- PDE Seminar
- Tuesday, February 21, 2023 - 15:00 for 1 hour (actually 50 minutes)
- Jonas Lührmann – Texas A&M University – email@example.com
For steady two-dimensional incompressible flows with a single eddy (i.e. nested closed streamlines), Prandtl (1905) and Batchelor (1956) proposed that in the limit of vanishing viscosity the vorticity is constant in an inner region separated from the boundary layer. By constructing higher order approximate solutions of the Navier-Stokes equations and establishing the validity of Prandtl boundary layer expansion, we give a rigorous proof of the existence of Prandtl-Batchelor flows on a disk with the wall velocity slightly different from the rigid-rotation. The leading order term of the flow is the constant vorticity solution (i.e. rigid rotation) satisfying the Batchelor-Wood formula. For an annulus with wall velocities slightly different from the rigid-rotation, we also constructed Prandtl-Batchelor flows, whose leading order terms are rotating shear flows. This is a joint work with Chen Gao, Mingwen Fei and Tao Tao.
It is well known that the wave operators cos(t (−∆)) and sin(t (−∆)) are not bounded on Lp(Rn), for n≥2 and 1≤p≤∞, unless p=2 or t=0. In fact, for 1 < p < ∞ these operators are bounded from W2s(p),p to Lp(Rn) for s(p) := (n−1)/2 | 1/p − 1/2 |, and this exponent cannot be improved. This phenomenon is symptomatic of the behavior of Fourier integral operators, a class of oscillatory operators which includes wave propagators, on Lp(Rn).
In this talk, I will introduce a class of Hardy spaces HFIOp (Rn), for p ∈ [1,∞],on which Fourier integral operators of order zero are bounded. These spaces also satisfy Sobolev embeddings which allow one to recover the optimal boundedness results for Fourier integral operators on Lp(Rn).
However, beyond merely recovering existing results, the invariance of these spaces under Fourier integral operators allows for iterative constructions that are not possible when working directly on Lp(Rn). In particular, we shall indicate how one can use this invariance to obtain the optimal fixed-time Lp regularity for wave equations with rough coefficients. We shall also mention the connection of these spaces to the phenomenon of local smoothing.
This talk is based on joint work with Andrew Hassell and Pierre Portal (Aus- tralian National University), and Zhijie Fan, Naijia Liu and Liang Song (Sun Yat- Sen University).
In this talk, we consider a finite-horizon optimal control problem of stochastic reaction-diffusion equations. First we apply the spike variation method which relies on introducing the first and second order adjoint state. We give a novel characterization of the second order adjoint state as the solution to a backward SPDE. Using this representation, we prove the maximum principle for controlled SPDEs.
As another application of our characterization of the second order adjoint state, we derive additional necessary optimality conditions in terms of the value function. These results generalize a classical relationship between the adjoint states and the derivatives of the value function to the case of viscosity differentials.
The last part of the talk is devoted to sufficient optimality conditions. We show how the necessary conditions lead us directly to a non-smooth version of the classical verification theorem in the framework of viscosity solutions.
This talk is based on joint work with Wilhelm Stannat: W. Stannat, L. Wessels, Peng's maximum principle for stochastic partial differential equations, SIAM J. Control Optim., 59 (2021), pp. 3552–3573 and W. Stannat, L. Wessels, Necessary and Sufficient Conditions for Optimal Control of Semilinear Stochastic Partial Differential Equations, https://arxiv.org/abs/2112.09639, 2022.
In a seminal work, Leray demonstrated the existence of global weak solutions to the Navier-Stokes equations in three dimensions. Are Leray's solutions unique? This is a fundamental question in mathematical hydrodynamics, which we answer in the negative, within the `forced' category, by exhibiting two distinct Leray solutions with zero initial velocity and identical body force. This is joint work with Elia Brué and Maria Colombo.