2018
DOI: 10.1088/1361-6544/aaa2e0
|View full text |Cite
|
Sign up to set email alerts
|

Global existence of weak solutions to dissipative transport equations with nonlocal velocity

Abstract: We consider 1D dissipative transport equations with nonlocal velocity field:where N is a nonlocal operator given by a Fourier multiplier. Especially we consider two types of nonlocal operators:(1) N = H, the Hilbert transform, (2) N = (1 − ∂xx) −α . In this paper, we show several global existence of weak solutions depending on the range of γ, δ and α. When 0 < γ < 1, we take initial data having finite energy, while we take initial data in weighted function spaces (in the real variables or in the Fourier variab… Show more

Help me understand this report
View preprint versions

Search citation statements

Order By: Relevance

Paper Sections

Select...
1
1
1

Citation Types

0
16
0

Year Published

2019
2019
2023
2023

Publication Types

Select...
7
2

Relationship

2
7

Authors

Journals

citations
Cited by 12 publications
(16 citation statements)
references
References 30 publications
0
16
0
Order By: Relevance
“…Hence, we only need to use some compactness argument to derive global weak solutions (see Section 2). In papers [1], global weak solutions to the following general models were studied:…”
mentioning
confidence: 99%
“…Hence, we only need to use some compactness argument to derive global weak solutions (see Section 2). In papers [1], global weak solutions to the following general models were studied:…”
mentioning
confidence: 99%
“…Our approach is very adaptable and can lead to advances in other systems of PDE. For instance, it has been used Bruell & Granero-Belinchón to study the evolution of thin films in Darcy and Stokes flows [3] by Córdoba and Gancedo [6], Constantin, Córdoba, Gancedo, Rodriguez-Piazza, & Strain [5] for the Muskat problem (see also [7] and [17]), by Burczak & Granero-Belinchón [4] to analyze the Keller-Segel system of PDE with diffusion given by a nonlocal operator and by Bae, Granero-Belinchón & Lazar [2] to prove several global existence results (with infinite L p energy) for nonlocal transport equations.…”
mentioning
confidence: 99%
“…• The model 2: N = −H(∂ xx ) −α , α > 0, ν = 1, and γ > 0. This is a regularized version of (1.2) which is also closely related to many equations as mentioned in [3]. In this case, we show the existence of weak solutions globally in time under weaker conditions on α and γ compared to [3].…”
Section: Introductionmentioning
confidence: 62%
“…Remark. Theorem 4.2 improves Theorem 1.4 in [3], where (α, γ) is assumed to satisfy α ≥ 1 2 − γ 4 . The main idea of taking weaker regularization in (4.1) is that the Hilbert transform in front of (1 − ∂ xx ) −α gives (4.6) which makes to obtain (4.7).…”
Section: The Modelmentioning
confidence: 76%