Radial standing and self-similar waves for the hyperbolic cubic NLS in 2D

In this note we propose a new set of coordinates to study the hyperbolic or non-elliptic cubic nonlinear Schrodinger equation in two dimensions. Based on these coordinates, we study the existence of bounded and continuous hyperbolically radial standing waves, as well as hyperbolically radial self-similar solutions. Many of the arguments can easily be adapted to more general nonlinearities.Comment: 19 pages, 1 Figure, to appear in Nonlinearit

Flagellin delays spontaneous human neutrophil apoptosis

Neutrophils are short-lived cells that rapidly undergo apoptosis. However, their survival can be regulated by signals from the environment. Flagellin, the primary component of the bacterial flagella, is known to induce neutrophil activation. In this study we examined the ability of flagellin to modulate neutrophil apoptosis. Neutrophils cultured for 12 and 24 h in the presence of flagellin from Salmonella thyphimurim at concentrations found in pathological situations underwent a marked prevention of apoptosis. In contrast, Helicobacter pylori flagellin did not affect neutrophil survival, suggesting that Salmonella flagellin exerts the antiapoptotic effect by interacting with TLR5. The delaying in apoptosis mediated by Salmonella flagellin was coupled to higher expression levels of the antiapoptotic protein Mcl-1 and lower levels of activated caspase-3. Analysis of the signaling pathways indicated that Salmonella flagellin induced the activation of the p38 and ERK1/2 MAPK pathways as well as the PI3K/Akt pathway. Furthermore, it also stimulated IBα degradation and the phosphorylation of the p65 subunit, suggesting that Salmonella flagellin also triggers NF-B activation. Moreover, the pharmacological inhibition of ERK1/2 pathway and NF-B activation partially prevented the antiapoptotic effects exerted by flagellin. Finally, the apoptotic delaying effect exerted by flagellin was also evidenced when neutrophils were cultured with whole heat-killed S. thyphimurim. Both a wild-type and an aflagellate mutant S. thyphimurim strain promoted neutrophil survival; however, when cultured in low bacteria/neutrophil ratios, the flagellate bacteria showed a higher capacity to inhibit neutrophil apoptosis, although both strains showed a similar ability to induce neutrophil activation. Taken together, our results indicate that flagellin delays neutrophil apoptosis by a mechanism partially dependent on the activation of ERK1/2 MAPK and NF-B. The ability of flagellin to delay neutrophil apoptosis could contribute to perpetuate the inflammation during infections with flagellated bacteria.Facultad de Ciencias Exacta

Energy dispersed large data wave maps in 2+1 dimensions

In this article we consider large data Wave-Maps from $\mathbb{R}^{2+1}$ into a compact Riemannian manifold $(\mathcal{M},g)$, and we prove that regularity and dispersive bounds persist as long as a certain type of bulk (non-dispersive) concentration is absent. In a companion article we use these results in order to establish a full regularity theory for large data Wave-Maps.Comment: 89 page

On the probabilistic Cauchy theory for nonlinear dispersive PDEs

In this note, we review some of the recent developments in the well-posedness theory of nonlinear dispersive partial differential equations with random initial data.Comment: 26 pages. To appear in Landscapes of Time-Frequency Analysis, Appl. Numer. Harmon. Ana

Flagellin delays spontaneous human neutrophil apoptosis

Neutrophils are short-lived cells that rapidly undergo apoptosis. However, their survival can be regulated by signals from the environment. Flagellin, the primary component of the bacterial flagella, is known to induce neutrophil activation. In this study we examined the ability of flagellin to modulate neutrophil apoptosis. Neutrophils cultured for 12 and 24 h in the presence of flagellin from Salmonella thyphimurim at concentrations found in pathological situations underwent a marked prevention of apoptosis. In contrast, Helicobacter pylori flagellin did not affect neutrophil survival, suggesting that Salmonella flagellin exerts the antiapoptotic effect by interacting with TLR5. The delaying in apoptosis mediated by Salmonella flagellin was coupled to higher expression levels of the antiapoptotic protein Mcl-1 and lower levels of activated caspase-3. Analysis of the signaling pathways indicated that Salmonella flagellin induced the activation of the p38 and ERK1/2 MAPK pathways as well as the PI3K/Akt pathway. Furthermore, it also stimulated IBα degradation and the phosphorylation of the p65 subunit, suggesting that Salmonella flagellin also triggers NF-B activation. Moreover, the pharmacological inhibition of ERK1/2 pathway and NF-B activation partially prevented the antiapoptotic effects exerted by flagellin. Finally, the apoptotic delaying effect exerted by flagellin was also evidenced when neutrophils were cultured with whole heat-killed S. thyphimurim. Both a wild-type and an aflagellate mutant S. thyphimurim strain promoted neutrophil survival; however, when cultured in low bacteria/neutrophil ratios, the flagellate bacteria showed a higher capacity to inhibit neutrophil apoptosis, although both strains showed a similar ability to induce neutrophil activation. Taken together, our results indicate that flagellin delays neutrophil apoptosis by a mechanism partially dependent on the activation of ERK1/2 MAPK and NF-B. The ability of flagellin to delay neutrophil apoptosis could contribute to perpetuate the inflammation during infections with flagellated bacteria.Facultad de Ciencias Exacta

Random data wave equations

Nowadays we have many methods allowing to exploit the regularising properties of the linear part of a nonlinear dispersive equation (such as the KdV equation, the nonlinear wave or the nonlinear Schroedinger equations) in order to prove well-posedness in low regularity Sobolev spaces. By well-posedness in low regularity Sobolev spaces we mean that less regularity than the one imposed by the energy methods is required (the energy methods do not exploit the dispersive properties of the linear part of the equation). In many cases these methods to prove well-posedness in low regularity Sobolev spaces lead to optimal results in terms of the regularity of the initial data. By optimal we mean that if one requires slightly less regularity then the corresponding Cauchy problem becomes ill-posed in the Hadamard sense. We call the Sobolev spaces in which these ill-posedness results hold spaces of supercritical regularity. More recently, methods to prove probabilistic well-posedness in Sobolev spaces of supercritical regularity were developed. More precisely, by probabilistic well-posedness we mean that one endows the corresponding Sobolev space of supercritical regularity with a non degenerate probability measure and then one shows that almost surely with respect to this measure one can define a (unique) global flow. However, in most of the cases when the methods to prove probabilistic well-posedness apply, there is no information about the measure transported by the flow. Very recently, a method to prove that the transported measure is absolutely continuous with respect to the initial measure was developed. In such a situation, we have a measure which is quasi-invariant under the corresponding flow. The aim of these lectures is to present all of the above described developments in the context of the nonlinear wave equation.Comment: Lecture notes based on a course given at a CIME summer school in August 201

Reciprocal Interaction between Macrophages and T cells Stimulates IFN-γ and MCP-1 Production in Ang II-induced Cardiac Inflammation and Fibrosis

Background: The inflammatory response plays a critical role in hypertension-induced cardiac remodeling. We aimed to study how interaction among inflammatory cells causes inflammatory responses in the process of hypertensive cardiac fibrosis. Methodology/Principal Findings: Infusion of angiotensin II (Ang II, 1500 ng/kg/min) in mice rapidly induced the expression of interferon c (IFN-c) and leukocytes infiltration into the heart. To determine the role of IFN-c on cardiac inflammation and remodeling, both wild-type (WT) and IFN-c-knockout (KO) mice were infused Ang II for 7 days, and were found an equal blood pressure increase. However, knockout of IFN-c prevented Ang II-induced: 1) infiltration of macrophages and T cells into cardiac tissue; 2) expression of tumor necrosis factor a and monocyte chemoattractant protein 1 (MCP-1), and 3) cardiac fibrosis, including the expression of a-smooth muscle actin and collagen I (all p,0.05). Cultured T cells or macrophages alone expressed very low level of IFN-c, however, co-culture of T cells and macrophages increased IFN-c expression by 19.860.95 folds (vs. WT macrophage, p,0.001) and 20.9 6 2.09 folds (vs. WT T cells, p,0.001). In vitro co-culture studies using T cells and macrophages from WT or IFN-c KO mice demonstrated that T cells were primary source for IFN-c production. Co-culture of WT macrophages with WT T cells, but not with IFN-c-knockout T cells, increased IFN-c production (p,0.01). Moreover, IFN-c produced by T cells amplified MCP-1 expression in macrophages and stimulated macrophag

Quasi-invariant Gaussian measures for the cubic fourth order nonlinear Schrödinger equation

We consider the cubic fourth order nonlinear Schr\"odinger equation on the circle. In particular, we prove that the mean-zero Gaussian measures on Sobolev spaces $H^s(\mathbb{T})$, $s > \frac34$, are quasi-invariant under the flow.Comment: 41 pages. To appear in Probab. Theory Related Field

A pedestrian approach to the invariant Gibbs measures for the 2-d defocusing nonlinear Schrödinger equations

42 pagesInternational audienceWe consider the defocusing nonlinear Schr\"odinger equations on the two-dimensional compact Riemannian manifold without boundary or a bounded domain in $\R^2$. Our aim is to give a pedagogic and self-contained presentation on the Wick renormalization in terms of the Hermite polynomials and the Laguerre polynomials and construct the Gibbs measures corresponding to the Wick ordered Hamiltonian. Then, we construct global-in-time solutions with initial data distributed according to the Gibbs measure and show that the law of the random solutions, at any time, is again given by the Gibbs measure