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A probabilistic model associated with the pressureless gas dynamics. (English) Zbl 1286.35154

The authors consider the non-viscous Burgers equation \(\partial _{t}u+(u,\nabla )u=0\) with the smooth initial condition \(u(x,0)=u_{0}(x)\in C^{1}(\mathbb{R}^{n})\cap C_{b}(\mathbb{R}^{n})\) which may be associated to the system \(\overset{.}{x}(t)=u(t,x(t))\) with the value \(\overset{.}{u} (t,x(t))=0\) along the characteristics \(x=x(t)\). They then consider the system of stochastic differential equations \(dX_{k}(t)=U_{k}(t)+\sigma d(W_{k})_{t}\), \(dU_{k}(t)=0\) with the initial conditions \(X_{k}(0)=x\) and \( U(0)=u\). Here \(\sigma >0\) and \((W)_{t}=(W_{k})_{t}\), \(k=1,\ldots ,n\), is a \( n\)-dimensional Brownian motion. This system may thus be considered as a stochastic perturbation of Burgers equation.
In the first part of their paper, the authors consider smooth initial data. Assuming that at \(t=0\) a density \( f(x)\) represents non-interacting particles, \(P(t,x,u)\) represents the density probability at \(t=0\) of position \(x\) and velocity \(u\). The authors introduce the quantity \[ \widehat{u}(x,t)=\frac{\int_{\mathbb{R} ^{n}}uP(t,x,u)du}{\int_{\mathbb{R}^{n}}P(t,x,u)du}. \] Choosing \( P_{0}(x,u)=\delta (u-u_{0}(x))f_{0}(x)\) the density \(P\) is proved to obey a Fokker-Planck equation, from which the authors compute another expression of \( \widehat{u}(t,x)=:\widehat{u}_{\sigma }(t,x)\).
The first result of the paper proves that \(\widehat{u}_{\sigma }\) converges to a solution of this Fokker-Planck equation when \(\sigma \) goes to 0, for fixed \((t,x)\in \mathbb{ R}^{n+1}\), with \(0<t<t_{\ast }(u_{0})\), where \(t_{\ast }(u_{0})\) is the moment of time where the solution of Burgers equation changes its smoothness. Defining \(\rho (t,x)=\int_{\mathbb{R}^{n}}P(t,x,u)du\) the authors prove that \((\rho ,\widehat{u})\) is the solution of a coupled evolution system and they describe the asymptotic behaviour of \(\rho =\rho _{\sigma }\) when \(\sigma \) goes to 0. Moving now to discontinuous initial data \((f_{0}(x),u_{0}(x))\) the authors define the notion of generalized solution, in the sense of free particles and that of monotonic approximation of these initial data. They give an example of such a generalized solution for which they discuss the Hugoniot conditions leading to a presence of a spurious pressure. The paper ends with a discussion on the notion of sticky particles in this generalized situation.

MSC:

35L45 Initial value problems for first-order hyperbolic systems
35Q84 Fokker-Planck equations
35L65 Hyperbolic conservation laws
35L67 Shocks and singularities for hyperbolic equations
35R60 PDEs with randomness, stochastic partial differential equations
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H30 Applications of stochastic analysis (to PDEs, etc.)
76N15 Gas dynamics (general theory)
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References:

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