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A global unique solvability of entropic weak solution to the one-dimensional pressureless Euler system with a flocking dissipation. (English) Zbl 1315.35151

The results presented in this article are related to the following Caucy problem for the quasi-equilibium Cucker-Smale equilibrium model \[ \displaystyle \partial_t \rho+\partial_x (\rho u)=0,~~x\in \mathbb{R},t>0, \]
\[ \displaystyle \partial_t (\rho u)+\partial_x (\rho u^2)=-K\rho \int_{\mathbb{R}}\psi(|x-y|) (u(x,t)-u(y,t))\rho(y,t)dy, \]
\[ \displaystyle (\rho,u)(x,0)=(\rho_0,u_0), \] where the unknowns are the local mass \(\rho\) and the velocity \(u\) and \(\psi\) is a given function with suitable properties.
When the initial mass density \(\rho_0\) is locally integrable and positive a.e. and the initial velocity \(u_0\) is uniformly bounded, the authors give explicit representations for global entropic weak solutions. The second main result is that any entropic weak solution (according to the definition given in this article) is unique.

MSC:

35Q31 Euler equations
35L60 First-order nonlinear hyperbolic equations
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35D30 Weak solutions to PDEs
35L45 Initial value problems for first-order hyperbolic systems
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