Bilinear oscillatory integrals and boundedness for new bilinear multipliers

Germain, Pierre

doi:10.1016/j.aim.2010.03.032

Cited by 21 publications

(30 citation statements)

References 32 publications

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“…The authors would like to acknowledge the recent work of Bernicot and Germain [2] on bilinear oscillatory integrals which seems to mildly overlap with our present work.…”

Section: Acknowledgmentsmentioning

confidence: 77%

“…To obtain the required estimate for the L p quasi-norm of the operator ∞ j=1 T σ j ( f 1 , f 2 ), due to the rapid convergence of the sums in 1,2,3,4 , it suffices to obtain the same estimate the L p quasi-norm of the expression ∞ j=1 2 jmc j,N 1 , 2 ,k 1 ,…”

Section: Proof Of Proposition 24mentioning

confidence: 99%

“…The results in this article are of local nature and for this reason the symbols we consider indeed have compact support in the variables x, y 1 , y 2 . Looking at the bilinear pseudodifferential operator written in the form (2), it is only a matter of introducing an appropriate oscillatory factor to create a bilinear Fourier integral operator. To set the framework for this theory, we first recall some definitions.…”

mentioning

confidence: 99%

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Bilinear Fourier integral operators

Grafakos

Peloso

2010

J. Pseudo-Differ. Oper. Appl.

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We study the boundedness of bilinear Fourier integral operators on products of Lebesgue spaces. These operators are obtained from the class of bilinear pseudodifferential operators of Coifman and Meyer via the introduction of an oscillatory factor containing a real-valued phase of five variables (x, y 1 , y 2 , ξ 1 , ξ 2 ) which is jointly homogeneous in the phase variables (ξ 1 , ξ 2 ). For symbols of order zero supported away from the axes and the antidiagonal, we show that boundedness holds in the local-L 2 case. Stronger conclusions are obtained for more restricted classes of symbols and phases.

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“…The authors would like to acknowledge the recent work of Bernicot and Germain [2] on bilinear oscillatory integrals which seems to mildly overlap with our present work.…”

Section: Acknowledgmentsmentioning

confidence: 77%

Section: Proof Of Proposition 24mentioning

confidence: 99%

mentioning

confidence: 99%

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Bilinear Fourier integral operators

Grafakos

Peloso

2010

J. Pseudo-Differ. Oper. Appl.

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“…The fact that for every fixed T > 0 and C * > 0 the sequence exp(j 2 C * T /2)/(Cj 2 ) diverges as j → ∞, shows that there exits some sufficiently large j * such that (4.36) holds, thereby concluding the proof of the theorem. is locally Lipschitz (X, Y ) well-posed, if there exist continuous functions T, K : [0, ∞) 2 → (0, ∞), the time of existence and the Lipschitz constant, so that for every pair of initial data Θ (1) (0, ·), Θ (2) (0, ·) ∈ Y there exist unique solutions Θ (1) , Θ (2) ∈ L ∞ (0, T ; X) of the initial value problem associated to (4.37)-(4.38), that additionally satisfy…”

Section: 1mentioning

confidence: 99%

“…Remark 4.5. Clearly the time of existence T and the Lipschitz constant K also depend on S L ∞ (0,∞;Y ) , and on κ, but we have omitted this dependence in Definition 4.4 since it is the same for both solutions Θ (1) and Θ (2) . For the purpose of our ill-posedness result, we shall let Θ (2) (t, x) be the steady state Θ(x 3 ) introduced earlier in (4.4).…”

Section: 1mentioning

confidence: 99%

On the supercritically diffusive magnetogeostrophic equations

2012

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ABSTRACT. We address the well-posedness theory for the magento-geostrophic equation, namely an active scalar equation in which the divergence-free drift velocity is one derivative more singular than the active scalar. In the presence of supercritical fractional diffusion given by (−∆) γ with 0 < γ < 1, we discover that for γ > 1/2 the equations are locally well-posed, while for γ < 1/2 they are ill-posed, in the sense that there is no Lipschitz solution map. The main reason for the striking loss of regularity when γ goes below 1/2 is that the constitutive law used to obtain the velocity from the active scalar is given by an unbounded Fourier multiplier which is both even and anisotropic. Lastly, we note that the anisotropy of the constitutive law for the velocity may be explored in order to obtain an improvement in the regularity of the solutions when the initial data and the force have thin Fourier support, i.e. they are supported on a plane in frequency space. In particular, for such well-prepared data one may prove the local existence and uniqueness of solutions for all values of γ ∈ (0, 1).

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Nonresonant bilinear forms for partially dissipative hyperbolic systems violating the Shizuta–Kawashima condition

Bianchini¹,

Natalini²

2022

J. Evol. Equ.

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In the context of hyperbolic systems of balance laws, the Shizuta–Kawashima coupling condition guarantees that all the variables of the system are dissipative even though the system is not totally dissipative. Hence it plays a crucial role in terms of sufficient conditions for the global in time existence of classical solutions. However, it is easy to find physically based models that do not satisfy this condition, especially in several space dimensions. In this paper, we consider two simple examples of partially dissipative hyperbolic systems violating the Shizuta–Kawashima condition (SK) in 3D, such that some eigendirections do not exhibit dissipation at all. We prove that if the source term is nonresonant (in a suitable sense) in the direction where dissipation does not play any role, then the formation of singularities is prevented, despite the lack of dissipation, and the smooth solutions exist globally in time. The main idea of the proof is to couple Green function estimates for weakly dissipative hyperbolic systems with the space–time resonance analysis for dispersive equations introduced by Germain, Masmoudi and Shatah. More precisely, the partially dissipative hyperbolic systems violating (SK) are endowed, in the nondissipative directions, with a special structure of the nonlinearity, the so-called nonresonant bilinear form for the wave equation (see Pusateri and Shatah, CPAM 2013).

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Bilinear oscillatory integrals and boundedness for new bilinear multipliers

Cited by 21 publications

References 32 publications

Bilinear Fourier integral operators

Bilinear Fourier integral operators

On the supercritically diffusive magnetogeostrophic equations

Nonresonant bilinear forms for partially dissipative hyperbolic systems violating the Shizuta–Kawashima condition

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