Some Further Results for a One-dimensional Transport Equation with Nonlocal Flux
نویسنده
چکیده
Here, we rigorously prove some of the results obtained with asymp-totic methods for a modiied Burgers' equation, obtained by replacing the ux term in Burgers' equation considered in its hyperbolic form by the Hilbert transform. We show that, under the asumptions ju x j 1 C ?1=3 , jH(u x)j 1 C ?1=3 , and R 1 0 jju x jj 2 dt C, H(u) being the Hilbert transform of u and C a constant independent of time, the smallest scale of the solution of the equation is essentially 2=3 , being the viscosity coeecient. The analytical results are connrmed with numerical ones. We also show that the solution of the equation, in the limit of zero viscosity, has a weak limit in the space of functions of bounded variation.
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