Global Weak Solutions for a Shallow Water Equation

Constantin, Adrian; Molinet, Luc

doi:10.1007/s002200050801

Cited by 468 publications

(232 citation statements)

References 18 publications

Supporting

Mentioning

230

Contrasting

Order By: Relevance

“…Moreover, Camassa-Holm equation not only has global strong solutions, but also admits finite time blow-up solutions [9,12,13,59,17], and the blow-up occurs in the form of breaking waves, namely, the solution remains bounded but its slope becomes unbounded in finite time. On the other hand, it also has global weak solutions in H 1 (see [3,14,20,79]). The advantage of the Camassa-Holm equation in comparison with the KdV equation lies in the fact that the Camassa-Holm equation has peaked solitons and models the peculiar wave breaking phenomena (cf.…”

Section: Shouming Zhou Chunlai Mu and Liangchen Wangmentioning

confidence: 96%

Well-posedness, blow-up phenomena and global existence for the generalized $b$-equation with higher-order nonlinearities and weak dissipation

Zhou¹,

Mu²,

Wang³

2013

DCDS-A

View full text Add to dashboard Cite

This paper deals with the Cauchy problem for a weakly dissipative shallow water equation with high-order nonlinearities yt + u m+1 yx + bu m uxy + λy = 0, where λ, b are constants and m ∈ N, the notation y :which includes the famous b-equation and Novikov equations as special cases. The local well-posedness of solutions for the Cauchy problem in Besov space B s p,r with 1 ≤ p, r ≤ +∞ and s > max{1 + 1 p , 3 2 } is obtained. Under some assumptions, the existence and uniqueness of the global solutions to the equation are shown, and conditions that lead to the development of singularities in finite time for the solutions are acquired, moreover, the propagation behaviors of compactly supported solutions are also established. Finally, the weak solution and analytic solution for the equation are considered. 2010 Mathematics Subject Classification. Primary: 35G25, 35L05; Secondary: 35Q50.

show abstract

Section: Shouming Zhou Chunlai Mu and Liangchen Wangmentioning

confidence: 96%

Well-posedness, blow-up phenomena and global existence for the generalized $b$-equation with higher-order nonlinearities and weak dissipation

Zhou¹,

Mu²,

Wang³

2013

DCDS-A

View full text Add to dashboard Cite

show abstract

“…More interestingly, it has global strong solutions [19,11,12] and also finite time blow-up solutions [19,20,11,12,21,13,16,17]. On the other hand, it has global weak solutions in H 1 (R) [22][23][24][25][26]. Finite propagation speed and persistence properties of solutions to the Camassa-Holm equation have been studied in [27,28].…”

Section: Introductionmentioning

confidence: 99%

Well-posedness, blow-up phenomena and persistence properties for a two-component water wave system

Guan

Yin

2015

Nonlinear Analysis: Real World Applications

View full text Add to dashboard Cite

“…To describe the peakons, one has to study weak solution to (1.1). In [9], it is proved that if u 0 ∈ H 1 (S) is such that (1−∂ 2 x )u 0 is a positive Radon measure with bounded total variation (e.g., u 0 = ϕ), then (1.1) has a unique solution u ∈ C([0, ∞); H 1 (S)) ∩ C 1 ([0, ∞); L 2 (S)), and H 0 , H 1 , and H 2 are conserved functionals. We would like to point out that if (1 − ∂ 2…”

Section: Commentsmentioning

confidence: 99%

Untitled

Lenells

2004

Internat. Math. Res. Notices

123

View full text Add to dashboard Cite

The nonlinear partial differential equationarises as a model for the unidirectional propagation of shallow water waves over a flat bottom, with u(x, t) representing the water's free surface in nondimensional variables. In this paper, we are concerned with the periodic solutions of (1.1), that is, u : S × [0, T ) → R where S denotes the unit circle and T > 0 is the maximal existence time of the solution. Equation (1.1) was first obtained [12] as an abstract bi-Hamiltonian equation with infinitely many conservation laws and was subsequently derived from physical principles [2]. For a discussion of the physical relevance of (1.1) in the context of water waves, we refer to [14, 15]. Equation (1.1) also arises as a model for nonlinear waves in cylindrical axially symmetric hyperelastic rods, with u(x, t) representing the radial stretch relative to a prestressed state [11]. Moreover, (1.1) is a re-expression of the geodesic flow in the group of compressible diffeomorphisms of the circle [18], just like the Euler equation is an expression of the geodesic flow in the group of incompressible diffeomorphisms of the torus [1]. This geometric interpretation leads to a proof that equation (1.1) satisfies the least action principle [7]: a state of the system is transformed to another nearby state through a uniquely determined flow that minimizes the energy. We also point out that for a large class of initial data, equation (1.1) is an infinite-dimensional completely integrable Hamiltonian system: by means of an isospectral problem, one can convert the equation into an infinite sequence of linear ordinary differential equations which can be

show abstract

Global Weak Solutions for a Shallow Water Equation

Cited by 468 publications

References 18 publications

Well-posedness, blow-up phenomena and global existence for the generalized $b$-equation with higher-order nonlinearities and weak dissipation

Well-posedness, blow-up phenomena and global existence for the generalized $b$-equation with higher-order nonlinearities and weak dissipation

Well-posedness, blow-up phenomena and persistence properties for a two-component water wave system

Untitled

Contact Info

Product

Resources

About