278 research outputs found
Instability and stability properties of traveling waves for the double dispersion equation
In this article we are concerned with the instability and stability
properties of traveling wave solutions of the double dispersion equation
for ,
. The main characteristic of this equation is the existence of two
sources of dispersion, characterized by the terms and . We
obtain an explicit condition in terms of , and on wave velocities
ensuring that traveling wave solutions of the double dispersion equation are
strongly unstable by blow up. In the special case of the Boussinesq equation
(), our condition reduces to the one given in the literature. For the
double dispersion equation, we also investigate orbital stability of traveling
waves by considering the convexity of a scalar function. We provide both
analytical and numerical results on the variation of the stability region of
wave velocities with , and and then state explicitly the conditions
under which the traveling waves are orbitally stable.Comment: 16 pages, 4 figure
A Comparison of Solutions of Two Convolution-Type Unidirectional Wave Equations
In this work, we prove a comparison result for a general class of nonlinear
dispersive unidirectional wave equations. The dispersive nature of
one-dimensional waves occurs because of a convolution integral in space. For
two specific choices of the kernel function, the Benjamin-Bona-Mahony equation
and the Rosenau equation that are particularly suitable to model water waves
and elastic waves, respectively, are two members of the class. We first prove
an energy estimate for the Cauchy problem of the nonlocal unidirectional wave
equation. Then, for the same initial data, we consider two distinct solutions
corresponding to two different kernel functions. Our main result is that the
difference between the solutions remains small in a suitable Sobolev norm if
the two kernel functions have similar dispersive characteristics in the
long-wave limit. As a sample case of this comparison result, we provide the
approximations to the hyperbolic conservation law.Comment: 12 pages, to appear in Applicable Analysi
The Camassa-Holm equation as the long-wave limit of the improved Boussinesq equation and of a class of nonlocal wave equations
In the present study we prove rigorously that in the long-wave limit, the
unidirectional solutions of a class of nonlocal wave equations to which the
improved Boussinesq equation belongs are well approximated by the solutions of
the Camassa-Holm equation over a long time scale. This general class of
nonlocal wave equations model bidirectional wave propagation in a nonlocally
and nonlinearly elastic medium whose constitutive equation is given by a
convolution integral. To justify the Camassa-Holm approximation we show that
approximation errors remain small over a long time interval. To be more
precise, we obtain error estimates in terms of two independent, small, positive
parameters and measuring the effect of nonlinearity and
dispersion, respectively. We further show that similar conclusions are also
valid for the lower order approximations: the Benjamin-Bona-Mahony
approximation and the Korteweg-de Vries approximation.Comment: 24 pages, to appear in Discrete and Continuous Dynamical System
The Cauchy problem for a class of two-dimensional nonlocal nonlinear wave equations governing anti-plane shear motions in elastic materials
This paper is concerned with the analysis of the Cauchy problem of a general
class of two-dimensional nonlinear nonlocal wave equations governing anti-plane
shear motions in nonlocal elasticity. The nonlocal nature of the problem is
reflected by a convolution integral in the space variables. The Fourier
transform of the convolution kernel is nonnegative and satisfies a certain
growth condition at infinity. For initial data in Sobolev spaces,
conditions for global existence or finite time blow-up of the solutions of the
Cauchy problem are established.Comment: 15 pages. "Section 6 The Anisotropic Case" added and minor changes.
Accepted for publication in Nonlinearit
Macrophage mal1 deficiency suppresses atherosclerosis in low-density lipoprotein receptor-null mice by activating peroxisome proliferator-activated receptor-γ-regulated genes
Objective- The adipocyte/macrophage fatty acid-binding proteins aP2 (FABP4) and Mal1 (FABP5) are intracellular lipid chaperones that modulate systemic glucose metabolism, insulin sensitivity, and atherosclerosis. Combined deficiency of aP2 and Mal1 has been shown to reduce the development of atherosclerosis, but the independent role of macrophage Mal1 expression in atherogenesis remains unclear. Methods and Results- We transplanted wild-type (WT), Mal1, or aP2 bone marrow into low-density lipoprotein receptor-null (LDLR) mice and fed them a Western diet for 8 weeks. Mal1→LDLR mice had significantly reduced (36%) atherosclerosis in the proximal aorta compared with control WT→LDLR mice. Interestingly, peritoneal macrophages isolated from Mal1-deficient mice displayed increased peroxisome proliferator-activated receptor-γ (PPARγ) activity and upregulation of a PPARγ-related cholesterol trafficking gene, CD36. Mal1 macrophages showed suppression of inflammatory genes, such as COX2 and interleukin 6. Mal1→LDLR mice had significantly decreased macrophage numbers in the aortic atherosclerotic lesions compared with WT→LDLR mice, suggesting that monocyte recruitment may be impaired. Indeed, blood monocytes isolated from Mal1→LDLR mice on a high-fat diet had decreased CC chemokine receptor 2 gene and protein expression levels compared with WT monocytes. Conclusion- Taken together, our results demonstrate that Mal1 plays a proatherogenic role by suppressing PPARγ activity, which increases expression of CC chemokine receptor 2 by monocytes, promoting their recruitment to atherosclerotic lesions. © 2011 American Heart Association, Inc
The Camassa-Holm equation as the long-wave limit of the improved Boussinesq equation and of a class of nonlocal wave equations
In the present study we prove rigorously that in the long-wave limit, the unidirectional solutions of a class of nonlocal wave equations to which the improved Boussinesq equation belongs are well approximated by the solutions of the Camassa-Holm equation over a long time scale. This general class of nonlocal wave equations model bidirectional wave propagation in a nonlocally and nonlinearly elastic medium whose constitutive equation is given by a convolution integral. To justify the Camassa-Holm approximation we show that approximation errors remain small over a long time interval. To be more precise, we obtain error estimates in terms of two independent, small, positive parameters is an element of and delta measuring the effect of nonlinearity and dispersion, respectively. We further show that similar conclusions are also valid for the lower order approximations: the Benjamin-Bona-Mahony approximation and the Korteweg-de Vries approximation
Macrophage Mal1 Deficiency Suppresses Atherosclerosis in Low-Density Lipoprotein Receptor -Null Mice by Activating Peroxisome Proliferator-Activated Receptor-g-Regulated Genes
Cataloged from PDF version of article.Objective-The adipocyte/macrophage fatty acid-binding proteins aP2 (FABP4) and Mal1 (FABP5) are intracellular lipid chaperones that modulate systemic glucose metabolism, insulin sensitivity, and atherosclerosis. Combined deficiency of aP2 and Mal1 has been shown to reduce the development of atherosclerosis, but the independent role of macrophage Mal1 expression in atherogenesis remains unclear.
Methods and Results-We transplanted wild-type (WT), Mal1(-/-), or aP2(-/-) bone marrow into low-density lipoprotein receptor-null (LDLR(-/-)) mice and fed them a Western diet for 8 weeks. Mal1(-/-)-> LDLR(-/-) mice had significantly reduced (36%) atherosclerosis in the proximal aorta compared with control WT -> LDLR(-/-) mice. Interestingly, peritoneal macrophages isolated from Mal1-deficient mice displayed increased peroxisome proliferator-activated receptor-gamma (PPAR gamma) activity and upregulation of a PPAR gamma-related cholesterol trafficking gene, CD36. Mal1(-/-) macrophages showed suppression of inflammatory genes, such as COX2 and interleukin 6. Mal1(-/-)-> LDLR(-/-) mice had significantly decreased macrophage numbers in the aortic atherosclerotic lesions compared with WT -> LDLR(-/-) mice, suggesting that monocyte recruitment may be impaired. Indeed, blood monocytes isolated from Mal1(-/-)-> LDLR(-/-) mice on a high-fat diet had decreased CC chemokine receptor 2 gene and protein expression levels compared with WT monocytes.
Conclusion-Taken together, our results demonstrate that Mal1 plays a proatherogenic role by suppressing PPAR gamma activity, which increases expression of CC chemokine receptor 2 by monocytes, promoting their recruitment to atherosclerotic lesions. (Arterioscler Thromb Vasc Biol. 2011;31:1283-1290.
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