 # Let |x| denote the euclidean norm of a vector x € Rn, meaning ...

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Let |x| denote the euclidean norm of a vector x € Rn, meaning that |x²² = Xixi (in Einstein notation). Let 1/21/2 denote the L² norm of a function u(x) on a specified domain n C Rd, meaning that : n The L² norm of a vector function u(x) is defined as ||ulll2, so 11ullz = / W Ui (x) 2 dx. 5 i The L² norm of the tensor Vu, for example, is defined as = W Oxj Oui (x) 2 dx. si,j Problem 1. Consider the incompressible Navier-Stokes equations (NSE) with no forcing, ut +u. Vu = -1/2 Vp + V u, P V.u=0, on a 3D domain where boundary conditions are either periodic (i.e. no boundaries) or no-slip (i.e. u = 0). . (a) Consider the total kinetic energy, K(t) = Show that d = -V ||Vullz. (b) What can you say about K(t) relative to K(0)? (c) What can you say about K ((t) in the incompressible Euler equations with no forcing? Problem 2. Consider the vorticity vector w = V X u. Note that the vector u must have 3 components for the cross product to be defined. (a) Show that if u satisfies the incompressible NSE with no forcing, then Wt + u. Vw = w.Vutusw. See the vector calculus identities (E1.1) and (E1.2) on page 22 of Doering & Gibbon. (b) Consider the total enstrophy, E(t) = 11w/12. Show that with periodic and/or no-slip boundary conditions, dt de = 2 / 5 w. Vu wdx - 2vllVwll2. (c) Suppose the flow is two dimensional. For concreteness suppose u = (u, U, 0) and ou/8z = 0. Show that instead of the vorticity vector we only need to consider a scalar vorticity. Give the PDE governing this scalar vorticity. (d) In 2D with periodic and/or no-slip boundaries, what can you say about €(t) relative to &(0)? (e) In the 2D incompressible Euler equations with no forcing and periodic and/or no-slip boundaries, what can you say about €(t) relative to €(0)? (f) For various boundary conditions on u including periodic and no-slip, the equality 11w/12 = (Vullz holds in 3D. Show by explicit calculation that it holds in 2D. (Notice this means that €(t) determines 1K(t().) Problem 3. Consider the two-dimensional NSE with x = (x,y) and u = (u,v). (a) Explain why there must exist a stream function 4 (x, y, t) such that u = -4y and U = x. (b) Show that, at all positions and times, u points along level sets of the stream function. (c) Derive a relation between 4 and the scalar vorticity W. (d) Write the scalar vorticity equation in terms of w and 4. To compress the notation of the u. Vw term, use the Poisson bracket defined by {f, g} = fxgy - fygx.

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