Global existence of large solutions to conservation laws with nonlocal dissipation-type terms

Abstract

We consider the global existence for large perturbation solutions of scalar conservation laws with nonlocal dissipative structures.At the very beginning, based on the Green function method and the Littlewood-Paley decomposition, we establish a new regularity criterion to improve the regularity of the solution. It is worth to point out that the key points in the regularity criterion are that $\|u\|_{L^\infty}<+\infty$ for $0\leq~s<1$ and $\|u\|_{C^\alpha}<+\infty$ for $s=1$. Thus, in the following part we devote ourselves to verify the two key points. For the subcritical case, we obtain the boundedness of solution in $L^p$ for any $p\in[2,~+\infty)$ by the maximum principle. Thanks to the nonlinear maximum principle, $C^\alpha$ boundedness is established for the critical case.Finally, the global existence of classical solutions to the scalar conservation with large initial data is obtained.