Loading [MathJax]/jax/output/CommonHTML/jax.js
Trudy Matematicheskogo Instituta imeni V.A. Steklova
RUS  ENG    JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB  
General information
Latest issue
Forthcoming papers
Archive
Impact factor
Guidelines for authors
License agreement

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Trudy Mat. Inst. Steklova:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2012, Volume 278, Pages 188–207 (Mi tm3418)  

This article is cited in 8 scientific papers (total in 8 papers)

Protter–Morawetz multidimensional problems

Nedyu Popivanova, Todor Popova, Rudolf Schererb

a Faculty of Mathematics and Informatics, University of Sofia, Sofia, Bulgaria
b Institute for Applied and Numerical Mathematics, Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany
Full-text PDF (258 kB) Citations (8)
References:
Abstract: About 50 years ago M. H. Protter introduced boundary value problems that are multidimensional analogues of the classical plane Morawetz problems for equations of mixed hyperbolic-elliptic type that model transonic fluid flows. Up to now there are no general existence results for the Protter–Morawetz multidimensional problems, and an understanding of the situation is not at hand. At the same time, Protter also formulated boundary value problems in the hyperbolic part of the domain – the nonhomogeneous wave equation is studied in a (3+1)-D domain bounded by two characteristic cones and a non-characteristic ball. These problems could be considered as multidimensional variants of the Darboux problem in R2. In the frame of classical solvability the hyperbolic Protter problem is not Fredholm, because it has an infinite-dimensional cokernel. On the other hand, it is known that the unique generalized solution of a Protter problem may have a strong power-type singularity even for some very smooth right-hand side functions. This singularity is isolated at the vertex O of the boundary light cone and does not propagate along the characteristic cone. In the general case of smooth right-hand side function, some necessary and sufficient conditions for the existence of a bounded solution are given and a priori estimates for the solution are found. The semi-Fredholm solvability of the problem is proved.
Received in February 2011
English version:
Proceedings of the Steklov Institute of Mathematics, 2012, Volume 278, Pages 179–198
DOI: https://doi.org/10.1134/S0081543812060181
Bibliographic databases:
Document Type: Article
UDC: 517.95
Language: English
Citation: Nedyu Popivanov, Todor Popov, Rudolf Scherer, “Protter–Morawetz multidimensional problems”, Differential equations and dynamical systems, Collected papers, Trudy Mat. Inst. Steklova, 278, MAIK Nauka/Interperiodica, Moscow, 2012, 188–207; Proc. Steklov Inst. Math., 278 (2012), 179–198
Citation in format AMSBIB
\Bibitem{PopPopSch12}
\by Nedyu~Popivanov, Todor~Popov, Rudolf~Scherer
\paper Protter--Morawetz multidimensional problems
\inbook Differential equations and dynamical systems
\bookinfo Collected papers
\serial Trudy Mat. Inst. Steklova
\yr 2012
\vol 278
\pages 188--207
\publ MAIK Nauka/Interperiodica
\publaddr Moscow
\mathnet{http://mi.mathnet.ru/tm3418}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=3058795}
\elib{https://elibrary.ru/item.asp?id=17928423}
\transl
\jour Proc. Steklov Inst. Math.
\yr 2012
\vol 278
\pages 179--198
\crossref{https://doi.org/10.1134/S0081543812060181}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=000309861500018}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84866982029}
Linking options:
  • https://www.mathnet.ru/eng/tm3418
  • https://www.mathnet.ru/eng/tm/v278/p188
  • This publication is cited in the following 8 articles:
    1. Nedyu Popivanov, Todor Popov, Ingo Witt, Trends in Mathematics, 1, Extended Abstracts MWCAPDE 2023, 2024, 177  crossref
    2. Nedyu Popivanov, Todor Popov, Ingo Witt, “Solutions with exponential singularity for (3 + 1)-D Protter problems”, J. Hyper. Differential Equations, 20:02 (2023), 475  crossref
    3. Nedyu Popivanov, Tsvetan Hristov, Rudolf Scherer, “TOPICAL ISSUES OF THERMOPHYSICS, ENERGETICS AND HYDROGASDYNAMICS IN THE ARCTIC CONDITIONS”: Dedicated to the 85th Birthday Anniversary of Professor E. A. Bondarev, 2528, “TOPICAL ISSUES OF THERMOPHYSICS, ENERGETICS AND HYDROGASDYNAMICS IN THE ARCTIC CONDITIONS”: Dedicated to the 85th Birthday Anniversary of Professor E. A. Bondarev, 2022, 030005  crossref
    4. Zeitsch P.J., “On the Riemann Function”, Mathematics, 6:12 (2018), 316  crossref  isi  scopus
    5. Popov T.P., “New Singular Solutions For the (3+1)-D Protter Problem”, Bull. Karaganda Univ-Math., 91:3 (2018), 61–68  crossref  isi
    6. Nedyu Popivanov, Todor Popov, Allen Tesdall, “Semi-Fredholm Solvability in the Framework of Singular Solutions for the (3+1)-D Protter-Morawetz Problem”, Abstract and Applied Analysis, 2014 (2014), 1  crossref
    7. N. Popivanov, T. Popov, R. Scherer, “Singular solutions with exponential growth to Protter's problems”, Sib. Adv. Math., 23:3 (2013), 219  crossref
    8. Dechevski L., Popivanov N., Popov T., “Exact Asymptotic Expansion of Singular Solutions for the (2+1)-D Protter Problem”, Abstract Appl. Anal., 2012, 278542  crossref  mathscinet  zmath  isi  elib  scopus
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Труды Математического института имени В. А. Стеклова Proceedings of the Steklov Institute of Mathematics
    Statistics & downloads:
    Abstract page:471
    Full-text PDF :65
    References:71
     
      Contact us:
    math-net2025_04@mi-ras.ru
     Terms of Use  Registration to the website  Logotypes © Steklov Mathematical Institute RAS, 2025