Abstract
We consider a sequence of approximate solutions to the compressible Euler system admitting uniform energy bounds and/or satisfying the relevant field equations modulo an error vanishing in the asymptotic limit. We show that such a sequence either (i) converges strongly in the energy norm, or (ii) the limit is not a weak solution of the associated Euler system. This is in sharp contrast to the incompressible case, where (oscillatory) approximate solutions may converge weakly to solutions of the Euler system. Our approach leans on identifying a system of differential equations satisfied by the associated turbulent defect measure and showing that it only has a trivial solution.
Highlights
In [28, Section 4], Greengard and Thomann constructed a sequence {vn}∞ n=1 of exact solutions to the incompressible Euler system in R2, compactly supported in the space variable, and converging weakly to the velocity field v = 0
We consider a sequence of approximate solutions to the compressible Euler system admitting uniform energy bounds and/or satisfying the relevant field equations modulo an error vanishing in the asymptotic limit
In the light of the recent results [10,11,12,13] indicating essential ill–posedness of the compressible Euler system, the vanishing viscosity limit might be seen as a sound selection criterion to identify the physically relevant solutions of systems describing inviscid fluids, this can be still arguable in view of the examples collected in the recent survey by Buckmaster and Vicol [6] and Constantin and Vicol [14]
Summary
The incompressible setting was further studied in space dimension two and for vortex sheet initial data by DiPerna and Majda [17,18] and Greengard and Thomann [28] Their results show that the set, where the approximate solutions do not converge strongly is either empty or its projection on the time axis is of positive measure. Strict convexity of E is nothing other than a formulation of the principle of thermodynamic stability, where the relevant phase variables are the density , the momentum m, and the total entropy S, cf Bechtel, Rooney, and Forrest [4] We consider both the full Euler system and its isentropic variant.
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