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Register a new userA vanishing dynamic capillarity limit equation with discontinuous flux
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We prove existence and uniqueness of a solution to the Cauchy problem corresponding to the equation begin{equation*} begin{cases} partial_t u_{varepsilon,delta} +mathrm{div} {mathfrak f}_{varepsilon,delta}({bf x}, u_{varepsilon,delta})=varepsilon Delta u_{varepsilon,delta}+delta(varepsilon) partial_t Delta u_{varepsilon,delta}, {bf x} in M, tgeq 0 u|_{t=0}=u_0({bf x}). end{cases} end{equation*} Here, ${mathfrak f}_{varepsilon,delta}$ and $u_0$ are smooth functions while $varepsilon$ and $delta=delta(varepsilon)$ are fixed constants. Assuming ${mathfrak f}_{varepsilon,delta} to {mathfrak f} in L^p( mathbb{R}^dtimes mathbb{R};mathbb{R}^d)$ for some $1<p<infty$, strongly as $varepsilonto 0$, we prove that, under an appropriate relationship between $varepsilon$ and $delta(varepsilon)$ depending on the regularity of the flux ${mathfrak f}$, the sequence of solutions $(u_{varepsilon,delta})$ strongly converges in $L^1_{loc}(mathbb{R}^+times mathbb{R}^d)$ towards a solution to the conservation law $$ partial_t u +mathrm{div} {mathfrak f}({bf x}, u)=0. $$ The main tools employed in the proof are the Leray-Schauder fixed point theorem for the first part and reduction to the kinetic formulation combined with recent results in the velocity averaging theory for the second.
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