Asymptotic Analysis - Volume 129, issue 2

Authors: Hunter, John K. | Moreno-Vasquez, Ryan C. | Shu, Jingyang | Zhang, Qingtian

Article Type: Research Article

Abstract: This paper proves that the motion of small-slope vorticity fronts in the two-dimensional incompressible Euler equations is approximated on cubically nonlinear timescales by a Burgers–Hilbert equation derived by Biello and Hunter (2010) using formal asymptotic expansions. The proof uses a modified energy method to show that the contour dynamics equations for vorticity fronts in the Euler equations and the Burgers–Hilbert equation are both approximated by the same cubically nonlinear asymptotic equation. The contour dynamics equations for Euler vorticity fronts are also derived.

Keywords: Incompressible Euler equations, vorticity fronts, asymptotic equations, modified energy

DOI: 10.3233/ASY-211724

Citation: Asymptotic Analysis, vol. 129, no. 2, pp. 141-177, 2022

Authors: Sheng, Shouqiong | Shao, Zhiqiang

Article Type: Research Article

Abstract: In this paper, we study the phenomenon of concentration and the formation of delta shock wave in vanishing adiabatic exponent limit of Riemann solutions to the Aw–Rascle traffic model. It is proved that as the adiabatic exponent vanishes, the limit of solutions tends to a special delta-shock rather than the classical one to the zero pressure gas dynamics. In order to further study this problem, we consider a perturbed Aw–Rascle model and proceed to investigate the limits of solutions. We rigorously proved that, as the adiabatic exponent tends to one, any Riemann solution containing two shock waves tends to a …delta-shock to the zero pressure gas dynamics in the distribution sense. Moreover, some representative numerical simulations are exhibited to confirm the theoretical analysis. Show more

Keywords: Aw–Rascle traffic model, Riemann solutions, delta shock wave, vanishing adiabatic exponent limit, zero pressure gas dynamics, weighted Dirac-measure, numerical simulations

DOI: 10.3233/ASY-211725

Citation: Asymptotic Analysis, vol. 129, no. 2, pp. 179-213, 2022

Authors: Benedetto, Dario | Caglioti, Emanuele | Rossi, Stefano

Article Type: Research Article

Abstract: We analyze the analytic Landau damping problem for the Vlasov-HMF equation, by fixing the asymptotic behavior of the solution. We use a new method for this “scattering problem”, closer to the one used for the Cauchy problem. In this way we are able to compare the two results, emphasizing the different influence of the plasma echoes in the two approaches. In particular, we prove a non-perturbative result for the scattering problem.

Keywords: Landau damping, HMF model, plasma echoes

DOI: 10.3233/ASY-211726

Citation: Asymptotic Analysis, vol. 129, no. 2, pp. 215-238, 2022

Authors: Nilsson, Dag

Article Type: Research Article

Abstract: For 0 < α < 1 , N ⩾ 2 and with initial data ‖ u 0 ‖ H N + α 2 = ε , sufficiently small, we show that the existence time for solutions of the fractional BBM equation ∂ t u + ∂ x u + u ∂ x u + | D | α ∂ t u = 0 …, can be extended from the hyperbolic existence time 1 ε to 1 ε 2 . For the proof we use a quasilinear modified energy method, based on a normal form transformation as in Hunter, Ifrim, Tataru, Wong (Proc. Am. Math. Soc. , 143 (8) (2015) 3407–3412). Show more

Keywords: fBBM, enhanced life span, dispersive equations, nonlinear equations

DOI: 10.3233/ASY-211727

Citation: Asymptotic Analysis, vol. 129, no. 2, pp. 239-259, 2022

Authors: Santos Junior, João R. | Siciliano, Gaetano

Article Type: Research Article

Abstract: We consider a boundary value problem in a bounded domain involving a degenerate operator of the form L ( u ) = − div ( a ( x ) ∇ u ) and a suitable nonlinearity f . The function a vanishes on smooth 1-codimensional submanifolds of Ω where it is not allowed to be C 2 . By using weighted Sobolev spaces we are still able to find existence of solutions which vanish, in the trace sense, on the set where a …vanishes. Show more

Keywords: Elliptic equations, degenerate operators, vanishing solutions

DOI: 10.3233/ASY-211728

Citation: Asymptotic Analysis, vol. 129, no. 2, pp. 261-272, 2022

Authors: Casado-Díaz, Juan

Article Type: Research Article

Abstract: We consider the homogenization of a non-linear elliptic system of two equations related to some models in chemotaxis and flows in porous media. One of the equations contains a convection term where the transport vector is only in L 2 and a right-hand side which is only in L 1 . This makes it necessary to deal with entropy or renormalized solutions. The existence of solutions for this system has been proved in reference (Comm. Partial Differential Equations 45 (7) (2020 ) 690–713). Here, we prove its stability …by homogenization and that the correctors corresponding to the linear diffusion terms still provide a corrector for the solutions of the non-linear system. Show more

Keywords: Nonlinear elliptic system, convection term, homogenization, entropy solutions, H-convergence, corrector

DOI: 10.3233/ASY-211729

Citation: Asymptotic Analysis, vol. 129, no. 2, pp. 273-288, 2022