PDEs
Partial Differential Equations and Applications
Tuesday at 10:50 (Sala: A259 , IME-USP)
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PAST SEMINARS
June 25, 2024 Pierluigi BenevieriIME-USPGlobal bifurcation results for a delay differential system representing a chemostat modelIn this talk, we explore a global bifurcation result for periodic solutions of the following delayed first-order system, dependent on a real parameter $\lambda \geq 0$, \begin{equation} \left\{ \begin{array}{lll} s'(t)=Ds^0(t)-Ds(t)-\dfrac{\lambda}{\gamma}\,\mu(s(t))x(t) && t \geq 0 \\ x'(t)=x(t)\big[\lambda\mu(s(t-\tau))-D\big] && t \geq 0, \end{array} \right. \end{equation} where the following conditions hold: |
June 05, 2024 Mostafa AdimyNRIA Lyon and Institut Camille Jordan, University LyonContinuous-Time Differential-Difference Models in Population Dynamics and EpidemiologyWe are interested in population dynamics and epidemiology models that involve two phases, one active and one inactive, with exchanges between the two phases. The durations of these two phases can be finite or infinite. Depending on the phenomenon under study, the active/inactive phases may have different interpretations. For example, in a population model composed of immature and mature individuals, the mature phase may be considered active and the immature phase inactive. In a predator-prey model, for example, the inactive phase may represent a refuge where the prey population is protected from predators. It may also represent a resting phase during which the predator does not hunt. In epidemiology, it may represent a period of immunity due to vaccination. For a population of cells, the inactive phase may correspond to the G1/G0 phase of the cell cycle, and the active phase to the S/G2/M stages leading to cell division. One of the questions addressed by this type of model is the impact of the inactive phase on the asymptotic behavior of the solutions. We will use age-structured PDE and continuous-time differential-difference models to investigate these issues. Show abstract |
May 21, 2024 Juliana Honda LopesIME-USPExistence of weak solutions for a nonhomogeneous incompressible cell-fluid Navier-Stokes model with chemotaxisThis work is concerned with the mathematical analysis of a general cell-fluid Navier-Stokes model with the inclusion of chemotaxis proposed recently by Y. Qiao and S. Evje. This general model relies on a mixture theory multi-phase formulation. It consists of two mass balance equations and two general momentum balance equations, respectively, for the cell and fluid phase, combined with a convection-diffusion-reaction equation for oxygen. We investigate the existence of weak solutions in a two or three-dimensional bounded domain when the fluids are assumed to be incompressible with constant volume fraction. Show abstract |
May 07, 2024 Gustavo de Paula RamosIME-USPSoluções ground state para equações tipo Hartree em $\mathbb{R}^3$ com potencial deltaConsidere a equação de tipo Hartree em $\mathbb{R}^3$ com um potencial delta descrita formalmente por $$\mathrm{i} \partial_t \psi=
- \Delta_x \psi+
\gamma \delta_0 \psi-
(I_\beta \ast |\psi|^p) |\psi|^{p - 2} \psi,
$$onde $\gamma \in \mathbb{R}$, $0 < \beta < 3$ e $I_\beta$ denota o potencial de Riesz. |
April 23, 2024 Sergey TikhomirovPUC-Rio de JaneiroReaction-diffusion equations with hysteresis operatorHysteresis naturally appears as a mechanism of self-organization and is often used in control theory. Important features of hysteresis operators are discontinuity and memory. We consider reaction-diffusion equations with hysteresis. Such equations describe processes in which diffusive and non-diffusive instances interact according to a hysteresis law. Due to the discontinuity of hysteresis, questions of well-posedness of such equations are highly non trivial. |
April 02, 2024 Estefani Moraes MoreiraIME-USPA non-local quasilinear parabolic problem: a study of bifurcation and hyperbolicityIn this talk, we will consider a nonlocal one-dimensional parabolic problem that depends on a parameter. For the problem, we will analyse the bifurcation and hyperbolicity of equilibria. We will also comment on a particular case for which we can describe the structure of the global attractor. Show abstract |
March 20, 2024 Alexander Muñoz GarciaIME-USPOn nonlinear dispersive equations in weighted Sobolev spacesIn this talk, we commence by examining the interplay between regularity and decay in the context of local well-posedness for various nonlinear dispersive equations posed in weighted Sobolev spaces with polynomial weights. We then proceed to discuss a unified approach while also addressing the optimality of these relationships, offering insights into the fundamental limits and constraints that govern the decay aspects of the dispersive equations. Show abstract |
November 08, 2023 Gustavo de Paula RamosIME-USPConstruindo soluções para problemas variacionais elípticos com Lyapunov–SchmidtEssa palestra se presta a apresentar, sem entrar em tecnicalidades, um método perturbativo derivado de Lyapunov–Schmidt que permite construir soluções para problemas variacionais semilineares elípticos através da redução ao problema de existência de ponto crítico para funcionais em espaços de dimensão finita. Terminamos apresentando como empregamos esse método recentemente para construir soluções de sistemas que modelam a auto-interação eletrostática de partículas quânticas, como o sistema Schrödinger–Bopp–Podolsky. Show abstract |
October 25, 2023 Marcone Corrêa PereiraIME-USPRoughness-induced effects on reaction-diffusion problems and thin domainsWe discuss a convection-diffusion-reaction problem in a thin domain endowed with the Robin-type boundary condition describing the reaction catalyzed by the upper wall. Motivated by the microfluidic applications, we allow the oscillating behavior of the upper boundary and analyze the resonant case where the amplitude and period of the oscillation have the same small order as the domain’s thickness. Depending on the magnitude of the reaction mechanism, we rigorously derive three different asymptotic models via the unfolding operator method. In particular, we identify the critical case in which the effects of the domain’s geometry and all physically relevant processes become balanced. It is a joint work with Igor Pazanin and J. C. Nakasato. Show abstract |
October 11, 2023 Phillipo LappicyUniversidad Complutense de MadridPerturbando o Big BangEsta será uma palestra introdutória sobre a dinâmica caótica do Big Bang. Este tópico tem atraído a atenção de matemáticos e físicos desde a abordagem heurística de Belinski, Khalatnikov e Lifshitz (conhecida como conjectura de BKL) e a construção do atrator Mixmaster de Misner. Veremos como uma perturbação específica da singularidade do Big Bang irá revelar algumas características dinâmicas bem conhecidas e outras totalmente novas. Por exemplo, veremos relações com dinâmica simbólica, conjuntos de Cantor, frações contínuas, etc. Além disso, veremos como tais perturbações produzem boas (ou não) aproximações da teoria geral da relatividade de Einstein. Estes resultados foram fruto de colaborações com K.E. Church (U Montreal), V.H. Daniel (Columbia U), J. Hell (FU Berlin), O. Hénot (McGill U), J.P. Lessard (McGill U), H. Sprink (FU Berlin) e C. Uggla (Karlstad U ) Show abstract |
June 20, 2023 Juliana Honda LopesIME-USPOn a non-isothermal incompressible Navier-Stokes-Allen-Cahn systemThis work is devoted to the study of a non-isothermal incompressible Navier-Stokes-Allen-Cahn system which can be considered as a model describing the motion of the mixture of two viscous incompressible fluids. This kind of models are physically relevant for the analysis of non- isothermal fluids. The governing system of nonlinear partial differential equations consists of the Navier-Stokes equations coupled with a phase- field equation, which is the convective Allen-Cahn equation type, and an energy transport equation for the temperature. More precisely, we consider the following system $$u_t + u · \nabla u − \nabla· (\nu(\theta)Du) + \nabla \rho = \lambda(−\varepsilon\Delta\phi + F'(\phi))\nabla\phi − \alpha\Delta \theta\nabla \theta$$ $$\nabla· u = 0$$ $$\phi_t + u ·\nabla\phi = \gamma( \varepsilon\Delta\phi - F'(\phi))$$ $$\theta_t + u · \nabla \theta = \nabla · (\kappa(\theta)\nabla \theta )$$ in $\Omega\times (0,\infty)$, where $\Omega$ is a bounded domain in $\mathbb{R},$ $ n = 2, 3$, with smooth boundary $\partial \Omega.$ We investigated the well-posedness of the nonlinear system. More precisely, existence and uniqueness of local strong solutions in two and three dimensions for any initial data are proved. Moreover, existence of global weak solutions and existence and uniqueness of global strong solution in dimension two, when the initial temperature is suitably small, are established. Show abstract |
June 06, 2023 Gaetano SicilianoIME-USPCritical points under the constraint of energyIn the talk we discuss the existence of critical points for a family of abstract and smooth functionals on Banach spaces under the energy constraint, in other words when the energy level at which we look for critical points is a priori given. By means of the Ljusternick-Schnirelmann theory and the fibering method of Pohozaev we show, under suitable assumptions, multiplicity results. The abstract framework is then applied to some partial differential equations depending on a parameter for which we obtain multiple solutions as well as some bifurcation results. Show abstract |
May 23, 2023 Fábio NataliUniversidade Estadual de MaringáTransversal spectral instability of periodic travelling waves for the generalized Zakharov-Kuznetsov equationIn this talk, we determine the transverse instability of periodic traveling wave solutions of the generalized Zakharov-Kuznetsov equation in two spatial dimensions. By adapting a well-known criterion used for solitary waves to the periodic context, we are able to prove that all positive and one-dimensional $L-$periodic waves are spectrally (transversally) unstable. Moreover, if periodic waves that change their sign exist, we can also establish the same property, provided that the associated projection operator defined in the zero-mean Sobolev space has only one negative eigenvalue. Show abstract |
May 02, 2023 João Fernando NariyoshiIME-USPVelocity averaging lemmas for general second-order equationsIn this talk, we will present and discuss some old and new theorems regarding the regularity of the so-called velocity averages $$\int_{\mathbb R_v} f(\mathbf x,v) \psi(v)\,dv.$$ Here, $\psi(v)$ is a given real function, and $f(\mathbf x,v)$ solves a general, multidimensional second-order equation of the form $$\sum_{j=1}^N\mathbf a_j(v) \frac{\partial f}{\partial \mathbf x_j}(\mathbf x,v) - \sum_{j,k=1}^N \mathbf{b}_{jk}(v) \frac{\partial^2 f}{\partial \mathbf x_j \partial \mathbf x_k}(\mathbf x,v) = g(\mathbf x,v).$$ As P.-L. Lions, B. Perthame, and E. Tadmor [J. Amer. Math. Soc., 7 (1994) 169-191] ingeniously demonstrated, these results, commonly known as ``velocity averaging lemmas'', have profound consequences in the theory of degenerate hyperbolic-parabolic equations such as $$\frac{\partial \varrho}{\partial t} + \sum_{j=1}^N \frac{\partial}{\partial x_j} \mathbf A_j(\varrho) - \sum_{j,k=1}^N \frac{\partial^2}{\partial x_j \partial x_k} \mathbf B_{jk}(\varrho) = 0.$$ We provide novel compactness and Sobolev regularity principles for such velocity averages, thus justifying and extending several propositions envisioned in the celebrated work of Lions-Perthame-Tadmor. Moreover, we also derive some criteria for technical hypotheses commonly known as ``non-degeneracy conditions''; in this fashion, we are able to generalize and correct all the examples of equations that satisfy such assumptions. Show abstract |
April 18, 2023 Jaime Angulo PavaIME-USPDynamic of soliton-profiles on metric graphsIn this talk we will give initially a brief summary of the orbital stability problem of soliton solutions for the Korteweg-de Vries equation (KdV) on the line. Afterwards, we will see how this model can be extended over branched structures (balanced metric graphs) and thus we study the existence and stability of soliton-profiles. The arguments presented in this talk have prospects for the study of the stability of other soliton-profiles for the KdV solutions and/or for other nonlinear evolution equations on branched systems. Show abstract |