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We are concerned with a new solution formula and its applications to theanalysis of properties of entropy solutions of the Cauchy problem forone-dimensional scalar hyperbolic conservation laws, wherein the flux functionsexhibit convexity degeneracy and the initial data are in $L^\infty$. We firstintroduce/validate the novel formula for entropy solutions for the Cauchyproblem, which generalizes the Lax-Oleinik formula. Then, by employing thisformula, we obtain a series of fine properties of entropy solutions anddiscover several new structures and phenomena, which include: (i) Series ofresults on the fine structures of entropy solutions, especially including thenew criteria for all six types of initial waves for the Cauchy problem, the newstructures of entropy solutions inside the backward characteristic triangle,and the new features of the formation and development of shocks such as allfive types of continuous shock generation points, along with their criteria andthe optimal regularities of the corresponding resulting shocks; (ii) Series ofresults on the global structures of entropy solutions, including the four newinvariants of entropy solutions, the new criteria for the locations and speedsof divides, and the exact determination of the global structures of entropysolutions; (iii) Series of new results on the asymptotic behaviors of entropysolutions, including the asymptotic profiles and decay rates of entropysolutions for initial data in $L^\infty$, respectively in the $L^\infty$--normand the $L^p_{{\rm loc}}$--norm. Through these results above, we obtain theglobal dynamic patterns of entropy solutions for scalar hyperbolic conservationlaws with the flux functions satisfying (1.3) and general initial data in$L^\infty$. Moreover, the new solution formula is also extended to more generalscalar hyperbolic conservation laws.

New Formula for Entropy Solutions for Scalar Hyperbolic Conservation  Laws with Flux Functions of Convexity Degeneracy and Global Dynamic Patterns  of Solutions