Long time existence results for bore-type initial data for BBM-Boussinesq systems

Burtea, Cosmin

doi:10.1016/j.jde.2016.07.014

Cited by 17 publications

(16 citation statements)

References 13 publications

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“…With these lemmas in hand, attention is returned to the functions (N, U, V ) which begin at the origin in function space at t = 0, exist in C b (0, t 0 ; H k (R 2 ) 3 for at least some positive time interval [0, t 0 ], and solve (42) on that interval. Local existence for (42) follows readily just as in [9], [10], [15] or [16] because of the regularity of the coefficients and the forcing functions established in Lemmas 4.2 and 4.3. To produce a solution on the time interval [0, 1 ε ] a priori bounds are now derived.…”

Section: 2mentioning

confidence: 93%

“…Subject to a size restriction R 1 , say, the initial-value problem for the twodimensional system (7)-(8) posed with H k (R 2 ) 3 initial data (η 0 , u 0 , v 0 ) is known to have a unique solution (η, u, v) on Ω ε , which is O(R). This follows from the early work [17], our previous paper [9] or the recent paper [15] of Burtea. This finite-energy solution lies in…”

mentioning

confidence: 92%

“…where c is an absolute constant and λ is as in (15). Thus W is bounded independently of ε > 0 on the time…”

mentioning

confidence: 99%

“…There is a substantial theory pertaining to the initial-value problem for these systems on the Boussinesq time scale O( 1 ε ) when the initial data is assumed to decay to zero (see [2], [20], [21], [26] for the long-crested situation where variations in the y-directions are ignored and [16] when the waves are three-dimensional). Work on bore-type problems for two-dimensional Boussinesq systems appeared in [9] and in the recent independent, but related work of Burtea [15].…”

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confidence: 99%

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Propagation of long-crested water waves. Ⅱ. Bore propagation

Bona¹,

Colin²,

Guillopé³

2019

Discrete &Amp; Continuous Dynamical Systems - A

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This essay is concerned with long-crested waves such as those arising in bore propagation. Such motions obtain on rivers when a surge of water invades an otherwise constantly flowing stretch and in the run-up of waves in the near-shore zone of large bodies of water. The dominating feature of the motion is that, in a standard xyz−coordinate system in which z increases in the direction opposite to which gravity acts and x increases in the principal direction of propagation, the depth of the fluid approaches a constant value h 0 > 0 as x → +∞ and another value h 1 > h 0 as x → −∞. In an earlier work, the authors developed theory for an idealized model for such waves based on a Boussinesq system of equations. The local well-posedness theory developed in that article applies to the sort of initial data arising in modeling bore propagation. However, well-posedness on the longer, Boussinesq time scale was not dealt with in the case of bore propagation, though such results were established for motions where h 1 = h 0 .We argue that without a well-posedness theory at least on the Boussinesq time scale, such models for bore-propagation may not be of any practical use. The issue of well-posedness is complicated by the fact that the total energy of the idealized initial data is infinite.The theory makes its way via the derivation of suitable approximations with which to compare the full solution. An interesting feature of the theory is the determination of dynamical boundary behavior that is not prescribed, but which the solution necessarily satisfies.2010 Mathematics Subject Classification. Primary: 58F15, 58F17; Secondary: 53C35.

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Section: 2mentioning

confidence: 93%

mentioning

confidence: 92%

“…where c is an absolute constant and λ is as in (15). Thus W is bounded independently of ε > 0 on the time…”

mentioning

confidence: 99%

mentioning

confidence: 99%

See 2 more Smart Citations

Propagation of long-crested water waves. Ⅱ. Bore propagation

Bona¹,

Colin²,

Guillopé³

2019

Discrete &Amp; Continuous Dynamical Systems - A

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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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Boussinesq, Schrödinger and Euler-Korteweg

Saut,

2024

Springer Proceedings in Mathematics &Amp; Statistics

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Long time existence results for bore-type initial data for BBM-Boussinesq systems

Cited by 17 publications

References 13 publications

Propagation of long-crested water waves. Ⅱ. Bore propagation

Propagation of long-crested water waves. Ⅱ. Bore propagation

Long Time Existence for a Strongly Dispersive Boussinesq System

Boussinesq, Schrödinger and Euler-Korteweg

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