ON THE TRANSPORT EQUATIONS WITH SINGULAR/REGULAR NONLOCAL VELOCITIES

초록

We consider the Cauchy problem for the transport equation theta(iota) + nu . del theta = 0 on R-N x [0, infinity), where nu = -(A(alpha)R(1)theta, ... ,A alpha R-N theta) with A = (-Delta)(1/2) and {Rj}(j=1)(N) the Riesz transforms. For alpha is an element of (0, 1] ( singular velocities) we show the local well-posedness in H-s(R-N) for all s > N/2 + 2, s is an element of R, while for a is an element of [-1, 0] ( regular velocities) we prove local well-posedness in the lower order Sobolev spaces H-s(R-N), s > N/2 + 1, and finite time blow-up criterion. In the case alpha is an element of (-N, -1) we prove the global well- posedness in H-s(R-N), s > N/2 + 1, for nonnegative initial data. We also consider the Cauchy problem for a similar transport equation with the velocity given by convolution of theta by even kernels, nu = -(Lambda(alpha)theta, Lambda(R1R2)-R-alpha theta, ... , Lambda(R1RN)-R-alpha theta). For alpha is an element of [-1, 0] the problem is locally well-posed in H-s(R-N), s > N/2 + 1, and we prove the blow-up criterion. In the case alpha is an element of (-N, -1) we obtain the global well-posedness in H-s(R-N), s > N/2 + 1, for single signed initial data.

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