Methods of constucting unbounded discontinuous solutions of scalar conservation laws

In a strip $\Pi_T=\{(t,\,x)\mid t \in (0,\,T),\ x \in \mathbb{R}\}$, where $0<T\le+\infty$, we consider the Cauchy problem
\begin{equation}\label{Cau} u_t+(f(u))_x=0, \ (t,\,x)\in\Pi_T, \qquad u|_{t=0}=u_0(x),\ x\in\mathbb{R}. \end{equation}
We suppose that the flux function is smooth, $f \in C^1(\mathbb{R})$, and the initial condition is unbounded, but locally bounded, $u_0\in L^\infty_\mathrm{loc}(\mathbb{R})$.
In the talk we discuss methods of constucting piecewise smooth entropy solutions of this problem.
In the first part of the talk we consider power flux function of the form $f(u)=\frac1\alpha |u|^{\alpha-1}u, \;\alpha>1$, and initial conditions either of power type, $u_0(x) = |x|^\beta, \;\beta(\alpha-1)>1$, or exponential, $u_0(x) = \exp(-x)$. Solutions of the Cauchy problem \eqref{Cau} are constructed on the basis of a group of symmetry associated with the problem. Symmetry allows us to reduce the original partial differential equation to an ordinary differential equation. We allow the solutions of this ODE to have points of discontinuity, at which we impose certain constraints dictated by the definition of the generalized entropy solution. This approach to the construction of solutions of such problems has been proposed by E. Yu. Panov.
In the second part of the talk we consider an odd flux function that has a single point of inflexion at zero. We propose a method for constructing sign-alternating discontinuous entropy solutions of the problem \eqref{Cau}, based on the Legendre transform.
The report is based on th articles.

• L. V. Gargyants, A. Yu. Goritsky, E. Yu. Panov, “Constructing unbounded discontinuous solutions of scalar conservation laws using the Legendre transform”, Sb. Math., 212:4 (2021), 475-–489.
• A. Yu. Goritsky, L. V. Gargyants, “Nonuniqueness of unbounded solutions of the Cauchy problem for scalar conservation laws”, J. Math. Sci. (N.Y.) 244:2 (2020), 183–197.