Non-Isothermal Creeping Flows in a Pipeline Network: Existence Results
Abstract
:1. Introduction and Problem Formulation
- , …, model pipes;
- , …, represent junctions in which pipes are connected.
- (A1)
- For each junction there exist exactly pipes , , ..., , where and , such that
- (A2)
- The intersection is a flat surface, for any and .
- (A3)
- For each pipe there exist exactly two junctions and such that
2. Preliminaries: Main Notation, Function Spaces, and Assumptions
- (B1)
- The function is continuous.
- (B2)
- There exist constants and such that
- (B3)
- The functions , , are measurable for any , , .
- (B4)
- The functions , , are continuous for each and almost every .
- (B5)
- There exist constants , , such that
- (B6)
- The function belongs to the Lebesgue space for each .
3. Functional Setting of the Problem and Main Results
4. Proof of Main Results
- the inclusion is valid;
- is symmetric in the following sense: if , then .
- for any pair
- is an odd mapping, i.e., for any vector .
- the function is measurable for every ;
- the function is continuous for almost every ;
- there exist constants , and a function such that the inequalityholds for every and for almost every .
5. Conclusions
- the task of proving the unique solvability under smallness of the data (as in the case of the Navier–Stokes equations);
- the study of the continuous dependence of solutions on the model data;
- the well-posedness analysis of non-steady problems;
- the numerical analysis of network models;
- the analysis of flow control problems and finding optimal solutions.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
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Baranovskii, E.S.; Provotorov, V.V.; Artemov, M.A.; Zhabko, A.P. Non-Isothermal Creeping Flows in a Pipeline Network: Existence Results. Symmetry 2021, 13, 1300. https://doi.org/10.3390/sym13071300
Baranovskii ES, Provotorov VV, Artemov MA, Zhabko AP. Non-Isothermal Creeping Flows in a Pipeline Network: Existence Results. Symmetry. 2021; 13(7):1300. https://doi.org/10.3390/sym13071300
Chicago/Turabian StyleBaranovskii, Evgenii S., Vyacheslav V. Provotorov, Mikhail A. Artemov, and Alexey P. Zhabko. 2021. "Non-Isothermal Creeping Flows in a Pipeline Network: Existence Results" Symmetry 13, no. 7: 1300. https://doi.org/10.3390/sym13071300
APA StyleBaranovskii, E. S., Provotorov, V. V., Artemov, M. A., & Zhabko, A. P. (2021). Non-Isothermal Creeping Flows in a Pipeline Network: Existence Results. Symmetry, 13(7), 1300. https://doi.org/10.3390/sym13071300