Quasi-potentials of the Entropy Functionals for Scalar Conservation Laws

where t ∈ [0, T ] for some T > 0, x ∈ T (the one-dimensional torus), and subscripts denote partial derivatives. Equation (1.1) does not admit in general classical solutions for the associated Cauchy problem, even if the initial datum is smooth. On the other hand, if f is non-linear, there exist in general infinitely many weak solutions. An admissibility condition, the so-called entropic condition, is then required to recover uniqueness for the Cauchy problem in the weak sense [6]. The unique solution satisfying such a condition is called the Kruzkhov solution. A classical result [6, Chap. 6.3] states that the Kruzkhov solution can be obtained as limit for ε ↓ 0 of the solution uε to the Cauchy problem associated with the equation