Transport equations with inflow boundary conditions

L. Ridgway Scott () and Sara Pollock
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L. Ridgway Scott: University of Chicago
Sara Pollock: University of Florida

Partial Differential Equations and Applications, 2022, vol. 3, issue 3, 1-20

Abstract: Abstract We provide bounds in a Sobolev-space framework for transport equations with nontrivial inflow and outflow. We give, for the first time, bounds on the gradient of the solution with the type of inflow boundary conditions that occur in Poiseuille flow. Following ground-breaking work of the late Charles Amick, we name a generalization of this type of flow domain in his honor. We prove gradient bounds in Lebesgue spaces for general Amick domains which are crucial for proving well posedness of the grade-two fluid model. We include a complete review of transport equations with inflow boundary conditions, providing novel proofs in most cases. To illustrate the theory, we review and extend an example of Bernard that clarifies the singularities of solutions of transport equations with nonzero inflow boundary conditions.

Keywords: 35M35; 65M60 (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s42985-022-00169-0

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