Pointwise constraints for scalar conservation laws with positive wave velocity
De Nitti N, Serre D, Zuazua Iriondo E (2024)
Publication Language: English
Publication Status: Submitted
Publication Type: Unpublished / Preprint
Future Publication Type: Journal article
Publication year: 2024
URI: http://cvgmt.sns.it/paper/6472/
Open Access Link: http://cvgmt.sns.it/paper/6472/
Abstract
We consider a conservation law with strictly positive wave velocity and study the well-posedness of a suitable notion of solution for the associated initial value problem under a pointwise flux constraint active in the half-line R+" role="presentation" style="display: inline-block; line-height: 0; font-size: 19.04px; overflow-wrap: normal; word-spacing: normal; text-wrap: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border-width: 0px; border-style: initial; padding-top: 1px; padding-bottom: 1px; position: relative;">R+�+.
The strict positivity of the wave velocity allows for the dynamics in the unconstrained region R−" role="presentation" style="display: inline-block; line-height: 0; font-size: 19.04px; overflow-wrap: normal; word-spacing: normal; text-wrap: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border-width: 0px; border-style: initial; padding-top: 1px; padding-bottom: 1px; position: relative;">R−�− to be fully determined by the restriction of the initial data to R−" role="presentation" style="display: inline-block; line-height: 0; font-size: 19.04px; overflow-wrap: normal; word-spacing: normal; text-wrap: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border-width: 0px; border-style: initial; padding-top: 1px; padding-bottom: 1px; position: relative;">R−�−.
On the other hand, the solution in the constrained region is dictated by the assumption that the total mass of the initial datum is conserved along the evolution. We formulate the transmission condition at the interface {x=0}" role="presentation" style="display: inline-block; line-height: 0; font-size: 19.04px; overflow-wrap: normal; word-spacing: normal; text-wrap: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border-width: 0px; border-style: initial; padding-top: 1px; padding-bottom: 1px; position: relative;">{x=0}{�=0} in such a way that the boundary datum for the initial boundary value problem posed on R+" role="presentation" style="display: inline-block; line-height: 0; font-size: 19.04px; overflow-wrap: normal; word-spacing: normal; text-wrap: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border-width: 0px; border-style: initial; padding-top: 1px; padding-bottom: 1px; position: relative;">R+�+ is given by the largest incoming flux that is admissible under the constraint, while the exceeding mass is accumulated in a " role="presentation" style="display: inline-block; line-height: 0; font-size: 19.04px; overflow-wrap: normal; word-spacing: normal; text-wrap: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border-width: 0px; border-style: initial; padding-top: 1px; padding-bottom: 1px; position: relative;">buffer'' (as an atomic measure concentrated at the interface).
Authors with CRIS profile
Nicola De Nitti
Lehrstuhl für Dynamics, Control, Machine Learning and Numerics (Alexander von Humboldt-Professur)
Enrique Zuazua Iriondo
Lehrstuhl für Dynamics, Control, Machine Learning and Numerics (Alexander von Humboldt-Professur)
Involved external institutions
France (FR)How to cite
APA:
De Nitti, N., Serre, D., & Zuazua Iriondo, E. (2024). Pointwise constraints for scalar conservation laws with positive wave velocity. (Unpublished, Submitted).
MLA:
De Nitti, Nicola, Denis Serre, and Enrique Zuazua Iriondo. Pointwise constraints for scalar conservation laws with positive wave velocity. Unpublished, Submitted. 2024.
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