Blow up and regularity for fractal Burgers equation

Kiselev, Alexander; Nazarov, Fëdor; Shterenberg, Roman

doi:10.4310/dpde.2008.v5.n3.a2

Cited by 170 publications

(292 citation statements)

References 17 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: 80%

“…Finally, in the case µ > 0, α = 1 and 0 < β < 1, we obtain the global existence for initial data satisfying a smallness condition in H In particular, our results do not apply to the case where max{α, β} < 1, µ > 0, and there is no large data, global results for the critical case with µ = 0, ν > 0, α = 1. In the context of the latter, let us observe that on one hand, there are certain new methods available for nonlinear problems with nonlocal critical dissipation, like the method of moduli of continuity by Kiselev, Nazarov & Shterenberg [25], the fine-tuned DeGiorgi method by Caffarelli & Vasseur [6] or the method of the nonlinear maximum principles by Constantin & Vicol [9] (see also Constantin, Tarfulea & Vicol [10]). But on the other hand, our nonlinearity is more complex than the typical ones.…”

Section: 2mentioning

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: 80%

Section: 2mentioning

confidence: 99%

On the generalized Buckley-Leverett equation

Burczak

Granero-Belinchón

Luli

2016

Journal of Mathematical Physics

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“…Additional pointwise bounds, available in the one-dimensional case, imply that the Burgers equation with fractional dissipation, u t + u · ∇u = −(− ) α u, admits global solutions for α > 1/2; the critical case, α = 1/2, was the subject of extensive recent studies [13][14][15]. 3.…”

Section: Fractional Dissipationmentioning

confidence: 99%

Critical thresholds in flocking hydrodynamics with non-local alignment

Tadmor

Tan

2014

Phil. Trans. R. Soc. A.

128

158

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We study the large-time behaviour of Eulerian systems augmented with non-local alignment. Such systems arise as hydrodynamic descriptions of agent-based models for self-organized dynamics, e.g. Cucker & Smale (2007 IEEE Trans. Autom. Control 52 , 852–862. (doi: 10.1109/TAC.2007.895842 )) and Motsch & Tadmor (2011 J. Stat. Phys. 144 , 923–947. (doi: 10.1007/s10955-011-0285-9 )) models. We prove that, in analogy with the agent-based models, the presence of non-local alignment enforces strong solutions to self-organize into a macroscopic flock. This then raises the question of existence of such strong solutions. We address this question in one- and two-dimensional set-ups, proving global regularity for subcritical initial data. Indeed, we show that there exist critical thresholds in the phase space of the initial configuration which dictate the global regularity versus a finite-time blow-up. In particular, we explore the regularity of non-local alignment in the presence of vacuum.

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“…[7,11]). Nevertheless, from the proof of Theorem 1.1, one can see that the approach also works for the following fully nonlinear equation:…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Well-posedness of fully nonlinear and nonlocal critical parabolic equations

Zhang

2012

J. Evol. Equ.

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Abstract. In this paper we prove the existence of smooth solutions to fully nonlinear and nonlocal parabolic equations with critical index. The proof relies on the apriori Hölder estimate for advection fractional-diffusion equation established by Silvestre [11]. Introduction and main resultIn this paper we are interested in solving the following fully nonlinear and nonlocal parabolic equation:where F (t, x, u, w, q) where F denotes the Fourier's transform, S(R d ) is the Schwartz class of smooth real-valued rapidly decreasing functions.Recently, in the sense of viscosity solutions, fully nonlinear and nonlocal elliptic and parabolic equations have been extensively studied (cf. [4,10,3,9], etc.). In [4], Caffarelli and Silvestre studied the following type of nonlocal equation:where α ∈ (0, 2), i, j ranges in arbitrary sets, c i j ∈ R and b i j ∈ R d , the kernel a i j (y) satisfiesThis type of equation appears in the stochastic control problems. In [4], the extremal Pucci operators are used to characterize the ellipticity, and the ABP estimate, Harnack inequality and interior C 1,β -regularity were obtained. In [11], Silvestre studied the following nonlocal parabolic equation with critical index α = 1:and established C 1,β -regularity of viscosity solutions. In particular, the following first order Hamilton-Jacobi equation is covered by the above equation when H is Lipschitz continuous:In [9], Lara and Davila extended Silvestre's result to the more general case, and in particular, focused on the uniformity of regularity as α → 2.However, it is not known how to solve the fully nonlinear and nonlocal equation (1.1) in Sobolev spaces. Let us fix the main idea of the present paper for solving (1.1). Assume that F does not depend on u. Taking the gradient with respect to x for equation (1.1), we have We make the following observation:

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Blow up and regularity for fractal Burgers equation

Cited by 170 publications

References 17 publications

On the generalized Buckley-Leverett equation

On the generalized Buckley-Leverett equation

Critical thresholds in flocking hydrodynamics with non-local alignment

Well-posedness of fully nonlinear and nonlocal critical parabolic equations

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