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Analysis of PDEs (math.AP)

Tue, 11 Apr 2023

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1.Non-isochoric stable granular models taking into account fluidisation by pore gas pressure

Authors:Laurent Chupin LMBP, Thierry Dubois LMBP

Abstract: In this paper, we study non-isochoric models for mixtures of solid particles, at high volume concentration, and a gas. One of the motivations of this work concerns geophysics and more particularly the pyroclastic density currents which are precisely mixtures of pyroclast and lithic fragments and air. They are extremely destructive phenomena, capable of devastating urbanised areas, and are known to propagate over long distances, even over almost flat topography. Fluidisation of these dense granular flows by pore gas pressure is one response that could explain this behaviour and must therefore be taken into account in the models. Starting from a gas-solid mixing model and invoking the compressibility of the gas, through a law of state, we rewrite the conservation of mass equation of the gas phase into an equation on the pore gas pressure whose net effect is to reduce the friction between the particles. The momentum equation of the solid phase is completed by generic constitutive laws, specified as in Schaeffer et al (2019, Journal of Fluid Mechanics, 874, 926-951) by a yield function and a dilatancy function. Therefore, the divergence of the velocity field, which reflects the ability of the granular flow to expand or compress, depends on the volume fraction, pressure, strain rate and inertial number. In addition, we require the dilatancy function to describe the rate of volume change of the granular material near an isochoric equilibrium state, i.e. at constant volume. This property ensures that the volume fraction, which is the solution to the conservation of mass equation, is positive and finite at all times. We also require that the non-isochoric fluidised model is linearly stable and dissipates energy (over time). In this theoretical framework, we derive the dilatancy models corresponding to classical rheologies such as Drucker-Prager and $\mu$(I) (with or without expansion effects). The main result of this work is to show that it is possible to obtain non-isochoric and fluidised granular models satisfying all the properties necessary to correctly account for the physics of granular flows and being well-posed, at least linearly stable.

2.Initial Data Identication in Space Dependent Conservation Laws and Hamilton-Jacobi Equations

Authors:Rinaldo M. Colombo IDP, Vincent Perrollaz IDP, Abraham Sylla UNIMIB

Abstract: Consider a Conservation Law and a Hamilton-Jacobi equation with a ux/Hamiltonian depending also on the space variable. We characterize rst the attainable set of the two equations and, second, the set of initial data evolving at a prescribed time into a prescribed prole. An explicit example then shows the deep dierences between the cases of x-independent and x-dependent uxes/Hamiltonians.

3.Strichartz estimates for the Schrödinger equation on negatively curved compact manifolds

Authors:Matthew D. Blair, Xiaoqi Huang, Christopher D. Sogge

Abstract: We obtain improved Strichartz estimates for solutions of the Schr\"odinger equation on negatively curved compact manifolds which improve the classical universal results results of Burq, G\'erard and Tzvetkov [11] in this geometry. In the case where the spatial manifold is a hyperbolic surface we are able to obtain no-loss $L^{q_c}_{t,x}$-estimates on intervals of length $\log \lambda\cdot \lambda^{-1} $ for initial data whose frequencies are comparable to $\lambda$, which, given the role of the Ehrenfest time, is the natural analog of the universal results in [11]. We are also obtain improved endpoint Strichartz estimates for manifolds of nonpositive curvature, which cannot hold for spheres.

4.Derivation of effective theories for thin 3D nonlinearly elastic rods with voids

Authors:Manuel Friedrich, Leonard Kreutz, Konstantinos Zemas

Abstract: We derive a dimension-reduction limit for a three-dimensional rod with material voids by means of $\Gamma$-convergence. Hereby, we generalize the results of the purely elastic setting [57] to a framework of free discontinuity problems. The effective one-dimensional model features a classical elastic bending-torsion energy, but also accounts for the possibility that the limiting rod can be broken apart into several pieces or folded. The latter phenomenon can occur because of the persistence of voids in the limit, or due to their collapsing into a {discontinuity} of the limiting deformation or its derivative. The main ingredient in the proof is a novel rigidity estimate in varying domains under vanishing curvature regularization, obtained in [32].

5.Sharp uniform-in-time mean-field convergence for singular periodic Riesz flows

Authors:Antonin Chodron de Courcel, Matthew Rosenzweig, Sylvia Serfaty

Abstract: We consider conservative and gradient flows for $N$-particle Riesz energies with mean-field scaling on the torus $\mathbb{T}^d$, for $d\geq 1$, and with thermal noise of McKean-Vlasov type. We prove global well-posedness and relaxation to equilibrium rates for the limiting PDE. Combining these relaxation rates with the modulated free energy of Bresch et al. and recent sharp functional inequalities of the last two named authors for variations of Riesz modulated energies along a transport, we prove uniform-in-time mean-field convergence in the gradient case with a rate which is sharp for the modulated energy pseudo-distance. For gradient dynamics, this completes in the periodic case the range $d-2\leq s<d$ not addressed by previous work of the second two authors. We also combine our relaxation estimates with the relative entropy approach of Jabin and Wang for so-called $\dot{W}^{-1,\infty}$ kernels, giving a proof of uniform-in-time propagation of chaos alternative to Guillin et al.

6.Identification of nonlinear beam-hardening effects in X-ray tomography

Authors:Yiran Wang

Abstract: We study streaking artifacts caused by beam-hardening effects in X-ray computed tomography (CT). The effect is known to be nonlinear. We show that the nonlinearity can be recovered from the observed artifacts for strictly convex bodies. The result provides a theoretical support for removal of the artifacts.

7.Optimal enhanced dissipation and mixing for a time-periodic, Lipschitz velocity field on $\mathbb{T}^2$

Authors:Tarek M. Elgindi, Kyle Liss, Jonathan C. Mattingly

Abstract: We consider the advection-diffusion equation on $\mathbb{T}^2$ with a Lipschitz and time-periodic velocity field that alternates between two piecewise linear shear flows. We prove enhanced dissipation on the timescale $|\log \nu|$, where $\nu$ is the diffusivity parameter. This is the optimal decay rate as $\nu \to 0$ for uniformly-in-time Lipschitz velocity fields. We also establish exponential mixing for the $\nu = 0$ problem.