Finite time singularities and global well-posedness for fractal Burgers equations

Dong, Hongjie; Du, Di; Dong, Li

doi:10.1512/iumj.2009.58.3505

Cited by 66 publications

(86 citation statements)

References 11 publications

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“…These numerics suggest also that in the critical case α = 1 the solution exists globally. This is in agreement with the results for the Burgers equation with fractional dissipation by Kiselev, Nazarov & Shterenberg, [25] and Dong, Du & Li [14]. Let us remark that, when the term µΛ β ∂ t u is added to the equation, even for α = β = 0.25 there is no evidence of blow-up.…”

Section: Introductionsupporting

confidence: 91%

“…However, we remark that it can also be obtained by means of pointwise arguments (see Castro & Córdoba [7] for an application of these pointwise arguments to prove blow up). These virial-type arguments have been used for several transport equations even in the case of nonlocal velocities (see Córdoba,Córdoba & Fontelos [12], Dong, Du & Li [14], Li & Rodrigo [28] and Li, Rodrigo & Zhang [29]). In this case, the transport term is highly nonlinear and this method fails in the case of viscosity ν > 0, 0 < α ≪ 1.…”

Section: Definition 2 Let γ and M Be Given Positive Constants Defimentioning

confidence: 99%

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On the generalized Buckley-Leverett equation

Burczak

Granero-Belinchón

Luli

2016

Journal of Mathematical Physics

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Abstract. In this paper we study the generalized Buckley-Leverett equation with nonlocal regularizing terms. One of these regularizing terms is diffusive, while the other one is conservative. We prove that if the regularizing terms have order higher than one (combined), there exists a global strong solution for arbitrarily large initial data. In the case where the regularizing terms have combined order one, we prove the global existence of solution under some size restriction for the initial data. Moreover, in the case where the conservative regularizing term vanishes, regardless of the order of the diffusion and under certain hypothesis on the initial data, we also prove the global existence of strong solution and we obtain some new entropy balances. Finally, we provide numerics suggesting that, if the order of the diffusion is 0 < α < 1, a finite time blow up of the solution is possible.

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Section: Introductionsupporting

confidence: 91%

Section: Definition 2 Let γ and M Be Given Positive Constants Defimentioning

confidence: 99%

On the generalized Buckley-Leverett equation

Burczak

Granero-Belinchón

Luli

2016

Journal of Mathematical Physics

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“…Unfortunately, the similar approach either the rescaling argument in [25] or the weighted norm method in [18] seems to be unsuccessful in the current problem to (1.4) for the case γ ∈ (0, 1 2 ). …”

Section: Remark 12 Thanks Tomentioning

confidence: 93%

“…On the other hand, Dong et al [18] recently considered well-posedness questions for all the values of γ in the periodic and non-periodic settings with the emphasis on the finite-time singularities in the supercritical case. In a recent paper, Chan and Czubak [8] showed existence of smooth solutions of the critical d-dimensional Burgers equation with any initial datum in L 2 (R d ) by using the parabolic De Giorgi's method of Caffarelli and Vasseur.…”

Section: Introductionmentioning

confidence: 99%

“…In a recent paper, Chan and Czubak [8] showed existence of smooth solutions of the critical d-dimensional Burgers equation with any initial datum in L 2 (R d ) by using the parabolic De Giorgi's method of Caffarelli and Vasseur. For further background and motivation for the fractional Burgers equation we refer our readers to [8,18,25,27], and references therein.…”

Section: Introductionmentioning

confidence: 99%

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Global well-posedness and blow-up of solutions for the Camassa–Holm equations with fractional dissipation

Gui

Liu

2015

Math. Z.

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Consideration in this paper is the effect of varying fractional dissipation with the dissipative operator power γ ≥ 0 on the well-posedness of the Camassa-Holm equations with fractional dissipation. It is shown that the zero-filter limit (α → 0) of the Camassa-Holm equation with fractional dissipation is the fractal Burgers equation. It is known that in the supercritical case γ ∈ [0, 1), the fractal Burgers equation blows up in finite time in H s (R) with s > 3 2 − γ . It is established here that the dissipative Camassa-Holm equation is globally well-posed in the critical Sobolev space H 3 2 −γ (R) with the fractional parameter γ ∈ [ 1 2 , 1). Moreover, it is also demonstrated that the solution of the dissipative Camassa-Holm equation blows up in finite time in the particular supercritical case when γ = 0.

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Asymptotic behavior of solutions to anisotropic conservation laws in two‐dimensional space

2019

Math Methods in App Sciences

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Finite time singularities and global well-posedness for fractal Burgers equations

Cited by 66 publications

References 11 publications

On the generalized Buckley-Leverett equation

On the generalized Buckley-Leverett equation

Global well-posedness and blow-up of solutions for the Camassa–Holm equations with fractional dissipation

Asymptotic behavior of solutions to anisotropic conservation laws in two‐dimensional space

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