Abstract
We introduce the new concept of maximally dissipative solutions for a general class of isothermal GENERIC systems. Under certain assumptions, we show that maximally dissipative solutions are well-posed as long as the bigger class of dissipative solutions is non-empty. Applying this result to the Navier–Stokes and Euler equations, we infer global well-posedness of maximally dissipative solutions for these systems. The concept of maximally dissipative solutions coincides with the concept of weak solutions as long as the weak solutions inherits enough regularity to be unique.
Highlights
Page 3 of 21 1 the existence proof, and minimizers of functionals often exhibit additional regularity, or are more amenable for regularity estimates. As it is the case for the Ericksen– Leslie equations, we hope that the new concept of maximally dissipative solutions may inspire stable numerical schemes for the Navier–Stokes or Euler equations
Supposing the existence of dissipative solutions, we prove the well-posedness of maximally dissipative solutions under some general assumptions
We consider a system, which can be seen as an isothermal GENERIC system [14] and demonstrate the general scheme of dissipative solutions
Summary
This section is devoted to a general approach to dissipative solutions. We consider a system, which can be seen as an isothermal GENERIC system [14] (general equations for non-equilibrium reversible and irreversible coupling) and demonstrate the general scheme of dissipative solutions.
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