### Browsing by Author "LIU, HAILIANG"

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Item CRITICAL THRESHOLDS IN 2D RESTRICTED EULER–POISSON EQUATIONS(Copyright: Society for Industrial and Applied Mathematics, 2003) LIU, HAILIANG; TADMOR, EITANWe provide a complete description of the critical threshold phenomenon for the twodimensional localized Euler–Poisson equations, introduced by the authors in [Comm. Math. Phys., 228 (2002), pp. 435–466]. Here, the questions of global regularity vs. finite-time breakdown for the two-dimensional (2D) restricted Euler–Poisson solutions are classified in terms of precise explicit formulae, describing a remarkable variety of critical threshold surfaces of initial configurations. In particular, it is shown that the 2D critical thresholds depend on the relative sizes of three quantities: the initial density, the initial divergence, and the initial spectral gap, that is, the difference between the two eigenvalues of the 2 × 2 initial velocity gradient.Item CRITICAL THRESHOLDS IN A CONVOLUTION MODEL FOR NONLINEAR CONSERVATION LAWS(Copyright: Society for Industrial and Applied Mathematics, 2001) LIU, HAILIANG; TADMOR, EITANIn 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.