Existence and selection of solutions in the energy-variational framework with applications in fluid dynamics

We provide a novel existence result for energy-variational solutions to a general class of evolutionary partial differential equations. Compared to previous works on this solution concept, the generalization is mainly twofold: a relaxation of the assumptions on the regularity weight and the admissibility of energies with merely linear growth. We apply the abstract theory to the Euler–Korteweg system and to the equation for binormal curvature flow, which serve as examples that require the first and second generalization, respectively. Moreover, we discuss criteria that are suitable for the selection of particular energy-variational solutions in the possibly multi-valued solution set.

💡 Research Summary

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The paper develops a comprehensive theory for energy‑variational solutions of evolutionary partial differential equations, extending previous frameworks in two major directions. First, the authors relax the regularity weight (K) from a strong continuity requirement to merely weak‑* lower semicontinuity, allowing lower‑order terms in the operator (A) to be handled by compactness arguments. Second, they permit the energy functional (E) to have only linear growth, instead of the super‑linear growth assumed in earlier works.

The abstract setting considers Banach spaces (Y\hookrightarrow V) and an evolution equation (\partial_t U + A(t,U)=0) with initial data (U(0)=U_0). An energy‑variational solution is a pair ((U,E)) where (U) is the state variable and (E: