Global solutions for the Muskat problem in the scaling invariant Besov space B˙∞,11

2022 ◽  
Vol 394 ◽  
pp. 108122
Author(s):  
Huy Q. Nguyen
2019 ◽  
Vol 149 (5) ◽  
pp. 1175-1188
Author(s):  
Léo Agélas

AbstractWe consider complex-valued solutions of the conserved Kuramoto–Sivashinsky equation which describes the coarsening of an unstable solid surface that conserves mass and that is parity symmetric. This equation arises in different aspects of surface growth. Up to now, the problem of existence and smoothness of global solutions of such equations remained open in ℝd and in the torus 𝕋d, d ⩾ 1. In this paper, we answer partially to this question. We prove the finite time blow-up of complex-valued solutions associated with a class of large initial data. More precisely, we show that there is complex-valued initial data that exists in every Besov space (and hence in every Lebesgue and Sobolev space), such that after a finite time, the complex-valued solution is in no Besov space (and hence in no Lebesgue or Sobolev space).


Author(s):  
Ajay Bhushan Prasad

21<sup>st</sup> Century is an era of stress and burnout. For the past few decades it has been a burning and hot topic of discussion for researchers, to evolve stress and burnout- from a global problem to global solutions, as it affects the life of individuals in an unprecedented manner and touches them at workplace across the globe. Everybody knows what stress is all about. It has become a part of life and perhaps, to some extent, necessary at work and outside work. Some people are more productive and creative when they work under stress. But if stress is intense and continuous, then it becomes a negative phenomenon leading to physical illness and psychological disorders. Stress and burnout in today's environment has become a well documented problem. Various researches have evaluated stress and burnout in workplaces. Stress is a non-specific response of body to any demand made on it. Many researchers have identified that stress and burnout has become an integral part of our daily life due to the negative aspects of job, such as, multiple responsibilities, disciplinary problems, employee's apathy, involuntary transfers, inadequate pay and perks, less chances of career advancement and lack of administrative support etc. As a result of these, individuals are likely to suffer from stress and may experience a sense of tiredness and frustration. When prolonged stress continues and it is not effectively managed, it can even lead to symptoms of burnout, a state of emotional exhaustion, depersonalization and diminished personal accomplishment in the workplace. Stress management can be done through six zones which we have discussed in detail, with a holistic approach. It includes health zone, intimate zone, family zone, work zone, social zone and spiritual zone. In this paper, an effort is made to discuss the solutions of stress and burnout in different and innovative ways through different zones which have been experienced by the author in his 20 years of professional career. Thus, stress and burnout are not a trivial problem but a major dysfunction of the 21<sup>st</sup> century, which has far reaching impact on quality and quantity of productivity. The present paper explores the concept of stress and burnout, the major differences between them, factors leading to genesis of the problem, various symptoms and how it is a serious quality concern for all professionals. Appropriate interventions for prevention and management of stress and burnout are also suggested.


2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Irena Lasiecka ◽  
Buddhika Priyasad ◽  
Roberto Triggiani

Abstract We consider the 𝑑-dimensional Boussinesq system defined on a sufficiently smooth bounded domain and subject to a pair { v , u } \{v,\boldsymbol{u}\} of controls localized on { Γ ~ , ω } \{\widetilde{\Gamma},\omega\} . Here, 𝑣 is a scalar Dirichlet boundary control for the thermal equation, acting on an arbitrarily small connected portion Γ ~ \widetilde{\Gamma} of the boundary Γ = ∂ ⁡ Ω \Gamma=\partial\Omega . Instead, 𝒖 is a 𝑑-dimensional internal control for the fluid equation acting on an arbitrarily small collar 𝜔 supported by Γ ~ \widetilde{\Gamma} . The initial conditions for both fluid and heat equations are taken of low regularity. We then seek to uniformly stabilize such Boussinesq system in the vicinity of an unstable equilibrium pair, in the critical setting of correspondingly low regularity spaces, by means of an explicitly constructed, finite-dimensional feedback control pair { v , u } \{v,\boldsymbol{u}\} localized on { Γ ~ , ω } \{\widetilde{\Gamma},\omega\} . In addition, they will be minimal in number and of reduced dimension; more precisely, 𝒖 will be of dimension ( d - 1 ) (d-1) , to include necessarily its 𝑑-th component, and 𝑣 will be of dimension 1. The resulting space of well-posedness and stabilization is a suitable, tight Besov space for the fluid velocity component (close to L 3 ⁢ ( Ω ) \boldsymbol{L}^{3}(\Omega) for d = 3 d=3 ) and a corresponding Besov space for the thermal component, q > d q>d . Unique continuation inverse theorems for suitably over-determined adjoint static problems play a critical role in the constructive solution. Their proof rests on Carleman-type estimates, a topic pioneered by M. V. Klibanov since the early 80s.


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