Regularity, Decay, Differentiability for Solutions to Conservation Laws and Structural Properties for Conservation Laws with Discontinuous Flux

In this thesis we study different problems related to the theory of conservation laws. Part I is composed of four chapters and deals with problems of regularity and decay of solutions, as well as differentiability properties of the solution operator. In the first chapter we introduce a Lagrangian representation for multidimensional scalar balance laws, in the framework of solutions with finite entropy production. We then use the representation to prove that, in the one-dimensional case and for a class of genuinely nonlinear $2\times2$ systems of conservation laws, including the isentropic system of gas dynamics with exponent $\gamma = 3$, the entropy dissipation measures are concentrated on a 1-rectifiable set. Moreover, regularity results are proved for the isentropic system. In chapter 2 we consider $2\times2$ systems of conservation laws. We observe that bounded vanishing viscosity solutions of $2\times 2$ systems obtained with the compensated compactness method satisfy a pair of (nonlocal) kinetic equations, and we use it to obtain a dispersive estimate in the case of genuinely nonlinear systems. In the third chapter we consider the problem of endowing the semigroup operator associated to a scalar conservation law with a differential structure. We prove that perturbations satisfy a continuity equation, and we observe that this is not enough to define a duality with integral functionals. We then introduce a finer framework, which is the correct one for computing variations of this type of functionals. In the last chapter of Part I we introduce a class of intermediate domains $\mathcal P_\alpha$, $0 < \alpha < 1$, lying between $\mathbf L^\infty$ and $BV$ for which the $BV$ norm of solutions decays like $t^{-\alpha}$. A key ingredient of the analysis is a ``Fourier-type" decomposition of functions of $\mc P_\alpha$ into components which oscillate more and more rapidly. The second part focuses on various aspects of scalar conservation laws with discontinuous flux. We introduce a notion of backward operator, and we completely classify the attainable states at a positive time $T > 0$. We also prove new regularity and stability properties of solutions.