On the generalized Buckley-Leverett equation

Burczak, Jan; Granero-Belinchón, Rafael; Luli, Garving K.

doi:10.1063/1.4945786

Cited by 5 publications

(10 citation statements)

References 33 publications

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“…Due to the definition of x t , we have v(x t ) − v(x t − η) ≤ 0, and, consequently, using (14), Λ β v(x t ) ≤ 0. Using this fact we get…”

Section: 1mentioning

confidence: 99%

“…Starting from our assumption, we have For the proof of the case p = 1, the interested reader can see [14] by the authors & Luli. The case p = ∞ was proved by Bae & Granero-Belinchón in [3].…”

Section: Proof Of Propositionmentioning

confidence: 99%

“…2 |.Recalling(14) together withc(α) = Γ(1 + α) cos((1 − α)π/2) π ,we haveΛ α h(x * ) = c(α) k P.V. T h(x * ) − h(x * − η) |η + k2π| 1+α dη ≥ c(α)P.V.…”

mentioning

confidence: 95%

“…Moreover, u 2+2s Ẇ α/(2+2s)−δ,1+s ≤ C(α, s, δ) u 1+s L 1+s T Λ α u(x)u s (x)dx.Proof. Let us defineI 1 = T u s (x)Λ α u(x)dx.Using(14) and changing variables, we computeI 1 = c α T k∈Z,k =0 T u s (x) u(x) − u(η) |x − η + 2kπ| 1+α dηdx + c α T P.V. T u s (x) u(x) − u(η) |x − η| 1+α dηdx = c α T k∈Z,k =0 T u s (η) u(η) − u(x) |η − x + 2kπ| 1+α dηdx + c α T P.V.…”

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confidence: 99%

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Global solutions for a supercritical drift–diffusion equation

Burczak

Granero-Belinchón

2016

Advances in Mathematics

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Abstract. We study the global existence of solutions to a one-dimensional drift-diffusion equation with logistic term, generalizing the classical parabolicelliptic Keller-Segel aggregation equation arising in mathematical biology. In particular, we prove that there exists a global weak solution, if the order of the fractional diffusion α ∈ (1 − c1, 2], where c1 > 0 is an explicit constant depending on the physical parameters present in the problem (chemosensitivity and strength of logistic damping). Furthermore, in the range 1 − c2 < α ≤ 2 with 0 < c2 < c1, the solution is globally smooth. Let us emphasize that when α < 1, the diffusion is in the supercritical regime.

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“…Due to the definition of x t , we have v(x t ) − v(x t − η) ≤ 0, and, consequently, using (14), Λ β v(x t ) ≤ 0. Using this fact we get…”

Section: 1mentioning

confidence: 99%

Section: Proof Of Propositionmentioning

confidence: 99%

“…2 |.Recalling(14) together withc(α) = Γ(1 + α) cos((1 − α)π/2) π ,we haveΛ α h(x * ) = c(α) k P.V. T h(x * ) − h(x * − η) |η + k2π| 1+α dη ≥ c(α)P.V.…”

mentioning

confidence: 95%

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confidence: 99%

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Global solutions for a supercritical drift–diffusion equation

Burczak

Granero-Belinchón

2016

Advances in Mathematics

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“…The estimate (21) follows for d = 2 from [19], Lemma 4.3 and for d = 1 from [17], Lemma 4. To justify the estimates (19), (20), we use for η…”

Section: Proof Of Theorem 1 (Classical Solvability and Uniform-in-tim...mentioning

confidence: 99%

Boundedness and homogeneous asymptotics for a fractional logistic Keller-Segel equations

Burczak¹,

Granero-Belinchón²

2020

Discrete &Amp; Continuous Dynamical Systems - S

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In this paper we consider a d-dimensional (d = 1, 2) parabolic-elliptic Keller-Segel equation with a logistic forcing and a fractional diffusion of order α ∈ (0, 2). We prove uniform in time boundedness of its solution in the supercritical range α > d (1 − c), where c is an explicit constant depending on parameters of our problem. Furthermore, we establish sufficient conditions for u(t) − u∞ L ∞ → 0, where u∞ ≡ 1 is the only nontrivial homogeneous solution. Finally, we provide a uniqueness result.1 1 Observe that the Patlak-Keller-Segel equation is often written for unknowns (u, −v)

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