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Global dissipative solutions of the 3D Naiver-Stokes and MHD equations
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abstract
For any divergence free initial data in $H^\frac12$, we prove the existence of infinitely many dissipative solutions to both the 3D Navier-Stokes and MHD equations, whose energy profiles are continuous and decreasing on $[0,\infty)$. If the initial data is only $L^2$, our construction yields infinitely many solutions with continuous energy, but not necessarily decreasing. Our theorem does not hold in the case of zero viscosity as this would violate the weak-strong uniqueness principle due to Lions. This was achieved by designing a convex integration scheme that takes advantage of the dissipative term.
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Cited by 1 Pith paper
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Sharp non-uniqueness for the Navier-Stokes equations in scaling critical spaces
Uniqueness of mild Navier–Stokes solutions in critical Besov spaces holds exactly for p<n (any q) or p=n (q≤2), and fails — even for zero initial data — for p=n, q>2 and for p>n.
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