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Existence of travelling waves in non-isothermal phase dynamics. (English) Zbl 1155.35061

The author of this interesting paper studies travelling wave solutions describing non-isothermal subsonic phase transitions in a van der Waals fluid. The motion of the fluid is governed by the following Euler equations supplemented with the viscosity coefficient \(\varepsilon \nu \geq 0\), the capillarity coefficient \(\varepsilon ^2 \geq 0\) and the heat conductivity \(\varepsilon k \geq 0\),
\[ \partial_t\rho + \partial_x(\rho u)=0, \qquad \partial_t(\rho u) + \partial_x(\rho u^2+p)= \varepsilon \nu \partial_x^2u - \varepsilon^2 \partial_x^3 v \]
\[ \partial_t(\rho((1/2)u^2+e))+\partial_x(\rho u((1/2)u^2+e+pv)) =\varepsilon \nu \partial_x(u \partial_x u)-\varepsilon^2 u \partial_x^3v +\varepsilon k \partial_x^2\theta , \] where \(\rho \) is the density, \(v\equiv 1/\rho \) the specific volume, \(u\) the velocity, \(\theta \) the temperature, \(p\) is the pressure satisfying the state equation \(p(v,\theta )= R\theta /(v-b)-a/v^2\) with \(R\) being the perfect gas constant and \(a,b\) being positive constants. The quantity \(e\) is the specific internal energy satisfying \(e(v,\theta )=C_v\theta -a/v\) with \(C_v>0\) being the heat capacity. When \(\varepsilon \to 0^{+}\) the above stated equations become the well-known Euler equations for inviscid fluid.
The author shows the existence and structural stability of travelling waves by using the center manifold method. The existence of viscosity-capillarity profiles for small viscosity, large heat conduction, and large heat capacity is discussed. An important assumption is that when the heat capacity goes to infinity, the heat conductivity also goes to infinity at the same order. This assumption makes the arguments of structural stability possible and it is physically acceptable. With this result on existence of planar waves, one can see that the fluid states on both sides of the phase boundary depend continuously on the parameters arising in the travelling wave equation.

MSC:

35L65 Hyperbolic conservation laws
35L45 Initial value problems for first-order hyperbolic systems
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
35Q35 PDEs in connection with fluid mechanics
35B25 Singular perturbations in context of PDEs
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References:

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