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Existence of a classical solution to complex material flow problems

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Abstract

An existence result of smooth solutions for a complex material flow problem is provided. The considered equations are of hyperbolic type including a nonlocal interaction term. The existence proof is based on a problem-adapted linear iteration scheme exploiting the structure conditions of the nonlocal term. 35Q70, 35L65 Copyright

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... The pressureless Euler system with non-local forces has been studied recently and very limited results have been done. The local existence of classical solutions of the complex material flow dynamics which has been derived in [11], under structural condition for the interaction force has been obtained in [6]. In [3], for the 1-D model with damping and non-local interaction, a critical threshold for the existence of classical solution by using the characteristic method is presented. ...
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... See [4,28,30] for threshold conditions for the pressureless systems, and [8] for isothermal pressure p(ρ) = ρ with small initial data. For the case where the communication weight is not positive, such as in the pedestrian and material flow case, an additional damping effect is required to prove the global smooth solution for small initial data in [6,32,33]. ...
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... The literature ranges from the individual computation of part trajectories (microscopic models) to rescaled models describing the evolution of the part density (macroscopic models). Depending on the type of application, numerical methods (Aggarwal et al. 2015;Göttlich and Schindler 2015;Evers et al. 2015), theoretical investigations (Che et al. 2016) and model hierarchies Yu 2002, 2005) are provided. However, optimization issues of such problems are typically considered for fluid problems in general such as Allain et al. (2014), McNamara et al. (2004), Schlitzer (2000) or Wojtan et al. (2006). ...
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