Existence of solutions to the transonic pressure-gradient equations of the compressible Euler equations in elliptic regions

Yuxi Zheng

doi:10.1080/03605309708821323

Existence of solutions to the transonic pressure-gradient equations of the compressible Euler equations in elliptic regions

Yuxi Zheng

Mathematics

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53 Scopus citations

Abstract

We establish the existence of a smooth solution in its elliptic region in the self-similar plane to the pressure-gradient system arisen from the wave-particle splitting of the two-dimensional compressible Euler system of equations. The pressure-gradient system takes the form ρu_t + p_x = 0, ρυ_t + p_y = 0, ρE_t + (up)_x + (υp)_y = 0. Here (u, υ) is the velocity, ρ is the density which is independent of time resulted from the splitting procedure, p is the pressure, and E = 1/2(u² + υ²) + 1/γ-1 p/ρ is the energy. The natural (parabolically degenerate) boundary value is used.

Original language	English (US)
Pages (from-to)	1849-1868
Number of pages	20
Journal	Communications in Partial Differential Equations
Volume	22
Issue number	11-12
DOIs	https://doi.org/10.1080/03605309708821323
State	Published - 1997

All Science Journal Classification (ASJC) codes

Analysis
Applied Mathematics

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10.1080/03605309708821323

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title = "Existence of solutions to the transonic pressure-gradient equations of the compressible Euler equations in elliptic regions",

abstract = "We establish the existence of a smooth solution in its elliptic region in the self-similar plane to the pressure-gradient system arisen from the wave-particle splitting of the two-dimensional compressible Euler system of equations. The pressure-gradient system takes the form ρut + px = 0, ρυt + py = 0, ρEt + (up)x + (υp)y = 0. Here (u, υ) is the velocity, ρ is the density which is independent of time resulted from the splitting procedure, p is the pressure, and E = 1/2(u2 + υ2) + 1/γ-1 p/ρ is the energy. The natural (parabolically degenerate) boundary value is used.",

author = "Yuxi Zheng",

note = "Funding Information: The author wishes to thank L. C. Evans, Taiping Liu, and Tong Zhang for stimulating discussions and suggestions. The author also wishes to thank a referee for his/her questions on an earlier presentation of the appendix and for calling his attention to a couple of references. This work is supported in part by NSF DMS-9303414 and an Alfred P. Sloan Research Fellows award.",

year = "1997",

doi = "10.1080/03605309708821323",

language = "English (US)",

volume = "22",

pages = "1849--1868",

journal = "Communications in Partial Differential Equations",

issn = "0360-5302",

publisher = "Taylor and Francis Ltd.",

number = "11-12",

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TY - JOUR

T1 - Existence of solutions to the transonic pressure-gradient equations of the compressible Euler equations in elliptic regions

AU - Zheng, Yuxi

N1 - Funding Information: The author wishes to thank L. C. Evans, Taiping Liu, and Tong Zhang for stimulating discussions and suggestions. The author also wishes to thank a referee for his/her questions on an earlier presentation of the appendix and for calling his attention to a couple of references. This work is supported in part by NSF DMS-9303414 and an Alfred P. Sloan Research Fellows award.

PY - 1997

Y1 - 1997

N2 - We establish the existence of a smooth solution in its elliptic region in the self-similar plane to the pressure-gradient system arisen from the wave-particle splitting of the two-dimensional compressible Euler system of equations. The pressure-gradient system takes the form ρut + px = 0, ρυt + py = 0, ρEt + (up)x + (υp)y = 0. Here (u, υ) is the velocity, ρ is the density which is independent of time resulted from the splitting procedure, p is the pressure, and E = 1/2(u2 + υ2) + 1/γ-1 p/ρ is the energy. The natural (parabolically degenerate) boundary value is used.

AB - We establish the existence of a smooth solution in its elliptic region in the self-similar plane to the pressure-gradient system arisen from the wave-particle splitting of the two-dimensional compressible Euler system of equations. The pressure-gradient system takes the form ρut + px = 0, ρυt + py = 0, ρEt + (up)x + (υp)y = 0. Here (u, υ) is the velocity, ρ is the density which is independent of time resulted from the splitting procedure, p is the pressure, and E = 1/2(u2 + υ2) + 1/γ-1 p/ρ is the energy. The natural (parabolically degenerate) boundary value is used.

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UR - https://www.scopus.com/pages/publications/0000277304#tab=citedBy

U2 - 10.1080/03605309708821323

DO - 10.1080/03605309708821323

M3 - Article

AN - SCOPUS:0000277304

SN - 0360-5302

VL - 22

SP - 1849

EP - 1868

JO - Communications in Partial Differential Equations

JF - Communications in Partial Differential Equations

IS - 11-12

ER -

Existence of solutions to the transonic pressure-gradient equations of the compressible Euler equations in elliptic regions

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