# Mathematical Methods in the Applied Sciences

Mathematical Methods in the Applied Sciences > 33 > 11 > 1350 - 1355

Mathematical Methods in the Applied Sciences > 34 > 3 > 360 - 372

Mathematical Methods in the Applied Sciences > 34 > 16 > 2060 - 2064

*∂*

_{1}

*u*

_{1}and

*∂*

_{2}

*u*

_{2}. Copyright © 2011 John Wiley & Sons, Ltd.

Mathematical Methods in the Applied Sciences > 36 > 1 > 86 - 98

^{α}. We prove the local existence in time and obtain a regularity criterion of solution for the generalized Boussinesq equations by means of the Littlewood–Paley theory and Bony's paradifferential...

Mathematical Methods in the Applied Sciences > 37 > 15 > 2320 - 2325

*BMO*spaces. We prove that if $\left(\stackrel{\u0303}{u},\stackrel{\u0303}{b}\right)$ belongs to the space ${L}^{2}\left(0,T;\mathrm{BMO}\left({R}^{3}\right)\right)$, then the solution (

*u*,

*b*) is regular. This extends recent results contained by Gala, Ji E and Lee J. Copyright © 2013 John...

Mathematical Methods in the Applied Sciences > 38 > 4 > 701 - 707

Mathematical Methods in the Applied Sciences > 38 > 11 > 2073 - 2080

Mathematical Methods in the Applied Sciences > 38 > 17 > 4019 - 4023

Mathematical Methods in the Applied Sciences > 38 > 18 > 5279 - 5283

*u*and (−Δ)

^{β}

*b*, with $\frac{5}{2}\le \beta <\infty $. Copyright © 2015 John Wiley & Sons, Ltd.

Mathematical Methods in the Applied Sciences > 40 > 1 > 279 - 285

*p*≥

*q*> 1 for the incompressible magnetohydrodynamics equations with zero magnetic diffusivity. Copyright © 2016 John Wiley & Sons, Ltd.

Mathematical Methods in the Applied Sciences > 40 > 3 > 546 - 563

Mathematical Methods in the Applied Sciences > 40 > 5 > 1497 - 1504

*L*

^{∞}‐norm of the vertical components. Copyright © 2016 John Wiley & Sons, Ltd.

Mathematical Methods in the Applied Sciences > 41 > 10 > 3672 - 3683

*v*,

*d*) is smooth, and our main observation is that the condition $\nabla d\in {L}^{\infty}(0,T;{L}^{3}({R}^{3}\left)\right)$ is not necessary in this situation. The proof is based on the blow‐up analysis and backward uniqueness for the parabolic operator developed by Escauriaza‐Seregin‐S̆verák.

Mathematical Methods in the Applied Sciences > 41 > 16 > 6702 - 6716

Mathematical Methods in the Applied Sciences > 41 > 17 > 7958 - 7970

*α*‐MHD in Campanato‐Morrey space. We show that if the magnetic field

*b*satisfies the following condition: ${\int}_{0}^{T}\Vert \nabla b(\tau ,\xb7){\Vert}_{{\dot{M}}_{q}^{p}}^{\frac{2p}{2p-3}}+\Vert {D}^{2}b(\tau ,\xb7){\Vert}_{{\dot{M}}_{q}^{p}}^{\frac{2p}{2p-3}}d\tau <\infty ,$ the strong solution (

*v*,

*b*) to the 3‐D Leray‐

*α*MHD equation can be extended beyond

*T*> 0. We also prove that if the magnetic field

*b*satisfies ${\int}_{0}^{T}\Vert \nabla b(\tau ,\xb7){\Vert}_{{\dot{M}}_{}^{}}^{}$...

Mathematical Methods in the Applied Sciences > 42 > 5 > 1652 - 1661

*ξ*|≈2

^{j}}.