CRITICAL THRESHOLDS IN A CONVOLUTION MODEL FOR NONLINEAR CONSERVATION LAWS
CRITICAL THRESHOLDS IN A CONVOLUTION MODEL FOR NONLINEAR CONSERVATION LAWS
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Date
2001
Authors
LIU, HAILIANG
TADMOR, EITAN
Advisor
Citation
H. Liu & E. Tadmor (2001). Critical thresholds in a convolution model for nonlinear conservation laws. SIAM Journal on Mathematical Analysis 33 (2001), 930-945.
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Abstract
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.