Multiple timescales in the evolution of fluids models
Connections for Women: Dispersive and Stochastic PDE August 19, 2015 - August 21, 2015
Location: MSRI: Simons Auditorium
1D Burger's equation
Navier-Stokes on torus
long range behavior
35Q30 - Navier-Stokes equations [See also 76D05, 76D07, 76N10]
35Qxx - Equations of mathematical physics and other areas of application [See also 35J05, 35J10, 35K05, 35L05]
37J15 - Symmetries, invariants, invariant manifolds, momentum maps, reduction [See also 53D20]
37-XX - Dynamical systems and ergodic theory [See also 26A18, 28Dxx, 34Cxx, 34Dxx, 35Bxx, 46Lxx, 58Jxx, 70-XX]
70H33 - Symmetries and conservation laws, reverse symmetries, invariant manifolds and their bifurcations, reduction
70Hxx - Hamiltonian and Lagrangian mechanics [See also 37Jxx]
The evolution of fluids is known (eg via experiments and numerical studies) to occur on multiple timescales. We discuss how this can be analyzed rigorously in two fundamental models of fluids: the 1D Burgers equation and the 2D Navier-Stokes equation. For Burgers equation, we provide a complete geometric explanation involving invariant manifolds in the phase space of the evolution. For Navier-Stokes on the 2D torus, we discuss two complementary approaches. The first involves the theory of hypocoercive operators, and the second involves invariant manifolds and geometric singular perturbation theory in the Fourier phase space.
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