Balanced-Viscosity solutions for multi-rate systems

7th International Workshop on MUlti-Rate Processes & HYSteresis; 2nd International Workshop on Hysteresis and Slow-Fast Systems

Mielke, Alexander^1,2 ; Rossi, Riccarda^3 ; Savaré, Giuseppe^4

Weierstra-Institut, Mohrenstrae 39, Berlin

D-10117, Germany^1

Institut für Mathematik, Humboldt-Universität zu Berlin, Rudower Chaussee 25, Berlin, Adlershof

D-12489, Germany^2

DICATAM - Sezione di Matematica, Università di Brescia, via Valotti 9, Brescia

I-25133, Italy^3

Dipartimento di Matematica F. Casorati, Università di Pavia, Via Ferrata, Pavia

I-27100, Italy^4

关键词: Finite dimensional;    Mechanical systems;    Multi-rate systems;    Rate-independent system;    Subdifferential inclusion;    Vanishing viscosity;    Viscosity solutions;    Viscous dissipation;   

Others  :  https://iopscience.iop.org/article/10.1088/1742-6596/727/1/012010/pdf
DOI  :  10.1088/1742-6596/727/1/012010

来源: IOP

【 摘 要 】

Several mechanical systems are modeled by the static momentum balance for the displacement u coupled with a rate-independent flow rule for some internal variable z. We consider a class of abstract systems of ODEs which have the same structure, albeit in a finite-dimensional setting, and regularize both the static equation and the rate-independent flow rule by adding viscous dissipation terms with coefficients αand , where 00 is a fixed parameter. Therefore for α ≠ 1 u and z have different relaxation rates. We address the vanishing-viscosity analysis as↓ 0 of the viscous system. We prove that, up to a subsequence, (reparameterized) viscous solutions converge to a parameterized curve yielding a Balanced Viscosity solution to the original rate-independent system, and providing an accurate description of the system behavior at jumps. We also give a reformulation of the notion of Balanced Viscosity solution in terms of a system of subdifferential inclusions, showing that the viscosity in u and the one in z are involved in the jump dynamics in different ways, according to whether α > 1, α =1, and α(0,1).