Abstract:
We introduce and analyze a mathematical model for activated fluids - incompressible flows that behave like ideal Euler fluids until the modulus of the symmetric part of the velocity gradient exceeds a prescribed (and arbitrarily large) critical threshold. Once this value is surpassed, a dissipation mechanism is activated, allowing the system to dissipate energy. Assuming a specific form of energy dissipation after activation, we establish global-in-time well-posedness of the corresponding initial-boundary value problem in the sense of Hadamard. In particular, we prove that for arbitrary sufficiently regular initial data there exists a unique weak solution to both the spatially periodic and the Dirichlet problem. Moreover, we show that this procedure naturally selects admissible initial data for which the problem is well-posed.