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تسجيل مستخدم جديدA microscopic model for a one parameter class of fractional laplacians with dirichlet boundary conditions
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We prove the hydrodynamic limit for the symmetric exclusion process with long jumps given by a mean zero probability transition rate with infinite variance and in contact with infinitely many reservoirs with density $alpha$ at the left of the system and $beta$ at the right of the system. The strength of the reservoirs is ruled by $kappa$N --$theta$ > 0. Here N is the size of the system, $kappa$ > 0 and $theta$ $in$. Our results are valid for $theta$ $le$ 0. For $theta$ = 0, we obtain a collection of fractional reaction-diffusion equations indexed by the parameter $kappa$ and with Dirichlet boundary conditions. Their solutions also depend on $kappa$. For $theta$ < 0, the hydrodynamic equation corresponds to a reaction equation with Dirichlet boundary conditions. The case $theta$ > 0 is still open. For that reason we also analyze the convergence of the unique weak solution of the equation in the case $theta$ = 0 when we send the parameter $kappa$ to zero. Indeed, we conjecture that the limiting profile when $kappa$ $rightarrow$ 0 is the one that we should obtain when taking small values of $theta$ > 0.
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