23 research outputs found

    Theory and numerical approximations for a nonlinear 1+1 Dirac system

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    We consider a nonlinear Dirac system in one space dimension with periodic boundary conditions. First, we discuss questions on the existence and uniqueness of the solution. Then, we propose an implicit-explicit finite difference method for its approximation, proving optimal order a priori error estimates in various discrete norms and showing results from numerical experiments

    Global existence for a kinetic model of chemotaxis via dispersion and Strichartz estimates

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    We investigate further the existence of solutions to kinetic models of chemotaxis. These are nonlinear transport-scattering equations with a quadratic nonlinearity which have been used to describe the motion of bacteria since the 80's when experimental observations have shown they move by a series of 'run and tumble'. The existence of solutions has been obtained in several papers [Chalub et al. Monatsh. Math. 142, 123--141 (2004), Hwang et al. SIAM J. Math. Anal. 36, 1177--1199 (2005)] using direct and strong dispersive effects. Here, we use the weak dispersion estimates of [Castella et al. C. R. Acad. Sci. Paris 322, 535--540 (1996)] to prove global existence in various situations depending on the turning kernel. In the most difficult cases, where both the velocities before and after tumbling appear, with the known methods, only Strichartz estimates can give a result, with a smallness assumption.Comment: 19 pages, 2 figure

    Critical mass phenomenon for a chemotaxis kinetic model with spherically symmetric initial data

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    The goal of this paper is to exhibit a critical mass phenomenon occuring in a model for cell self-organization via chemotaxis. The very well known dichotomy arising in the behavior of the macroscopic Keller-Segel system is derived at the kinetic level, being closer to microscopic features. Indeed, under the assumption of spherical symmetry, we prove that solutions with initial data of large mass blow-up in finite time, whereas solutions with initial data of small mass do not. Blow-up is the consequence of a virial identity and the existence part is derived from a comparison argument. Spherical symmetry is crucial within the two approaches. We also briefly investigate the drift-diffusion limit of such a kinetic model. We recover partially at the limit the Keller-Segel criterion for blow-up, thus arguing in favour of a global link between the two models

    The one-dimensional Keller-Segel model with fractional diffusion of cells

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    12 pagesWe investigate the one-dimensional Keller-Segel model where the diffusion is replaced by a non-local operator, namely the fractional diffusion with exponent 0<α20<\alpha\leq 2. We prove some features related to the classical two-dimensional Keller-Segel system: blow-up may or may not occur depending on the initial data. More precisely a singularity appears in finite time when α<1\alpha<1 and the initial configuration of cells is sufficiently concentrated. On the opposite, global existence holds true for α1\alpha\leq1 if the initial density is small enough in the sense of the L1/αL^{1/\alpha} norm

    Global well-posedness for the massless cubic Dirac equation

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    We show that the cubic Dirac equation with zero mass is globally well-posed for small data in the scale invariant space H^{\frac{n-1}{2}}(R^n) for n=2, 3. The proof proceeds by using the Fierz identities to rewrite the equation in a form where the null structure of the system is readily apparent. This null structure is then exploited via bilinear estimates in spaces based on the null frame spaces of Tataru. We hope that the spaces and estimates used here can be applied to other nonlinear Dirac equations in the scale invariant setting. Our work complements recent results of Bejenaru-Herr who proved a similar result for n=3 in the massive case.Comment: Error in the statement and proof of Theorem 4.1 has been correcte

    The one-dimensional Keller-Segel model with fractional diffusion of cells

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    We investigate the one-dimensional Keller-Segel model where the diffusion is replaced by a non-local operator, namely the fractional diffusion with exponent 0<α20<\alpha\leq 2. We prove some features related to the classical two-dimensional Keller-Segel system: blow-up may or may not occur depending on the initial data. More precisely a singularity appears in finite time when α<1\alpha<1 and the initial configuration of cells is sufficiently concentrated. On the opposite, global existence holds true for α1\alpha\leq1 if the initial density is small enough in the sense of the L1/αL^{1/\alpha} norm.Comment: 12 page

    Mathematical description of bacterial traveling pulses

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    The Keller-Segel system has been widely proposed as a model for bacterial waves driven by chemotactic processes. Current experiments on E. coli have shown precise structure of traveling pulses. We present here an alternative mathematical description of traveling pulses at a macroscopic scale. This modeling task is complemented with numerical simulations in accordance with the experimental observations. Our model is derived from an accurate kinetic description of the mesoscopic run-and-tumble process performed by bacteria. This model can account for recent experimental observations with E. coli. Qualitative agreements include the asymmetry of the pulse and transition in the collective behaviour (clustered motion versus dispersion). In addition we can capture quantitatively the main characteristics of the pulse such as the speed and the relative size of tails. This work opens several experimental and theoretical perspectives. Coefficients at the macroscopic level are derived from considerations at the cellular scale. For instance the stiffness of the signal integration process turns out to have a strong effect on collective motion. Furthermore the bottom-up scaling allows to perform preliminary mathematical analysis and write efficient numerical schemes. This model is intended as a predictive tool for the investigation of bacterial collective motion

    A note on the Chern-Simons-Dirac equations in the Coulomb gauge

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    We prove that the Chern-Simons-Dirac equations in the Coulomb gauge are locally well-posed from initial data in H^s with s > 1/4 . To study nonlinear Wave or Dirac equations at this regularity generally requires the presence of null structure. The novel point here is that we make no use of the null structure of the system. Instead we exploit the additional elliptic structure in the Coulomb gauge together with the bilinear Strichartz estimates of Klainerman-Tataru.Comment: Preliminary version. Final version will appear in Discrete and Continuous Dynamical Systems - Series
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