Mathematics

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    CRITICAL THRESHOLDS IN A CONVOLUTION MODEL FOR NONLINEAR CONSERVATION LAWS
    (Copyright: Society for Industrial and Applied Mathematics, 2001) LIU, HAILIANG; TADMOR, EITAN
    In this work we consider a convolution model for nonlinear conservation laws.Due to the delicate balance between the nonlinear convection and the nonlocal forcing, this model allows for narrower shock layers than those in the viscous Burgers’ equation and yet exhibits the conditional finite time breakdown as in the damped Burgers’ equation.W e show the critical threshold phenomenon by presenting a lower threshold for the breakdown of the solutions and an upper threshold for the global existence of the smooth solution.The threshold condition depends only on the relative size of the minimum slope of the initial velocity and its maximal variation.W e show the exact blow-up rate when the slope of the initial profile is below the lower threshold.W e further prove the L1 stability of the smooth shock profile, provided the slope of the initial profile is above the critical threshold.
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    Stability analysis of finite-difference, pseudospectral and Fourier-Galerkin approximations for time-dependent problems
    (Copyright: Society for Industrial and Applied Mathematics, 1987-12) Tadmor, Eitan
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    Stability analysis of spectral methods for hyperbolic initial-boundary value systems
    (Copyright: Society for Industrial and Applied Mathematics, 1987-04) Gottlieb, David; Lustman, Liviu; Tadmor, Eitan
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    Convergence of spectral methods for hyperbolic initial-boundary value systems
    (Copyright: Society for Industrial and Applied Mathematics, 1987-06) Gottlieb, David; Lustman, Liviu; Tadmor, Eitan
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    Convergence of spectral methods for hyperbolic initial-boundary value systems
    (Copyright: Society for Industrial and Applied Mathematics, 1987-06) Gottlieb, David; Lustman, Liviu; Tadmor, Eitan
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    THE WELL-POSEDNESS OF THE KURAMOTO-SIVASHINSKY EQUATION
    (copyright: Society for Industrial and Applied Mathematics, 1986-07) Tadmor, Eitan
    The Kuramoto-Sivashinsky equation arises in a variety of applications, among which are modeling reaction-diffusion systems, flame-propagation and viscous flow problems. It is considered here, as a prototype to the larger class of generalized Burgers equations: those consist of quadratic nonlinearity and arbitrary linear parabolic part. We show that such equations are well-posed, thus admitting a unique smooth solution, continuously dependent on its initial data. As an attractive alternative to standard energy methods, existence and stability are derived in this case, by "patching" in the large short time solutions without "loss of derivatives".