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The Euler equations for an inviscid incompressible 2-D fluid flow are given by

'quot(D .oper ?nu?,D .oper t) = _'grad(p), x .eltof 'Real^2, t > 0
'diverg(?nu?) = 0, ?nu?(x,0) = ?nu?;0(x)

where ?nu? = 'transp(?nu?;1,?nu;2) is the fluid velocity, p is the scalar pressure, 'quot(D .oper ?nu?,D .oper t) = 'ptderiv(?nu?,t) + 'oper(?nu? .dot 'Grad, ?nu?), and ?nu?;0 is an initial incompressible velocity field, i.e. 'diverg(?nu?;0) = 0.

In this paper, we study the detailed limiting behaviour of approximate solution sequences for 2-D Euler with vortex sheet initial data. A sequence of smooth velocity fields (?nu? .upidx ?epsilon?)(x,t) is an approximate solution sequence for 2-D Euler provided that the ?nu? is incompressible, i.e. 'diverg(?nu?) = 0, and satisfies the following properties:

  1. The velocity fields ?nu? .upidx ?epsilon? have uniformly bounded local kinetic energy, i.e.

    'max('integ('abs((?nu? .upidx ?epsilon?)(x,t))^2,x,'abs(x) <= R), 0 <= t <= T) <= C

    for any R, T > 0.

  2. The corresponding vorticity, ?omega? .upidx ?epsilon? = 'curl(?nu? .upidx ?epsilon?), is uniformly bounded in L .upidx 1, i.e.

    'max('integ('abs((?omega? .upidx ?epsilon?)(x,t)),x), 0 <= t <= T) <= C

    for any T > 0.

  3. The vortex field ?nu? .upidx ?epsilon? is weakly consistent with 2-D Euler, i.e. for all smooth test functions, ?phi? .eltof (C .upidx 'inf)('Real^2 .cartprod 'openopen(0,'inf)) with 'diverg(?phi?) = 0,

    'lim('integ('integ(?phi?;t .dot ?nu? .upidx ?epsilon? + 'matprod('Grad .oper ?phi?, ?nu? .upidx ?epsilon? .tensorprod ?nu? .upidx ?epsilon?), x), t), ?epsilon? .approach 0) = 0.

    Here ?nu? .tensorprod ?nu? = 'pt(v;i,v;j), 'Grad .oper ?phi? = 'ptderiv(?phi?;i,x;j), and 'matprod(A,B) denotes the matrix product 'Sum(a;i;j*b;i;j,'tuple(i,j)). We remark in passing...