A notion of L p -exact controllability is introduced for linear controlled (forward) stochastic differential equations, for which several sufficient conditions are established. Further, it is proved that the L p -exact controllability, the validity of an observability inequality for the adjoint equation, the solvability of an optimization problem, and the solvability of an L p -type norm optimal control problem are all equivalent.
In this paper, by invoking the appropriate decomposition of pressure to exploit the energy hidden in pressure, we present some new ε-regularity criteria for suitable weak solutions of the 3D Navier-Stokes equations at one scale: for any p, q ∈ [1, ∞] satisfyingThis is an improvement of corresponding results recently proved by Guevara and Phuc in [7, Calc. Var. 56:68, 2017]. As an application of these ε-regularity criteria, we improve the known upper box dimension of the possible interior singular set of suitable weak solutions of the Navier-Stokes system from 975/758(≈ 1.286) [16] to 2400/1903(≈ 1.261).MSC (2000): 35B65, 35D30, 76D05
In this paper, we derive regular criteria via pressure or gradient of the velocity in Lorentz spaces to the 3D Navier-Stokes equations. It is shown that a Leray-Hopf weak solution is regular on (0, T ] provided that either the norm Π L p,∞ (0,T ;L q,∞ (R 3 )) with 2/p + 3/q = 2 (3/2 < q < ∞) or ∇Π L p,∞ (0,T ;L q,∞ (R 3 )) with 2/p + 3/q = 3 (1 < q < ∞) is small. This gives an affirmative answer to a question proposed by Suzuki in [26, Remark 2.4, p.3850]. Moreover, regular conditions in terms of ∇u obtained here generalize known ones to allow the time direction to belong to Lorentz spaces. MSC(2000): 76D03, 76D05, 35B33, 35Q35
In this paper, we are concerned with the partial regularity of the suitable weak solutions to the fractional MHD equations in R n for n = 2, 3. In comparison with the work of the 3D fractional Navier-Stokes equations obtained by Tang and Yu in [24, Commun. Math. Phys. 334: 1455-1482, our results include their endpoint case α = 3/4 and the external force belongs to more general parabolic Morrey space. Moreover, we prove some interior regularity criteria just via the scaled mixed norm of the velocity for the suitable weak solutions to the fractional MHD equations. MSC(2000): 76D03, 76D05, 35B33, 35Q35
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