Skip to content
DraconiticResearch
Search Sources API
203,234 studies31 sourcesUpdated 26 min ago
Peer-reviewedCC BY-NC-NDJournal de Mathématiques Pures et Appliquées

Global weak solutions of the Navier-Stokes-Korteweg equations in one dimension

  • Paolo Antonelli1,2
  • Didier Bresch3,4
  • Stefano Spirito5,6
6 affiliations
  1. 1Gran Sasso Science Institute, Viale Francesco Crispi, 7, 67100, L'Aquila, Italy
  2. 2GSSI - Gran Sasso Science Institute (Viale F. Crispi 7, 67100 L'Aquila, Italy - Italy)
  3. 3Université Savoie Mont Blanc, UMR5127 CNRS, Laboratoire de Mathématiques, 73376, Le Bourget-du-Lac, France
  4. 4LAMA - Laboratoire de Mathématiques (Université Savoie Mont Blanc, Campus Scientifique, Bâtiment Le Chablais 73376 Le Bourget du Lac, France - France)
  5. 5DISIM – Dipartimento di Ingegneria e Scienze dell'Informazione e Matematica, Università degli Studi dell'Aquila, Via Vetoio, 67100, L'Aquila, Italy
  6. 6UNIVAQ - Università degli Studi dell'Aquila = University of L'Aquila = Université de L'Aquila (Via Giovanni Di Vincenzo 16/B, 67100 L'Aquila - Italy)
Open at sourcePDFDOICite

Abstract

We prove the global existence of weak solutions of the one-dimensional Navier-Stokes-Korteweg (NSK) equations when the viscosity and the capillarity coefficients are power functions of the density, which may be zero on a set with positive measure. The proofs are based on a truncation argument combined with the Energy estimate and BD Entropy. Notably, we do not require any upper bound on the exponent of the power of the viscosity coefficient. In particular, we are able to consider very degenerate viscosity coefficient and to substantially improve previous results.

Related

  1. Homogeneous solutions of the stationary Navier-Stokes equations in arbitrary dimension
    Qi Ma
    arXivPreprintarXiv licenceFull textAnalysis & PDEsarXiv
  2. Global existence of smooth solutions to the Navier-Stokes equations for initial data with infinite energy
    Alexey Cheskidov, Hedong Hou
    arXivPreprintCC BY 4.0Full textAnalysis & PDEsarXiv
  3. Global weak solutions with vacuum for isentropic compressible Navier-Stokes and Navier-Stokes-Poisson equations in the unit disk under Dirichlet boundary conditions
    Shuai Wang, Xin Zhong
    arXivPreprintarXiv licenceFull textAnalysis & PDEsarXiv
  4. An Lp-theory for global weak solutions to the Navier-Stokes equations in exterior domains
    Filippo Palma
    arXivPreprintarXiv licenceFull textAnalysis & PDEsarXiv
  5. Local and Global Existence of Strong Solutions to the One-Dimensional Compressible Navier--Stokes Equations with General Pressure Law
    Qingsong Zhao
    arXivPreprintarXiv licenceFull textAnalysis & PDEsarXiv

Cite

BibTeXRIS
BibTeX
@article{antonelli2026global,
  title = {{Global weak solutions of the Navier-Stokes-Korteweg equations in one dimension}},
  author = {Paolo Antonelli and Didier Bresch and Stefano Spirito},
  journal = {Journal de Mathématiques Pures et Appliquées},
  year = {2026},
  month = jun,
  doi = {10.1016/j.matpur.2026.103952},
  url = {https://doi.org/10.1016/j.matpur.2026.103952},
  note = {CC BY-NC-ND},
}
RIS
TY  - JOUR
TI  - Global weak solutions of the Navier-Stokes-Korteweg equations in one dimension
AU  - Paolo Antonelli
AU  - Didier Bresch
AU  - Stefano Spirito
PY  - 2026
DA  - 2026/06/18
JO  - Journal de Mathématiques Pures et Appliquées
DO  - 10.1016/j.matpur.2026.103952
UR  - https://doi.org/10.1016/j.matpur.2026.103952
AB  - We prove the global existence of weak solutions of the one-dimensional Navier-Stokes-Korteweg (NSK) equations when the viscosity and the capillarity coefficients are power functions of the density, which may be zero on a set with positive measure. The proofs are based on a truncation argument combined with the Energy estimate and BD Entropy. Notably, we do not require any upper bound on the exponent of the power of the viscosity coefficient. In particular, we are able to consider very degenerate viscosity coefficient and to substantially improve previous results.
SN  - 0021-7824
SN  - 1776-3371
ER  -