Gradient estimates for elliptic systems in non-smooth domains

We obtain the global W ^1,p , 1 < p < ∞, estimate for the weak solution of an elliptic system with discontinuous coefficients in non-smooth domains without using maximal function approach. It is assumed that the boundary of a bounded domain is well approximated by hyperplanes at every point and at every scale, and that the tensor coefficients belong to BMO space with their BMO semi-norms sufficiently small. S.-S. Byun was supported in part by KRF-2006-C00034 and L. Wang was supported in part by NSF Grant 0701392.     

A new class of Gelfand pairs

The well-known result for rank 1 real symmetric spacesX=G/H, stating the multiplicity-free decomposition of the natural action ofG onL ²(X), is extended to the classical rank 1p-adic symmetric spaces. The absence of ap-adic analogue of the Laplace-Beltrami operator causes additional complications and necessitates a global study of the invariant distributions on the quotient space.     

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.     

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.     

Functional calculi of second-order elliptic partial differential operators with bounded measurable coefficients

Consider a second-order elliptic partial differential operatorL in divergence form with real, symmetric, bounded measurable coefficients, under Dirichlet or Neumann conditions on the boundary of a strongly Lipschitz domain Ω. Suppose that 1 <p < ∞ and μ > 0. ThenL has a bounded H_∞ functional calculus in L^p(Ω), in the sense that ¦¦f (L +cI)u¦¦_p ≤C sup_¦arλ¦<μ ^¦f¦ ¦‖u¦‖_p for some constantsc andC, and all bounded holomorphic functionsf on the sector ¦ argλ¦ < μ that contains the spectrum ofL +cI. We prove this by showing that the operatorsf(L + cI) are Calderón-Zygmund singular integral operators.     

On explicit L∞(QT)‐estimates for a model of tissue invasion by solid tumors

In this paper, we prove that if the initial data is small enough, we obtain an explicit L^∞(Q_T)‐estimate for a two‐dimensional mathematical model of cancer invasion, proving an explicit bound with respect to time T for the estimate of solutions. Copyright © 2017 John Wiley & Sons, Ltd.     

Regularity of radial minimizers of reaction equations involving the <Emphasis Type="Italic">p</Emphasis>-Laplacian

We consider semi-stable, radially symmetric, and decreasing solutions of  − Δ _p u = g(u) in the unit ball of , where p > 1, Δ _p is the p-Laplace operator, and g is a locally Lipschitz function. For this class of radial solutions, which includes local minimizers, we establish pointwise, L ^q , and W ^1,q estimates which are optimal and do not depend on the specific nonlinearity g. Among other results, we prove that every radially decreasing and semi-stable solution u belonging to W ^1,p (B ₁) is bounded whenever n < p + 4p/(p − 1). Under standard assumptions on the nonlinearity g(u) = λf (u), where λ > 0 is a parameter, it is proved that the corresponding extremal solution u ^* is semi-stable, and hence, it enjoys the regularity stated in our main result.     

SUPERCRITICAL SEMILINEAR WAVE EQUATION WITH NON-NEGATIVE POTENTIAL

We prove a weighted L ^∞ estimate for the solution to the linear wave equation with a smooth positive time independent potential. The proof is based on application of generalized Fourier transform for the perturbed Laplace operator and a finite dependence domain argument. We apply this estimate to prove the existence of global small data solution to supercritical semilinear wave equations with potential.     

Primarite DeL p(L r), 1<p,r<∞

In this article we show thatL ^p(L ^r) is primary forp andr in ]1,+∞[. If (h _k) _k≧1 denote the Haar basis, we begin with a study of the sequence (h _k ⊗h _i) and, in particular, the space generated by a subsequence of this sequence. In the first part we study the base ofL ^p(L ^r) and in the second part we show that this space is primary.     

The Banach–Saks index of rearrangement invariant spaces on [0,1]

The set of all rearrangement invariant function spaces on [0,1] having the p-Banach–Saks property has a unique maximal element for all p∈(1,2]. For p=2 this is L₂, for p∈(1,2) this is L_p,∞⁰. We compute the Banach–Saks index for the families of Lorentz spaces $\text{L}{}_{\mathrm{p,q}}\text{,}\text{1<p<∞}$, 1?q?∞, and Lorentz–Zygmund spaces L(p,α), $\text{1?p<∞,}\text{α∈}\text{R}$, extending the classical results of Banach–Saks and Kadec–Pelczynski for L_p-spaces. Our results show that the set of rearrangement invariant spaces with Banach–Saks index p∈(1,2] is not stable with respect to the real and complex interpoltaion methods. To cite this article: E.M. Semenov, F.A. Sukochev, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-maximal regularity for second order Cauchy problems

We introduce the concept of L^p-maximal regularity for second order Cauchy problems. We prove L^p-maximal regularity for an abstract model problem and we apply the abstract results to prove existence, uniqueness and regularity of solutions for nonlinear wave equations. The author acknowledges with thanks the support provided by the Department ofApplied Analysis, University of Ulm, and the travel grants provided by NBMH India and MSF Delhi, India.     

A global existence result for a semilinear scale‐invariant wave equation in even dimension

In the present article a, semilinear scale‐invariant wave equation with damping and mass is considered. The global (in time) existence of radial symmetric solutions in even spatial dimension n is proved by using weighted L^∞ ? L^∞ estimates, under the assumption that the multiplicative constants, which appear in the coefficients of damping and of mass terms, fulfill an interplay condition, which yields somehow a “wave‐like” model. In particular, combining this existence result with a recently proved blow‐up result, a suitable shift of Strauss exponent is proved to be the critical exponent for the considered model. Moreover, the still open part of a conjecture done by D'Abbicco‐Lucente‐Reissig is proved to be true in the massless case.     

Maximum principle and existence of <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-viscosity solutions for fully nonlinear uniformly elliptic equations with measurable and quadratic terms

We study L^p-viscosity solutions of fully nonlinear, second-order, uniformly elliptic partial differential equations (PDE) with measurable terms and quadratic nonlinearity. We present a sufficient condition under which the maximum principle holds for L^p-viscosity solution. We also prove stability and existence results for the equations under consideration.     

Property (<Emphasis Type="Italic">T</Emphasis>) and rigidity for actions on Banach spaces

We study property (T) and the fixed-point property for actions on L ^p and other Banach spaces. We show that property (T) holds when L ² is replaced by L ^p (and even a subspace/quotient of L ^p ), and that in fact it is independent of 1≤p<∞. We show that the fixed-point property for L ^p follows from property (T) when 1<p< 2+ε. For simple Lie groups and their lattices, we prove that the fixed-point property for L ^p holds for any 1< p<∞ if and only if the rank is at least two. Finally, we obtain a superrigidity result for actions of irreducible lattices in products of general groups on superreflexive spaces. Bader partially supported by ISF grant 100146; Furman partially supported by NSF grants DMS-0094245 and DMS-0604611; Gelander partially supported by NSF grant DMS-0404557 and BSF grant 2004010; Monod partially supported by FNS (CH) and NSF (US).     

Strong solutions to 3D compressible magnetohydrodynamic equations with Navier‐slip condition

We consider the short time strong solutions to the compressible magnetohydrodynamic equations with initial vacuum, in which the velocity field satisfies the Navier‐slip condition. The Navier‐slip condition differs in many aspects from no‐slip conditions, and it has attracted considerable attention in nanoscale and microscale flows research. Inspired by Kato and Lax's idea, we use the Lax–Milgram theorem and contraction mapping argument to prove local existence. Moreover, under the Navier‐slip condition, we establish a criterion for possible breakdown of such solutions at finite time in terms of the temporal integral of L^∞ norm of the deformation tensor D(u). Copyright © 2015 John Wiley & Sons, Ltd.     

M ‐elliptic pseudo‐differential operators on L p (ℝn )

We construct the minimal and maximal extensions in L ^p (?ⁿ ), 1 < p < ∞, for M ‐elliptic pseudo‐differential operators initiated by Garello and Morando. We prove that they are equal and determine the domains of the minimal, and hence maximal, extensions of M ‐elliptic pseudo‐differential operators. For M ‐elliptic pseudodifferential operators with constant coefficients, the spectra and essential spectra are computed. An application to quantization is given. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Shallow water model for lakes with friction and penetration

We deduce a shallow water model, describ‐ ing the motion of the fluid in a lake, assuming inflow–outflow effects across the bottom. This model arises from the asymptotic analysis of the 3D dimensional Navier–Stokes equations. We prove the global in time existence result for this model in a bounded domain taking the nonlinear slip/friction boundary conditions to describe the inflows and outflows of the porous coast and the rivers. The solvability is shown in the class of solutions with L_p‐bounded vorticity for any given p∈(1,∞]. Copyright © 2009 John Wiley & Sons, Ltd.     

<Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis>,<Emphasis Type="Italic">q</Emphasis></Subscript>-Cohomology and Normal Solvability

We consider some problems concerning the L _p,q -cohomology of Riemannian manifolds. In the first part, we study the question of the normal solvability of the operator of exterior derivation on a surface of revolution M considered as an unbounded linear operator acting from L_p^k (M) into L^k+1_q (M). In the second part, we prove that the first L _p,q-cohomology of the general Heisenberg group is nontrivial, provided that p < q. Received: 17 January 2006 Supported by INTAS (Grant 03–51–3251) and the State Maintenance Program for the Leading Scientific Schools of the Russian Federation (Grants NSh 311.2003.1, NSh 8526.2006.1).     

On Quasi-static Non=Newtonian Fluids with Power-law

We discuss certain classes of quasi-static non-Newtonian fluids for which a power-law of the form σ^D=∇ϕ(ℰv) holds. Here σ^D is the stress deviator, v the velocity field, ℰv its symmetric derivative and ϕ is the function \[ \phi ({\cal E}v)=\frac 12\mu _\infty \left| {\cal E}v\right| ⁁2+\frac 1p\mu _0\left\{ \begin{array}{c} \left( 1+\left| {\cal E}v\right| ⁁2\right) ⁁{p/2} \\ \text{or} \\ \left| {\cal E}v\right| ⁁p \end{array} \right\}, \] ϕ(ℰv)=1 2 μ_∞∣ℰv∣²+1 p μ₀ (1+∣ℰv∣²)^p/2 or ∣ℰv∣^p, μ_∞⩾0, μ₀⩾0, μ_∞+μ₀>0, 1<p<∞. We then prove various regularity results for the velocity field v, for example differentiability almost everywhere and local boundedness of the tensor ℰv.