I have a problem understanding how both these statements can be true simultaneously:

Statement 1: (Corollary 3.1 of [1]). Let $v_1$ and $v_2$ be two smooth solutions to Eqs. 3.2 on $[0,T]$ with the same initial $v_0$ data and external force. Then $v_1=v_2$. Eqs. 3.2 are: Navier-Stokes eq. with external force; $\text{div} v_i =0$ and $v_i|_{t=0}=v_0$ for $i \in \{1,2\}$.

Statement 2: (Thm 1.2 in [2]) There exists a weak solution $u(x,t) \in L^2(\mathbb{T}^2 \times \mathbb{R})$ of the Euler equation (+ $\text{div}u=0$) (without external force) and $C>0$ s.t. $u(x,t) \equiv 0$ if $t \in \mathbb{R}\backslash [-C,C]$.

My problem: The proof of statement 1 also covers the case $\nu=0$, as one drops this term. Thus this statement should still hold for Euler-equation. Now I think I can drop the condition of smoothness in the hypothesis and only request that $v_1,v_2$ must be weak solutions of Navier-Stokes as well (more precisely the 'basic energy identity'). Accepting this I dont see how Statement 2 can ensure existence of such strikingly non-unique solutions of Euler eq. I understand that I need some continuity for Gronwalls inequality and the 'pulling out $\frac{\partial}{\partial t}$ from an integral but I would be surprised if this is the reason that both statements hold.

References: [1]: A.Majda, A.Bertozzi: Vorticity and Incompressible Flow, Cambridge.

[2]: A.Shnirelman: On the Nonuniqueness of Weak Solutions of the Euler Equations, Comm. on Pure & Appl. Math, Vol L, 1261-1286 (1997).

(Sorry for bad tags.)

  • 1
    $\begingroup$ Uniqueness of strong solution and uniqueness of weak solution do not go hand in hand in general. I don't have another example for discrepancy about uniqueness, but a classic example for discrepancy about existence is the eikonal equation. Consider for instance the ODE BVP $|u'|=1,u(0)=u(1)=0$. This has many weak solutions and no strong solutions. $\endgroup$ – Ian Jun 7 '17 at 13:37

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