All Questions

How can we compute the "N-wave" source-solution of the conservation law $$u_t + (u - u^2)_x = 0, $$ that is, the entropy solution of this conservation law with the initial data $u(0,\cdot) = ...

Let us consider the transport PDE $$ u^\epsilon_t + u^\epsilon_x= -\frac{1}{\epsilon} W'(u^\epsilon) $$ where $W$ is a double-well potential -- for example, $W(x)=\frac{1}{4}(x^2-1)^2$ so that the PDE ...

What is a concrete example of BV vector field $v$ with $\mathrm{div}\, v = 0$ that makes Ambrosio's theory of regular Lagrangian flow relevant? With concrete I mean that we can compute the flow ...

What is the difference between the notions of Regular Lagrangian flow Filippov solution Caratheodory solution of an ODE $\dot \Phi(t,x) = b(t,\Phi(t,x))$, with initial condition $\Phi(0,x) = x$, ...

Consider the transport equations $$ (1) \qquad \partial_t u + \operatorname{div}(bu) = 0$$ and $$ (2) \qquad \partial_t u + b \cdot \nabla u= 0$$ Can we define a notion of entropy solutions for (1) ...