixed positive a, b > A, B ⊂ R of corresponding sizeinf | A | = a, | B | = b Z A Z B ( x − y ) dxdy is assumed precisely when A, B are intervals with the same midpoint (potentially ignoring Lebesguenull sets in the process). Let
A, B be hypothetical counterexamples, then there exist constants c , c such that a neighbourhood of c is not contained in A and both A := A ∩ { x : x > c } and A := A ∩ { x : x < c } are nonempty and likewise for c and B . Let us then replace A and B by A + ε and B + ε for sufficiently small ε such that no overlap occurs. Then the integrationbetween A and B as well as between A and B remains unchanged while it decreases between A and B as well as A and B . This shows that the infimum can only be assumed a pair ofintervals (up to Lebesgue null sets) and an explicit calculation yields the midpoint property. (cid:3) For symmetric functions, the functional simplifies to Z R Z R u ( x ) ( x − y ) u ( y ) dxdy = 2 (cid:18)Z R u ( x ) x dx (cid:19) (cid:18)Z R u ( x ) dx (cid:19) . Assuming a lower bound of the type I & M α E β , we may use the fact that the gKdV scaling actsnicely on M and E to derive the necessary condition4 − p − α (cid:18) − p − (cid:19) + β (cid:18) − pp − (cid:19) . A standard scaling u ( · ) → au ( b · ) with a, b > α + ( p + 1) β ≤ ≤ α + β ≤ α, β ) = (3 , − k u x k L ). The other endpoint issimply Proposition 2.4.3. Proof of Proposition 2.
Here we give a proof of Proposition 2. The argument first shows ∀ p > ∃ c p > ∀ u ∈ L ( R ) (cid:18)Z R u x dx (cid:19) (cid:18)Z R | u | p +1 dx (cid:19) p − ≥ c p (cid:18)Z R u dx (cid:19) p +1 p − . It is based on using invariance under scaling and dilations and a characterization of compactnessin L . It implies the existence of a minimizer but no numerical bounds on c p . The second step,based on an observation of Soonsik Kwon, is that the Euler-Lagrange functional has a closed formsolution. Proof.
It suffices to consider functions u invariant under symmetric decreasing rearrangement. Wepick a minimizing sequence u n ∈ H ( R ) of (cid:0)R R u x dx (cid:1) p − R R u p +1 dx (cid:0)R R u dx (cid:1) p +1 p − and use invariance under scaling and dilations to prescribe u (0) = 1 and R u x = 1. Trivially,the sequence is then bounded in L by Z R u dx ≤ Z | x |≥ u x dx = 3 . We use an observation that is usually ascribed to either Feichtinger [5] or Pego [13]. A boundedset K ⊂ L is compact in L if there exists a function C : R + → R + giving uniform control ondecay ∀ f ∈ K Z | x | >C ( ε ) | f ( x ) | dx + Z | ξ | >C ( ε ) | ˆ f ( ξ ) | dξ < ε. The first part is easy and follows from the normalization. The symmetry of our functions impliesthat their Fourier transform is real-valued and thus (cid:12)(cid:12)(cid:12) ˆ f ( ξ ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z ∞−∞ f ( x ) cos ξxdx (cid:12)(cid:12)(cid:12)(cid:12) . | ξ | , due to the fact that the monotonicity gives rise to an alternating series. Hence the minimizingsequence is compact and there exists a minimizer. We fix Z R u x dx = 1 = Z R u and want to minimize Z R | u | p +1 dx under these constraints. The Lagrange multiplier theorem implies that u p = λ x u + λ u for some constants λ , λ from which the statement follows. (cid:3) References [1] J. Colliander, M. Grillakis and N. Tzirakis, Improved interaction Morawetz inequalities for the cubic nonlinearSchrdinger equation on R . Int. Math. Res. Not. IMRN 2007, no. 23, Art. ID rnm090, 30 pp.[2] A. de Bouard and Y. Martel, Non existence of L -compact solutions of the Kadomtsev-Petviashvili II equation,Math. Ann. (2004), 525–544.[3] B. Dodson, Global well-posedness for the defocusing, quintic nonlinear Schrdinger equation in one dimensionfor low regularity data. Int. Math. Res. Not. IMRN 2012, no. 4, 870–893.[4] B. Dodson, Global well-posedness and scattering for the defocusing, mass - critical generalized KdV equation,arXiv:1304.8025[5] H. Feichtinger, Compactness in translation invariant Banach spaces of distributions and compact multipliers.J. Math. Anal. Appl. 102 (1984), no. 2, 289–327.[6] J. Ginibre and G. Velo, Quadratic Morawetz inequalities and asymptotic completeness in the energy space fornonlinear Schrdinger and Hartree equations. Quart. Appl. Math. 68 (2010), no. 1, 113–134.[7] R. Killip, S. Kwon, S. Shao and M. Visan, On the mass-critical generalized KdV equation. Discrete Contin.Dyn. Syst. (2012), no. 1, 191–221.[8] S. Kwon and S. Shao, Nonexistence of soliton-like solutions for defocusing generalized KdV equations,arXiv:1205.0849[9] C. Laurent and Y. Martel, Smoothness and exponential decay for L − compact solutions of the generalizedKorteweg-de Vries equation, Comm. Partial Diff. Eq. (2004), 157–171.[10] H. Lindblad and T. Tao, Asymptotic decay for a one-dimensional nonlinear wave equation. Anal. PDE 5 (2012),no. 2, 411–422.[11] Y. Martel and F. Merle, A Liouville theorem for the critical generalized Korteweg-de Vries equation. J. Math.Pures Appl. (9) 79 (2000), no. 4, 339–425.[12] C. Morawetz, The decay of solutions of the exterior initial-boundary value problem for the wave equation.Comm. Pure Appl. Math., Vol. 14, (1961), p. 561–568.[13] R. Pego, Compactness in L and the Fourier transform, Proc. Amer. Math. Soc. , 2 (1985), 252–254.[14] F. Planchon and L. Vega, Bilinear virial identities and applications. Ann. Sci. Ec. Norm. Super. (4) 42 (2009),no. 2, 261–290.[15] T. Tao, Two remarks on the generalised Korteweg-de Vries equation. Discrete Contin. Dyn. Syst. (2007),no. 1, 1–14.[16] T. Tao, Nonlinear Dispersive Equations – Local and Global Analysis, CBMS Regional Conference Series inMathematics , 2006., 2006.