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Global dissipative solutions of the 3D Naiver-Stokes and MHD equations (2503.05692v1)
Published 7 Mar 2025 in math.AP and physics.flu-dyn
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 $L2$, 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.