Speaker
Description
I prove global in time existence of solutions to the full revised Enskog equation. This equation generalizes the Boltzmann theory to dense gases in two ways:
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by taking into account the fact that the centers of two colliding spheres are at a distance a, equal to the diameter of hard spheres.
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by increasing the collision frequency by a factor $Y_0$ which nowadays is identified with the radial pair correlation function $g_2$ for the system of hard spheres at a uniform equilibrium.
In contrast to the dilute gas mode described by the Boltzmann equation, the Enskog equation includes spatial pair correlation function for hard-spheres potential and depends in a highly non linear way on the local density of dense gas. The full revised Enskog equation refers to the case where $g_2$, the pair the correlation function (for non-uniform equilibrium of hard-spheres) is in general form. In terms of the virial expansion (in local density $n$, spatially dependent) at contact value, $g_2$ reads:
$g_2(n)=1+V_1(n)+V_2(n)+....+V_N(n)+...$,
where the term $V_1(n)$ depends on $n$ linearly, $V_2(n)$ depends on $n$ quadratically, $V_N(n)$ depends on $n$ as $n^N$, and so on.
Circa 30 years ago Arkeryd-Cercignani proved the result for the truncated $g_2$, i.e., when $g_2=1$ (no density dependence). The case with $g_2=1$ refers to the so called Boltzmann-Enskog equation. It differs from the Boltzmann equation only by existence of the shifts in the spatial variable in the collisional integral.
Since then many researchers tried/wanted to prove the result for general form of $g_2$. Dependence of $g_2$ on $n$ requires a different approach and new tools, as compared to Arkeryd-Cercignani proof ([1]). Additionally, this result finally completes and fulfills the existence result for the revised Enskog Equation.
The proof of existence of solutions to the revised Enskog equation is based on two constructions:
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Construction of an H-functional (see [2]), where the full expansion of $g_2$ is used, but convergence of the series was not addressed.
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Construction of a special sequence of stochastic kinetic equations (studied in [3]) and the proof that their solutions converge to weak solutions of the revised Enskog equation.
References
[1] L. Arkeryd and C. Cercignani, Global existence in L1 for the Enskog equation and convergence of solutions to solutions of the Boltzmann equation, J. Stat. Phys. 59 (1990), 845–867.
[2] M. Mareschal, J. Bławzdziewicz, and J. Piasecki, Local entropy production from the revised Enskog equation:
General formulation for inhomogeneous fluids, Phys. Rev. Lett. 52 (1984), 1169–1172.
[3] J. Polewczak and G. Stell,. Transport coefficients in some stochastic models of the revised Enskog equation, J. Stat. Phys. 109, (2002) 569–590.
