Energy Equality and Uniqueness of Weak Solutions to MHD Equations in L∞（0, T; L^n（Ω））

In this paper, we study the energy equality and the uniqueness of weak solutions to the MHD equations in the critical space L∞（0, T; L^n（Ω）. We prove that if the velocity u belongs to the critical space L∞（0, T; L^n（Ω）, the energy equality holds. On the basis of the energy equality, we further prove that the weak solution to the MHD equations is unique.     

Decay properties of solutions to the incompressible magnetohydrodynamics equations in a half space

We consider the asymptotic behavior of the strong solution to the incompressible magnetohydrodynamics (MHD) equations in a half space. The L^r‐decay rates of the strong solution and its derivatives with respect to space variables and time variable, including the L¹ and L ^∞ decay rates of its first order derivatives with respect to space variables, are derived by using L^q ? L^r estimates of the Stokes semigroup and employing a decomposition for the nonlinear terms in MHD equations. In addition, if the given initial data lie in a suitable weighted space, we obtain more rapid decay rates than observed in general. Similar results are known for incompressible Navier–Stokes equations in a half space under same assumption. Copyright © 2012 John Wiley & Sons, Ltd.     

Global well‐posedness for the compressible magnetohydrodynamic system in the critical Lp framework

The present work is dedicated to the well‐posedness issue of strong solutions (away from vacuum) to the compressible viscous magnetohydrodynamic (MHD) system in (d ≥ 2). We aim at extending those results in previous studies to more general L^p critical framework. Precisely, by recasting the whole system in Lagrangian coordinates, we prove the local existence and uniqueness of solutions by means of Banach fixed‐point theorem. Furthermore, with the aid of effective velocity, we employ the energy argument to establish global a priori estimates, which lead to the unique global solution near constant equilibrium. Our results hold in case of small data but large highly oscillating initial velocity and magnetic field.     

A regularity criterion for two‐and‐half‐dimensional magnetohydrodynamic equations with horizontal dissipation and horizontal magnetic diffusion

We study the global regularity of classical solution to two‐and‐half‐dimensional magnetohydrodynamic equations with horizontal dissipation and horizontal magnetic diffusion. We prove that any possible finite time blow‐up can be controlled by the L^∞‐norm of the vertical components. Copyright © 2016 John Wiley & Sons, Ltd.     

A logarithmic regularity criterion for the 3D generalized MHD system

We prove a logarithmic regularity criterion for the 3D generalized magnetohydrodynamics (MHD) system with diffusion terms ?Δu and (?Δ)^βb, with . Copyright © 2015 John Wiley & Sons, Ltd.     

Global solutions to isentropic planar magnetohydrodynamic equations with density‐dependent viscosity

In this paper, we study the isentropic compressible planar magnetohydrodynamic equations with viscosity depending on density and with free boundaries. Precisely, when the viscosity coefficient λ(ρ) is proportional to ρ^θ with θ > 0, where ρ is the density, we establish the existence of global solutions under certain assumptions on the initial data by deriving some new a priori estimates.     

Large time behavior for the incompressible magnetohydrodynamic equations in half‐spaces

The weighted L^r‐asymptotic behavior of the strong solution and its first‐order spacial derivatives to the incompressible magnetohydrodynamic (MHD) equations is established in a half‐space. Further, the L^∞‐decay rates of the second‐order spatial derivatives of the strong solution are derived by using the Stokes solution formula and employing a decomposition for the nonlinear terms in MHD equations. Copyright © 2014 John Wiley & Sons, Ltd.     

Liouville‐type theorem for the steady compressible Hall‐MHD system

In this paper, we prove a Liouville‐type theorem for the steady compressible Hall‐magnetohydrodynamics system in Π, where Π is whole space or half space . We show that a smooth solution (ρ, u , B ,P) satisfying 1/C₀<ρ<C₀, , and B ∈L^9/2(Π) for some constant C₀>0 is indeed trivial. This generalizes and improves 2 results of Chae.     

Time decay rate for two 3D magnetohydrodynamics‐ α models

We consider the long time behavior of solutions to the magnetohydrodynamics‐ α model in three spatial dimensions. Time decay rate in L²‐norm of the solution is obtained. Similar results for a generalized Leray‐ α‐magnetohydrodynamics model are also established. As a by‐product, an optimal time decay rate for the Navier–Stokes‐ α model is achieved. Copyright © 2013 John Wiley & Sons, Ltd.     

Diffusion vanishing limit of the nonlinear pipe Magnetohydrodynamic flow with fixed viscosity

We establish magnetic diffusion vanishing limit of the nonlinear pipe Magnetohydrodynamic flow by the mathematical validity of the Prandtl boundary layer theory with fixed viscosity. The convergence is verified under various Sobolev norms, including the L~∞(L~2)and L~∞(H~1) norm.     

On regularity criterion to the 3D axisymmetric incompressible MHD equations

This paper is concerned with the regularity criterion for a class of axisymmetric solutions to 3D incompressible magnetohydrodynamic equations. More precisely, for the solutions that have the form of u = u_re_r+u_θe_θ+u_ze_z and b = b_θe_θ, we prove that if |ru(x,t)|≤C holds for ?1≤t < 0, then (u,b) is regular at time zero. This result can be thought as a generalization of recent results in for the 3D incompressible Navier‐Stokes equations. Copyright © 2016 John Wiley & Sons, Ltd.     

Higher order commutator estimates and local existence for the non-resistive MHD equations and related models

This paper establishes the local-in-time existence and uniqueness of strong solutions in H^s$H^s$ for s>n/2$s>n/2$ to the viscous, non-resistive magnetohydrodynamics (MHD) equations in Rⁿ${\mathbb{R}}^n$, n=2,3$n=2,3$, as well as for a related model where the advection terms are removed from the velocity equation. The uniform bounds required for proving existence are established by means of a new estimate, which is a partial generalisation of the commutator estimate of Kato and Ponce (1988) [13].     

An unexpected separation result in Linearly Bounded Arithmetic

The theories Sⁱ₁(α) and Tⁱ₁(α) are the analogues of Buss' relativized bounded arithmetic theories in the language where every term is bounded by a polynomial, and thus all definable functions grow linearly in length. For every i, a Σ^b_i+1(α)‐formula TOPⁱ(a), which expresses a form of the total ordering principle, is exhibited that is provable in Sⁱ⁺¹₁ (α), but unprovable in Tⁱ₁(α). This is in contrast with the classical situation, where Sⁱ⁺¹₂ is conservative over Tⁱ₂ w. r. t. Σ^b_i+1‐sentences. The independence results are proved by translations into propositional logic, and using lower bounds for corresponding propositional proof systems. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A lower bound on the number of Semi‐Boolean quadruple systems

A Steiner quadruple system of order 2ⁿ is Semi‐Boolean (SBQS(2ⁿ) in short) if all its derived triple systems are isomorphic to the point‐line design associated with the projective geometry PG(n?1, 2). We prove by means of explicit constructions that for any n, up to isomorphism, there exist at least 2^{? 3(n?4)/2?} regular and resolvable SBQS(2ⁿ). © 2003 Wiley Periodicals, Inc. J Combin Designs 11: 229–239, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10050     

Further results on overlarge sets of Kirkman triple systems

In this paper, we introduce a new concept -- overlarge sets of generalized Kirkman systems （OLGKS）, research the relation between it and OLKTS, and obtain some new results for OLKTS. The main conclusion is： If there exist both an OLKF（6^k） and a 3-OLGKS（6^k-1,4） for all k ∈{6,7,...,40}／{8,17,21,22,25,26}, then there exists an OLKTS（v） for any v ≡ 3 （mod 6）, v ≠ 21. As well, we obtain the following result： There exists an OLKTS（6u ＋ 3） for u = 2^2n-1 - 1, 7^n, 31^n, 127^n, 4^r25^s, where n ≥ 1,r＋s≥ 1.     

Positive solution for asymptotically linear elliptic systems

In this paper, we show that semilinear elliptic systems of the form (1) possess at least one positive solution pair (u, v)∈H¹(?^N)×H¹(?^N), where λ and µ are nonnegative numbers, f(x, t) and g(x, t) are continuous functions on ?^N×? and asymptotically linear as t→+∞. Copyright © 2009 John Wiley & Sons, Ltd.     

Remarks on the global regularity and time decay of the 2D MHD equations with partial dissipation

This paper focuses on a system of the two‐dimensional (2D) magnetohydrodynamic (MHD) equations with the partial kinematic dissipation (?_yyu₁,?_xxu₂) and the partial magnetic diffusion (?_yyb₁,?_xxb₂). Based on the basic energy estimates only, we are able to show that this system always possesses a unique global smooth solution when the initial data are sufficiently smooth. Moreover, we obtain optimal large‐time decay rates of both solutions and their higher order derivatives by developing the classic Fourier splitting methods together with the auxiliary decay estimates of the first derivative of solutions and induction technique.     

On number system constructions

We investigate various number system constructions. After summarizing earlier results we prove that for a given lattice Λ and expansive matrix M: Λ → Λ if ρ(M ⁻¹) < 1/2 then there always exists a suitable digit set D for which (Λ, M, D) is a number system. Here ρ means the spectral radius of M ⁻¹. We shall prove further that if the polynomial f(x) = c ₀ + c ₁ x + ··· + c _k x ^k ∈ Z[x], c _k = 1 satisfies the condition |c ₀| > 2 Σ _i=1 ^k |c _i | then there is a suitable digit set D for which (Z ^k , M, D) is a number system, where M is the companion matrix of f(x). The research was supported by OTKA-T043657 and Bolyai Fellowship Committee.