**Title:** The boundary Riemann solver coming from the real vanishing viscosity approximation

**Speaker:** Laura Spinolo

**Speaker Info:** SISSA ( Italy)

**Brief Description:**

**Special Note**:

**Abstract:**

The seminar will deal with the parabolic approximation \begin{equation*} \left\{ \begin{array}{lll} u^{\varepsilon}_t + {A} \big( u^{\varepsilon}, \, \varepsilon u^{\varepsilon}_x \big) = \varepsilon {B}(u^{\varepsilon} ) u^{\varepsilon}_x \qquad u^{\varepsilon} \in \mathbb{R}^N\\ u^{\varepsilon} (t, \, 0) \equiv \bar{u}_b \\ u^{\varepsilon} (0, \, x) \equiv \bar{u}_0 \\ \end{array} \right. \end{equation*} of an hyperbolic boundary Riemann problem. The conservative case is, in particular, included in the previous formulation.The main result is the complete characterization of the boundary Riemann solver induced in the hyperbolic limit when the difference between the boundary and the initial datum is small.

The hypotheses imposed on the matrices ${A}$ and ${B}$ are essentially those introduced by Kawashima. In particular, the boundary characteristic case is allowed, i.e. one eigenvalue of $A$ is allowed to be close to zero. Moreover, no hypothesis of invertibility is made on the viscosity matrix ${B}$.

It is also assumed that the approximating solutions $u^{\varepsilon}$ converge in $L^1_{loc}$ to a unique limit, which is supposed to depend continuously in $L^1$ with respect to the initial and the boundary data.

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