Bounded nonnegative weak solutions to anisotropic parabolic double phase problems with variable growth

2021 ◽  
pp. 1-14
Author(s):  
Hamid EL Bahja
Keyword(s):  
Weak Solutions ◽  
Double Phase

scholarly journals On the uniqueness for weak solutions of steady double-phase fluids

2021 ◽  
Vol 11 (1) ◽  
pp. 454-468
Author(s):  
Mohamed Abdelwahed ◽  
Luigi C. Berselli ◽  
Nejmeddine Chorfi
Keyword(s):  
Stress Tensor ◽  
Newtonian Fluid ◽  
Weak Solutions ◽  
Convective Term ◽  
Anisotropic Stress ◽  
Stress Tensors ◽  
Double Phase ◽  
Wide Range

Abstract We consider a double-phase non-Newtonian fluid, described by a stress tensor which is the sum of a p-Stokes and a q-Stokes stress tensor, with 1 < p<2 < q<∞. For a wide range of parameters (p, q), we prove the uniqueness of small solutions. We use the p < 2 features to obtain quadratic-type estimates for the stress-tensor, while we use the improved regularity coming from the term with q > 2 to justify calculations for weak solutions. Results are obtained through a careful use of the symmetries of the convective term and are also valid for rather general (even anisotropic) stress-tensors.

scholarly journals On a degenerate parabolic equation from double phase convection

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Huashui Zhan
Keyword(s):  
Parabolic Equation ◽  
Weak Solutions ◽  
Degenerate Parabolic Equation ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Boundary Value Condition ◽  
Double Phase ◽  
Degenerate Parabolic ◽  
Initial Boundary Value

AbstractThe initial-boundary value problem of a degenerate parabolic equation arising from double phase convection is considered. Let $a(x)$ a ( x ) and $b(x)$ b ( x ) be the diffusion coefficients corresponding to the double phase respectively. In general, it is assumed that $a(x)+b(x)>0$ a ( x ) + b ( x ) > 0 , $x\in \overline{\Omega }$ x ∈ Ω ‾ and the boundary value condition should be imposed. In this paper, the condition $a(x)+b(x)>0$ a ( x ) + b ( x ) > 0 , $x\in \overline{\Omega }$ x ∈ Ω ‾ is weakened, and sometimes the boundary value condition is not necessary. The existence of a weak solution u is proved by parabolically regularized method, and $u_{t}\in L^{2}(Q_{T})$ u t ∈ L 2 ( Q T ) is shown. The stability of weak solutions is studied according to the different integrable conditions of $a(x)$ a ( x ) and $b(x)$ b ( x ) . To ensure the well-posedness of weak solutions, the classical trace is generalized, and that the homogeneous boundary value condition can be replaced by $a(x)b(x)|_{x\in \partial \Omega }=0$ a ( x ) b ( x ) | x ∈ ∂ Ω = 0 is found for the first time.

scholarly journals Local subestimates of solutions to double-phase parabolic equations via nonlinear parabolic potentials

2019 ◽  
Vol 16 (1) ◽  
pp. 28-45
Author(s):  
Kateryna Buryachenko
Keyword(s):  
Weak Solutions ◽  
Parabolic Equations ◽  
Growth Conditions ◽  
Local Boundedness ◽  
Nonstandard Growth ◽  
Right Hand ◽  
Double Phase ◽  
Nonlinear Parabolic ◽  
Nonstandard Growth Conditions ◽  
The Right

For parabolic equations with nonstandard growth conditions, we prove local boundedness of weak solutions in terms of nonlinear parabolic potentials of the right-hand side of the equation.

scholarly journals Weak solutions for a (p(z),q(z))-Laplacian Dirichlet problem

Filomat ◽  
2020 ◽  
Vol 34 (3) ◽  
pp. 999-1011
Author(s):  
Antonella Nastasi
Keyword(s):  
Dirichlet Problem ◽  
Weak Solutions ◽  
Nonnegative Solution ◽  
Laplacian Operator ◽  
Potential Term ◽  
Reaction Term ◽  
Double Phase

We establish the existence of a nontrivial and nonnegative solution for a double phase Dirichlet problem driven by a (p(z); q(z))-Laplacian operator plus a potential term. Our approach is variational, but the reaction term f need not satisfy the usual in such cases Ambrosetti-Rabinowitz condition.

scholarly journals Self-Improving inequalities for bounded weak solutions to nonlocal double phase equations

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
James M. Scott ◽  
Tadele Mengesha
Keyword(s):  
Differential Equations ◽  
Weak Solutions ◽  
Zero Set ◽  
Rough Kernels ◽  
Sobolev Regularity ◽  
Double Phase ◽  
Higher Sobolev Regularity ◽  
Bounded Weak Solutions

<p style='text-indent:20px;'>We prove higher Sobolev regularity for bounded weak solutions to a class of nonlinear nonlocal integro-differential equations. The leading operator exhibits nonuniform growth, switching between two different fractional elliptic "phases" that are determined by the zero set of a modulating coefficient. Solutions are shown to improve both in integrability and differentiability. These results apply to operators with rough kernels and modulating coefficients. To obtain these results we adapt a particular fractional version of the Gehring lemma developed by Kuusi, Mingione, and Sire in their work "Nonlocal self-improving properties" <i>Analysis &amp; PDE</i>, 8(1):57–114 for the specific nonlinear setting under investigation in this manuscript.</p>

Quantitative nano-characterization of heterogeneous catalysts

1995 ◽  
Vol 53 ◽  
pp. 398-399
Author(s):  
P.A. Crozier ◽  
M. Pan
Keyword(s):  
Heterogeneous Catalysts ◽  
Low Dose ◽  
Imaging Techniques ◽  
Structural Features ◽  
Catalyst Characterization ◽  
New Developments ◽  
Double Phase ◽  
Phase Systems ◽  
Dose Imaging

Heterogeneous catalysts can be of varying complexity ranging from single or double phase systems to complicated mixtures of metals and oxides with additives to help promote chemical reactions, extend the life of the catalysts, prevent poisoning etc. Although catalysis occurs on the surface of most systems, detailed descriptions of the microstructure and chemistry of catalysts can be helpful for developing an understanding of the mechanism by which a catalyst facilitates a reaction. Recent years have seen continued development and improvement of various TEM, STEM and AEM techniques for yielding information on the structure and chemistry of catalysts on the nanometer scale. Here we review some quantitative approaches to catalyst characterization that have resulted from new developments in instrumentation.HREM has been used to examine structural features of catalysts often by employing profile imaging techniques to study atomic details on the surface. Digital recording techniques employing slow-scan CCD cameras have facilitated the use of low-dose imaging in zeolite structure analysis and electron crystallography. Fig. la shows a low-dose image from SSZ-33 zeolite revealing the presence of a stacking fault.

ECHANGE THERMIQUE EN DOUBLE PHASE DANS DES TUBES VERTICAUX OU HORIZONTAUX

1970 ◽  
Author(s):  
R. Ricque ◽  
R. Roumy
Keyword(s):  
Double Phase

Probing the Versatility of Shape-Persistent Tetraphenylmethane Dendrimers by Modification of the Skeleton

2019 ◽  
Author(s):  
Julio Ignacio Urzúa ◽  
Sandra Campana ◽  
Massimo Lazzari ◽  
Mercedes Torneiro
Keyword(s):  
Design Principle ◽  
Focal Point ◽  
Sonogashira Coupling ◽  
Terminal Alkyne ◽  
Double Phase ◽  
Rigid Rod ◽  
Convergent Approach ◽  
First And Second Generation ◽  
Oxidative Homocoupling ◽  
Hypercrosslinked Polymers

Tetraphenylmethane has emerged as a recurrent building block for advanced porous materials such as COFs, PAFs and hypercrosslinked polymers. Guided by a similar design principle, we have previously synthesized shape-persistent dendrimers with tetraphenylmethane nodes and ethynylene linkers. Here we report the generality of our approach by describing new dendritic architectures built from tetraphenylmethane. First, we prepared expanded dendrimers where the tetrahedral units are bonded through larger rigid rod spacers. Among the different synthetic strategies tested, the convergent route, with alternating steps of Pd-catalyzed Sonogashira coupling and alkyne activation by removal of TMS masking groups, efficiently afforded the first- and second-generation dendrimers. A second type of compounds having a linear diyne at the core is also described. The dendrimers of generations 1-2 were also synthesized by a convergent approach, with the diyne being assembled in the last step of the synthesis by a Glaser oxidative homocoupling of the corresponding dendrons bearing a terminal alkyne at the focal point. A third-generation dendrimer was also successfully prepared by a double-phase strategy.<br>

Differentiation of Tuberculous and Pyogenic Spondylitis Using Double Phase F-18 FDG PET

2008 ◽  
Vol 2 (1) ◽  
pp. 1-6 ◽  
Author(s):  
Jung Sub Lee ◽  
Seong-Jang Kim ◽  
Kuen Tak Suh ◽  
In-Ju Kim ◽  
Yong-Ki Kim
Keyword(s):  
Fdg Pet ◽  
Pyogenic Spondylitis ◽  
Double Phase
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