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A system of the Hamilton--Jacobi and the continuity equations in the vanishing viscosity limit

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  • We study the following system of the viscous Hamilton--Jacobi and the continuity equations in the limit as $\varepsilon \downarrow 0$:

    $ S^\varepsilon_t+\frac{1}{2}|D S^\varepsilon|^2+V(x)-\varepsilon\Delta S^\varepsilon =0$ in $Q_T$, $S^\varepsilon(0,x)=S_0(x)$ in $R^n;$

    $ \rho^\varepsilon_t+$ Div$(\rho^\varepsilon D S^\varepsilon)=0$ in $Q_T$, $\rho^\varepsilon(0,x)=\rho_0(x)$ in $R^n$.

    Here $Q_T=(0,T]\times R^n$. The potential $V$ and the initial function $S_0$ are allowed to grow quadratically while $\rho_0$ is a Borel measure. The paper justifies and describes the vanishing viscosity transition to the corresponding inviscid system. The notion of weak solution employed for the inviscid system is that of a viscosity--measure solution $(S,\rho)$.

    Mathematics Subject Classification: 35F25, 35L67, 35R05.


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