Multidimensional Compressible Flows with Symmetry

David Hoff () and Helge Kristian Jenssen ()
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David Hoff: Indiana University, Department of Mathematics
Helge Kristian Jenssen: Indiana University, Department of Mathematics

A chapter in Hyperbolic Problems: Theory, Numerics, Applications, 2003, pp 493-498 from Springer

Abstract: Abstract We prove the global existence of weak solutions of the Navier-Stokes equations for compressible, heat-conducting flow, (1) $$ \left\{ {\begin{array}{*{20}{c}} {{{\rho }_{t}} + div(\rho {\text{U}}) = 0} \hfill \\ {{{{(\rho {{{\text{U}}}^{j}})}}_{t}} + div(\rho {{{\text{U}}}^{j}}{\text{U}}) + P{{{(\rho ,\theta )}}_{{{{x}_{j}}}}} = \mu \Delta {{{\text{U}}}^{j}} + (\lambda + \mu )div{{{\text{U}}}_{{{{x}_{j}}}}} + \rho {{{\text{F}}}^{j}}} \hfill \\ {{{{(\rho E)}}_{t}} + div(\rho E{\text{U}} + P(\rho ,\theta ){\text{U}}) = \Delta (\kappa \theta + \tfrac{1}{2}\mu |{\text{U}}{{|}^{2}})} \hfill \\ { + \mu div((\nabla {\text{U}}){\text{U}}) + \lambda div((div{\text{U}}){\text{U}}) + \rho {\text{U}}\cdot {\text{F}},} \hfill \\ \end{array} } \right. $$ with initial data (ρ0, U 0, θ0) and external force F which are large, discontinuous, and spherically or cylindrically symmetric. Here x ∈, ℝ3 is the spatial coordinate, t > 0 is time, and ρ, U, θ, and P=Kρθ are respectively the density, velocity, temperature, energy density, and pressure of an ideal fluid (with unit specific heat), μ and λ are viscosity constants assumed to satisfy μ > and, and κ is a positive heat conduction coefficient.

Keywords: Global Existence; Compressible Fluid; Annular Region; Isothermal Flow; Entropy Bound (search for similar items in EconPapers)
Date: 2003
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-642-55711-8_45

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DOI: 10.1007/978-3-642-55711-8_45

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