Renormalizing Partial Differential Equations

In this review paper, we explain how to apply Renormalization Group ideas to the analysis of the long-time asymptotics of solutions of partial differential equations. We illustrate the method on several examples of nonlinear parabolic equations. We discuss many applications, including the stability of profiles and fronts in the Ginzburg-Landau equation, anomalous scaling laws in reaction-diffusion equations, and the shape of a solution near a blow-up point.Comment: 34 pages, Latex; [email protected]; [email protected]

Nonlinear localized waves in a periodic medium

We analyze the existence and stability of nonlinear localized waves in a periodic medium described by the Kronig-Penney model with a nonlinear defect. We demonstrate the existence of a novel type of stable nonlinear band-gap localized states, and also reveal an important physical mechanism of the oscillatory wave instabilities associated with the band-gap resonances.Comment: 4 pages, 5 figure

Normal scaling in globally conserved interface-controlled coarsening of fractal clusters

Globally conserved interface-controlled coarsening of fractal clusters exhibits dynamic scale invariance and normal scaling. This is demonstrated by a numerical solution of the Ginzburg-Landau equation with a global conservation law. The sharp-interface limit of this equation is volume preserving motion by mean curvature. The scaled form of the correlation function has a power-law tail accommodating the fractal initial condition. The coarsening length exhibits normal scaling with time. Finally, shrinking of the fractal clusters with time is observed. The difference between global and local conservation is discussed.Comment: 4 pages, 3 eps figure

The phase shift of line solitons for the KP-II equation

The KP-II equation was derived by [B. B. Kadomtsev and V. I. Petviashvili,Sov. Phys. Dokl. vol.15 (1970), 539-541] to explain stability of line solitary waves of shallow water. Stability of line solitons has been proved by [T. Mizumachi, Mem. of vol. 238 (2015), no.1125] and [T. Mizumachi, Proc. Roy. Soc. Edinburgh Sect. A. vol.148 (2018), 149--198]. It turns out the local phase shift of modulating line solitons are not uniform in the transverse direction. In this paper, we obtain the $L^\infty$-bound for the local phase shift of modulating line solitons for polynomially localized perturbations

Existence and stability of viscoelastic shock profiles

We investigate existence and stability of viscoelastic shock profiles for a class of planar models including the incompressible shear case studied by Antman and Malek-Madani. We establish that the resulting equations fall into the class of symmetrizable hyperbolic--parabolic systems, hence spectral stability implies linearized and nonlinear stability with sharp rates of decay. The new contributions are treatment of the compressible case, formulation of a rigorous nonlinear stability theory, including verification of stability of small-amplitude Lax shocks, and the systematic incorporation in our investigations of numerical Evans function computations determining stability of large-amplitude and or nonclassical type shock profiles.Comment: 43 pages, 12 figure

Solitary waves in the Nonlinear Dirac Equation

In the present work, we consider the existence, stability, and dynamics of solitary waves in the nonlinear Dirac equation. We start by introducing the Soler model of self-interacting spinors, and discuss its localized waveforms in one, two, and three spatial dimensions and the equations they satisfy. We present the associated explicit solutions in one dimension and numerically obtain their analogues in higher dimensions. The stability is subsequently discussed from a theoretical perspective and then complemented with numerical computations. Finally, the dynamics of the solutions is explored and compared to its non-relativistic analogue, which is the nonlinear Schr{\"o}dinger equation. A few special topics are also explored, including the discrete variant of the nonlinear Dirac equation and its solitary wave properties, as well as the PT-symmetric variant of the model

A pilot study of a phenomenological model of adipogenesis in maturing adipocytes using Cahn–Hilliard theory

We consider the accumulation and formation of lipid droplets in an adipocyte cell. The process incorporates adipose nucleation (adipogenesis) and growth. At later stages, there will be merging of droplets and growth of larger droplets at the expense of the smaller droplets, which will essentially undergo lipolysis. The process is modeled by the use of the Cahn–Hilliard equation, which is mass-conserving and allows the formation of secondary phases in the context of spinodal decomposition. The volume of fluid (VOF) method is used to determine the total area that is occupied by the lipids in a given cross section. Further, we present an algorithm, applicable to all kinds of grids (structured or unstructured) in two spatial dimensions, to count the number of lipid droplets and the portion of the domain of computation that is occupied by the lipid droplets as a function of time during the process. The results are preliminary and are validated from a qualitative point using experiments carried out on cell cultures. It turns out that the Cahn–Hilliard theory can model many of the features during adipogenesis qualitatively

Signaling Role of Fructose Mediated by FINS1/FBP in Arabidopsis thaliana

Sugars are evolutionarily conserved signaling molecules that regulate the growth and development of both unicellular and multicellular organisms. As sugar-producing photosynthetic organisms, plants utilize glucose as one of their major signaling molecules. However, the details of other sugar signaling molecules and their regulatory factors have remained elusive, due to the complexity of the metabolite and hormone interactions that control physiological and developmental programs in plants. We combined information from a gain-of-function cell-based screen and a loss-of-function reverse-genetic analysis to demonstrate that fructose acts as a signaling molecule in Arabidopsis thaliana. Fructose signaling induced seedling developmental arrest and interacted with plant stress hormone signaling in a manner similar to that of glucose. For fructose signaling responses, the plant glucose sensor HEXOKINASE1 (HXK1) was dispensable, while FRUCTOSE INSENSITIVE1 (FINS1), a putative FRUCTOSE-1,6-BISPHOSPHATASE, played a crucial role. Interestingly, FINS1 function in fructose signaling appeared to be independent of its catalytic activity in sugar metabolism. Genetic analysis further indicated that FINS1–dependent fructose signaling may act downstream of the abscisic acid pathway, in spite of the fact that HXK1–dependent glucose signaling works upstream of hormone synthesis. Our findings revealed that multiple layers of controls by fructose, glucose, and abscisic acid finely tune the plant autotrophic transition and modulate early seedling establishment after seed germination

Associated factors to serious infections in a large cohort of juvenile-onset systemic lupus erythematosus from Lupus Registry (RELESSER).

Objective: To assess the incidence of serious infection (SI) and associated factors in a large juvenile-onset systemic lupus erythematosus (jSLE) retrospective cohort. Methods: All patients in the Spanish Rheumatology Society Lupus Registry (RELESSER) who meet =4 ACR-97 SLE criteria and disease onset <18 years old (jSLE), were retrospectively investigated for SI (defined as either the need for hospitalization with antibacterial therapy for a potentially fatal infection or death caused by the infection). Standardized SI rate was calculated per 100 patient years. Patients with and without SI were compared. Bivariate and multivariate logistic and Cox regression models were built to calculate associated factors to SI and relative risks. Results: A total of 353 jSLE patients were included: 88.7% female, 14.3 years (± 2.9) of age at diagnosis, 16.0 years (± 9.3) of disease duration and 31.5 years (±10.5) at end of follow-up. A total of 104 (29.5%) patients suffered 205 SI (1, 55.8%; 2-5, 38.4%; and =6, 5.8%). Incidence rate was 3.7 (95%CI: 3.2–4.2) SI per 100 patient years. Respiratory location and bacterial infections were the most frequent. Higher number of SLE classification criteria, SLICC/ACR DI score and immunosuppressants use were associated to the presence of SI. Associated factors to shorter time to first infection were higher number of SLE criteria, splenectomy and immunosuppressants use. Conclusions: The risk of SI in jSLE patients is significant and higher than aSLE. It is associated to higher number of SLE criteria, damage accrual, some immunosuppressants and splenectomy

Lack of the Long Pentraxin PTX3 Promotes Autoimmune Lung Disease but not Glomerulonephritis in Murine Systemic Lupus Erythematosus

The long pentraxin PTX3 has multiple roles in innate immunity. For example, PTX3 regulates C1q binding to pathogens and dead cells and regulates their uptake by phagocytes. It also inhibits P-selectin-mediated recruitment of leukocytes. Both of these mechanisms are known to be involved in autoimmunity and autoimmune tissue injury, e.g. in systemic lupus erythematosus, but a contribution of PTX3 is hypothetical. To evaluate a potential immunoregulatory role of PTX3 in autoimmunity we crossed Ptx3-deficient mice with Fas-deficient (lpr) C57BL/6 (B6) mice with mild lupus-like autoimmunity. PTX3 was found to be increasingly expressed in kidneys and lungs of B6lpr along disease progression. Lack of PTX3 impaired the phagocytic uptake of apoptotic T cells into peritoneal macrophages and selectively expanded CD4/CD8 double negative T cells while other immune cell subsets and lupus autoantibody production remained unaffected. Lack of PTX3 also aggravated autoimmune lung disease, i.e. peribronchial and perivascular CD3+ T cell and macrophage infiltrates of B6lpr mice. In contrast, histomorphological and functional parameters of lupus nephritis remained unaffected by the Ptx3 genotype. Together, PTX3 specifically suppresses autoimmune lung disease that is associated with systemic lupus erythematosus. Vice versa, loss-of-function mutations in the Ptx3 gene might represent a genetic risk factor for pulmonary (but not renal) manifestations of systemic lupus or other autoimmune diseases