1. (i) Let h = h(r) be a harmonic function on R, n 2, and...

 

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1. (i) Let h = h(r) be a harmonic function on R", n ≥ 2, and T = T(t) be a smooth function of time. Then, show that the vector field v(r,t)= Vh(r)T(t) is a solution of the Euler equations. (ii) Find the pressure p = p(x, t) in terms of h(r) and T(t) in the problem (i). (iii) Given T > 0, using the result of (i), construct an initial velocity vo(r) such that for a solution v(x, t) of the Euler equations with v(x, 0) = vo(r) we have Vo = 2. Let (v,p) is a solution of the Euler equations in R" with the initial velocity vo(r), and let X (a, t) be defined as the solution of the ordinary differential equations: Show that lim sup Vv(x, t)| = +∞. t-T TER¹ OX(a,t) Ət = v(X(a, t),t); X(a,0) = a. = X(a,t) = a + w(a)t — f" ["Vp(X(a. 17), 7) drds. 1. (i) Let h = h(r) be a harmonic function on R", n ≥ 2, and T = T(t) be a smooth function of time. Then, show that the vector field v(r,t)= Vh(r)T(t) is a solution of the Euler equations. (ii) Find the pressure p = p(x, t) in terms of h(r) and T(t) in the problem (i). (iii) Given T > 0, using the result of (i), construct an initial velocity vo(r) such that for a solution v(x, t) of the Euler equations with v(x, 0) = vo(r) we have Vo = 2. Let (v,p) is a solution of the Euler equations in R" with the initial velocity vo(r), and let X (a, t) be defined as the solution of the ordinary differential equations: Show that lim sup Vv(x, t)| = +∞. t-T TER¹ OX(a,t) Ət = v(X(a, t),t); X(a,0) = a. = X(a,t) = a + w(a)t — f" ["Vp(X(a. 17), 7) drds.

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