Non-classical generalized solutions for some hyperbolic systems (Q2747391)

The authors consider some hyperbolic systems of conservation laws whose solutions are not usual functions but measures (and can include so-called \(\delta\)-shock waves). In particular the authors study the Cauchy problem NEWLINE\[NEWLINE u_t+f(u)_x=0, \quad v_t+(g(u)v)_x=0, \quad (u,v)|_{t=0}=(u_0,v_0),NEWLINE\]NEWLINE and using a new notion of generalized solutions they introduce methods which allow to prove existence and uniqueness of generalized solutions. These results also include the important case of gas dynamical equations with zero pressure. Then the authors expand their methods to some two-dimensional problems and prove existence and uniqueness of generalized solutions to the problem NEWLINE\[NEWLINE u_t+(u^2/2)_x=0, \quad v_t+(v^2/2)_y=0, \quad \rho_t+(\rho u)_x+(\rho v)_y=0, NEWLINE\]NEWLINE NEWLINE\[NEWLINE u(0,x,y)=u_0(x), \quad v(0,x,y)=v_0(y), \quad \rho(0,x,y)=\rho_0(x,y). NEWLINE\]NEWLINE Besides they study the Riemann problem for simplified Euler equations NEWLINE\[NEWLINE u_t+(u^2)_x+(uv)_y=0, \quad v_t+(uv)_x+(v^2)_y=0,NEWLINE\]NEWLINE where the initial data have two different constant states separated by an initial discontinuity, and construct solutions of this problem. Some new phenomena of 2D solutions are also discussed.