Global dissipative solutions of the 3D Navier–Stokes and MHD equations

Abstract

For any nontrivial divergence‐free initial data in H12$H^\frac{1}{2}$ , 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,∞)$[0,\infty)$ . If the initial data is only L2$L^2$ , 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.