New long time existence results for a class of Boussinesq-type systems

Burtea, Cosmin

doi:10.1016/j.matpur.2016.02.008

Cited by 28 publications

(32 citation statements)

References 12 publications

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“…(13) b = d = 0, a < 0, c = 0. Actually, the same scheme of proof (by symmetrization) is used in [36] but because of the many different cases to be dealt with (the technical details cannot be treated in an unified way), we only provided a complete proof in [36] for cases (4) ("generic case"), (1) and (11), which are "strongly dispersive". The other cases can be dealt with by similar symmetrization techniques but the proofs for some of them need more explanations that we detail below.…”

Section: A Review Of Long Time Well-posed Boussinesq Systemsmentioning

confidence: 99%

“…During the completion of the present paper we were informed of the very interesting paper [11] where an alternative proof of long time existence for most of the Boussinesq systems is provided (excluding the "strongly dispersive" ones b = d = 0, thus the "difficult case" a = b = d = 0, c < 0). This proof also relaxes the noncavitation condition on the initial data η 0 .…”

Section: Introductionmentioning

confidence: 99%

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The Cauchy Problem on Large Time for Surface-Waves-Type Boussinesq Systems II

Saut¹,

Wang²,

Xu³

2017

SIAM J. Math. Anal.

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This paper is a continuation of a previous work by two of the Authors [36] on long time existence for Boussinesq systems modeling the propagation of long, weakly nonlinear water waves. We provide proofs on examples not considered in [36] in particular we prove a long time well-posedness result for a delicate "strongly dispersive" Boussinesq system.where τ ≥ 0 is the surface tension parameter (Bond number).Recall also [6] that (1.2) is linearly well-posed whenand when a = c, b ≥ 0, d ≥ 0. An important step to justify rigorously (1.2) as an asymptotic model for water waves is to establish the well-posedness of the Cauchy problem on time scales of order 1/ǫ, with uniform bounds in suitable Sobolev spaces, the error estimate being then (see [5,29]).This step has been established in [36] (see also [34]) for most of Boussinesq systems with and without surface tension. The idea in [36] is to find an appropriate symmetrization of the system and this is not a straightforward task since one cannot obviously use the classical symmetrizer of the underlying Saint-Venant (shallow water) hyperbolic system. This will be reviewed in the first section of this paper. A complete proof of cases that were not fully developed in [36] will be given in Section 4. 1 Introducing surface tension enlarges the range of physically admissible parameters (a,b,c,d) and so even a local theory 2 for a few linearly well -posed systems is still missing, for instance the cases b = d = 0, a < 0, c = 0 and b = d = 0, a = 0, c < 0). Both cases will be considered here but the later leads to serious difficulties and the long time existence for it is the main result of the present paper.Note also that the (linearly well-posed) "exceptional KdV-KdV" case b = d = 0, a = c > 0 which is studied in [30] leading to well-posedness on time scales of order 1/ √ ǫ in Sobolev spaces H s (R 2 ), s > 3/2 which are larger than the "hyperbolic" one H s (R 2 ), s > 2 is not covered neither in [36] nor in the present paper so that a long time existence is still open in this case 3 .An important mathematical issue concerning Boussinesq systems (1.2) is that despite they describe the same dynamics of water waves, their mathematical properties are rather different, due essentially to their different linear dispersion relations. Of course those dispersion relations all coincide in the long wave limit but there are quite different in the short wave limit. A convenient way to classify the system is according to the order of the Fourier multiplier operator given by the eigenvalues of the linearized operator (see [6]). The order can be −1, 0, 1, 2 or 3. The two last cases are referred to as the strongly dispersive ones.1 Due to the large number of cases to be considered, we chosed in [36] to give complete proofs for a limited number of cases.2 That is not taking care of the dependence of the lifespan of the solution with respect to ǫ 3 However, the case b = d = 0, a < 0, c < 0 that can only occur with a strong surface tension is covered by the theory in [36].

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Section: A Review Of Long Time Well-posed Boussinesq Systemsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

The Cauchy Problem on Large Time for Surface-Waves-Type Boussinesq Systems II

Saut¹,

Wang²,

Xu³

2017

SIAM J. Math. Anal.

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“…We do not consider here the system in the long-wave regime as it was done in [19], since our method of proof does not seem to provide, at least directly, good lower bounds for the existence time with respect to the small parameter ǫ measuring the size of the dispersive and nonlinear effects, which are of the same order in this regime. It remains nevertheless an interesting issue to prove that systems (1.1) and (1.2) are locally well posed over large time as it was done for some of the (a, b, c, d)-Boussinesq systems [27,3,28].…”

Section: Introductionmentioning

confidence: 99%

“…Finally, it is worth noticing that the non-cavitation condition on the 2 We refer to [19] for an heuristic argument of this fact. 3 The scaling H s × H s+ 1 2 is needed to cancel out the linear terms 4 Note that the technique may work for some other systems with a nonlocal dispersion. We refer for example to [31] for a nonlocal dispersive system in the context of internal wave.…”

Section: Introductionmentioning

confidence: 99%

On the local well-posedness for a full-dispersion Boussinesq system with surface tension

Kalisch

Pilod

2019

Proc. Amer. Math. Soc.

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In this note, we prove local-in-time well-posedness for a fully dispersive Boussinesq system arising in the context of free surface water waves in two and three spatial dimensions. Those systems can be seen as a weak nonlocal dispersive perturbation of the shallow-water system. Our method of proof relies on energy estimates and a compactness argument. However, due to the lack of symmetry of the nonlinear part, those traditional methods have to be supplemented with the use of a modified energy in order to close the a priori estimates.

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“…The existence of solutions of the Boussinesq systems on time scales of order O(1/ǫ) has been established in [10,11,21,23,26] for all the locally-well posed Boussinesq systems except the case b = d = 0, a = c > 0 which is in some sense special since the "generic" case b = d = 0, a, c > 0, a = c is linearly ill-posed. We also refer to [25] for the case of Full-Dispersion Boussinesq systems.…”

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confidence: 99%

Long Time Existence for a Strongly Dispersive Boussinesq System

Saut¹,

Xu²

2020

SIAM J. Math. Anal.

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We prove a long time existence result for the solutions of a two-dimensional Boussinesq system modeling the propagation of long, weakly nonlinear water waves. This system is exceptional in the sense that it is the only linearly well-posed system in the (abcd) family of Boussinesq systems whose eigenvalues of the linearized system have nontrivial zeroes. This new difficulty is solved by the use of "good unknowns " and of normal form techniques.

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New long time existence results for a class of Boussinesq-type systems

Cited by 28 publications

References 12 publications

The Cauchy Problem on Large Time for Surface-Waves-Type Boussinesq Systems II

The Cauchy Problem on Large Time for Surface-Waves-Type Boussinesq Systems II

On the local well-posedness for a full-dispersion Boussinesq system with surface tension

Long Time Existence for a Strongly Dispersive Boussinesq System

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