Particle dynamics inside shocks in Hamilton-Jacobi equations

Characteristics of a Hamilton-Jacobi equation can be seen as action minimizing trajectories of fluid particles. For nonsmooth "viscosity" solutions, which give rise to discontinuous velocity fields, this description is usually pursued only up to the moment when trajectories hit a shock and cease to minimize the Lagrangian action. In this paper we show that for any convex Hamiltonian there exists a uniquely defined canonical global nonsmooth coalescing flow that extends particle trajectories and determines dynamics inside the shocks. We also provide a variational description of the corresponding effective velocity field inside shocks, and discuss relation to the "dissipative anomaly" in the limit of vanishing viscosity.Comment: 15 pages, no figures; to appear in Philos. Trans. R. Soc. series

A kinetic formulation for multidimensional scalar conservation laws with boundary conditions and applications

International audienceWe state a kinetic formulation of weak entropy solutions of a general multidimensional scalar conservation law with initial and boundary conditions. We first associate with any weak entropy solution a entropy defect measure; the analysis of this measure at the boundary of the domain relies on the study of weak entropy sub and supersolutions and implies the introduction of the notion of sided boundary defect measures. As a first application, we prove that any weak entropy subsolution of the initial-boundary value problem is bounded above by any weak entropy supersolution (Comparison Theorem). We next study a BGK-like kinetic model that approximates the scalar conservation law. We prove that such a model converges by adapting the proof of the Comparison Theorem

Non-uniqueness of weak solutions for the fractal Burgers equation

The notion of Kruzhkov entropy solution was extended by the first author in 2007 to conservation laws with a fractional laplacian diffusion term; this notion led to well-posedness for the Cauchy problem in the $L^\infty$-framework. In the present paper, we further motivate the introduction of entropy solutions, showing that in the case of fractional diffusion of order strictly less than one, uniqueness of a weak solution may fail.Comment: 23 page

MeSLAM: Memory Efficient SLAM based on Neural Fields

Existing Simultaneous Localization and Mapping (SLAM) approaches are limited in their scalability due to growing map size in long-term robot operation. Moreover, processing such maps for localization and planning tasks leads to the increased computational resources required onboard. To address the problem of memory consumption in long-term operation, we develop a novel real-time SLAM algorithm, MeSLAM, that is based on neural field implicit map representation. It combines the proposed global mapping strategy, including neural networks distribution and region tracking, with an external odometry system. As a result, the algorithm is able to efficiently train multiple networks representing different map regions and track poses accurately in large-scale environments. Experimental results show that the accuracy of the proposed approach is comparable to the state-of-the-art methods (on average, 6.6 cm on TUM RGB-D sequences) and outperforms the baseline, iMAP$^*$. Moreover, the proposed SLAM approach provides the most compact-sized maps without details distortion (1.9 MB to store 57 m$^3$) among the state-of-the-art SLAM approaches.Comment: Accepted paper at IEEE Systems, Man, and Cybernetics 2022 (IEEE SMC 2022), IEEE copyrigh

Smoothing properties for the higher order nonlinear Schr\"{o}dinger equation with constant coefficients

We study local and global existence and smoothing properties for the initial value problem associated to a higher order nonlinear Schr\"odinger equation with constant coefficients which appears as a model for propagation of pulse in optical fiber

Approximating the vanishing capillarity limit of two-phase flow in multi-dimensional heterogeneous porous medium

International audienceNeglecting capillary pressure effects in two-phase flow models for porous media may lead to non-physical solutions: indeed, the physical solution is obtained as limit of the parabolic model with small but non-zero capillarity. In this paper, we propose and compare several numerical strategies designed specifically for approximating physically relevant solutions of the hyperbolic model with neglected capillarity, in the multi-dimensional case. It has been shown in [Andreianov&Canc'es, Comput. Geosci., 2013, to appear] that in the case of the one-dimensional Buckley-Leverett equation with distinct capillary pressure properties of adjacent rocks, the interface may impose an upper bound on the transmitted flux. This transmission condition may reflect the oil trapping phenomenon. We recall the theoretical results for the one-dimensional case which are used to motivate the construction of multi- dimensional finite volume schemes. We describe and compare a coupled scheme resulting as the limit of the scheme constructed in [Brenner & Canc'es & Hilhorst, HAL preprint no.00675681, 2012) and two IMplicit Pressure - Explicit Saturation (IMPES) schemes with different level of coupling

A theory of L1L^1-dissipative solvers for scalar conservation laws with discontinuous flux

We propose a general framework for the study of $L^1$ contractive semigroups of solutions to conservation laws with discontinuous flux. Developing the ideas of a number of preceding works we claim that the whole admissibility issue is reduced to the selection of a family of "elementary solutions", which are certain piecewise constant stationary weak solutions. We refer to such a family as a "germ". It is well known that (CL) admits many different $L^1$ contractive semigroups, some of which reflects different physical applications. We revisit a number of the existing admissibility (or entropy) conditions and identify the germs that underly these conditions. We devote specific attention to the anishing viscosity" germ, which is a way to express the "$\Gamma$-condition" of Diehl. For any given germ, we formulate "germ-based" admissibility conditions in the form of a trace condition on the flux discontinuity line $x=0$ (in the spirit of Vol'pert) and in the form of a family of global entropy inequalities (following Kruzhkov and Carrillo). We characterize those germs that lead to the $L^1$-contraction property for the associated admissible solutions. Our approach offers a streamlined and unifying perspective on many of the known entropy conditions, making it possible to recover earlier uniqueness results under weaker conditions than before, and to provide new results for other less studied problems. Several strategies for proving the existence of admissible solutions are discussed, and existence results are given for fluxes satisfying some additional conditions. These are based on convergence results either for the vanishing viscosity method (with standard viscosity or with specific viscosities "adapted" to the choice of a germ), or for specific germ-adapted finite volume schemes

Regularity of a kind of marginal functions in Hilbert spaces

We study well-posedness of some mathematical programming problem depending on a parameter that generalizes in a certain sense the metric projection onto a closed nonconvex set. We are interested in regularity of the set of minimizers as well as of the value function, which can be seen, on one hand, as the viscosity solution to a Hamilton-Jacobi equation, while, on the other, as the minimal time in some related optimal time control problem. The regularity includes both the Fréchet differentiability of the value function and the Hölder continuity of its (Fréchet) gradient

Very Singular Diffusion Equations-Second and Fourth Order Problems

This paper studies singular diffusion equations whose diffusion effect is so strong that the speed of evolution becomes a nonlocal quantity. Typical examples include the total variation flow as well as crystalline flow which are formally of second order. This paper includes fourth order models which are less studied compared with second order models. A typical example of this model is an H−1 gradient flow of total variation. It turns out that such a flow is quite different from the second order total variation flow. For example, we prove that the solution may instantaneously develop jump discontinuity for the fourth order total variation flow by giving an explicit example

Mathematical Models of Incompressible Fluids as Singular Limits of Complete Fluid Systems

A rigorous justification of several well-known mathematical models of incompressible fluid flows can be given in terms of singular limits of the scaled Navier-Stokes-Fourier system, where some of the characteristic numbers become small or large enough. We discuss the problem in the framework of global-in-time solutions for both the primitive and the target system. © 2010 Springer Basel AG