We are concerned with singular elliptic problems of the form $-Δu\pm p(d(x))g(u)=\la f(x,u)+μ|\nabla u|^a$ in $Ω,$ where $Ω$ is a smooth bounded domain in $\RR^N$, $d(x)={\rm dist}(x,\partialΩ),$ $\la>0,$ $μ\in\RR$, $0<a\leq 2$, and $f,k$ are nonnegative and nondecreasing functions. We assume that $p(d(x))$ is a positive weight with possible singular behavior on the boundary of $Ω$ and that the nonlinearity $g$ is unbounded around the origin. Taking into account the competition between the anisotropic potential $p(d(x))$, the convection term $|\nabla u|^a$, and the singular nonlinearity $g$, we establish various existence and nonexistence results.
We show the existence of positive solutions for a class of singular elliptic systems with convection term. The approach combines pseudomonotone operator theory, sub and supersolution method and perturbation arguments involving singular terms.
The present paper concerns with the existence of blow-up solution for a class of elliptic system with convection term. Here, we prove a result involving sub and supersolution for a class of elliptic system whose nonlinearity can depend of the gradient of the solution. This result permits to study the existence of blow-up solution for a large class of systems.
We study radial symmetry of large solutions of the semi-linear elliptic problem Δu + \nabla h.\nabla u = f(|x|,u), and we provide sharp conditions under which the problem has a radial solution. The result is independent of the rate of growth of the solution at infinity.
We establish several results related to existence, nonexistence or bifurcation of positive solutions for a Dirichlet boundary value problem with in a smooth bounded domain. The main feature of this paper consists in the presence of a singular nonlinearity, combined with a convection term. Our approach takes into account both the sign of the potential and the decay rate around the origin of the singular nonlinearity. The proofs are based on various techniques related to the maximum principle for elliptic equations.
In this paper we present some very recent results regarding existence, uniqueness, and multiplicity of solutions for quasilinear elliptic equations and systems, exhibiting both singular and convective reaction terms. The importance of boundary conditions (Dirichlet, Neumann, or Robin) is also discussed. Existence is achieved via sub-supersolution and truncation techniques, fixed point theory, nonlinear regularity, and set-valued analysis, while uniqueness and multiplicity are obtained by monotonicity arguments.
In this paper we study quasilinear elliptic systems driven by so-called double phase operators and nonlinear right-hand sides depending on the gradients of the solutions. Based on the surjectivity result for pseudomonotone operators we prove the existence of at least one weak solution of such systems. Furthermore, under some additional conditions on the data, the uniqueness of weak solutions is shown.
In this note we present $L^\infty$-results for problems of the form $A(x,u,Du)=B(x,u,Du)$ in $Ω$, $u>0$ in $Ω$, $u=0$ on $\partialΩ$, where the growth condition for the function $B\colon Ω\times \mathbb{R}\times \mathbb{R}^N\to \mathbb{R}$ contains both a singular and a convection term. We use ideas from the works of Giacomoni-Schindler-Takac (2007) and the authors (2019) to prove the boundedness of weak solutions for such general problem by applying appropriate bootstrap arguments.
We establish the existence of positive solutions for a nonlinear elliptic Dirichlet problem in dimension $N$ involving the $N$-Laplacian. The nonlinearity considered depends on the gradient of the unknown function and an exponential term. In such case, variational methods cannot be applied. Our approach is based on approximation scheme, where we consider a new class of normed spaces of finite dimension. As a particular case, we extended the result achieved by De Araujo and Montenegro [2016] for any $N>2$.
The weak and strong comparison principles, respectively, are investigated for quasi-linear elliptic boundary value problems with the $p$-Laplacian in one space dimension. We treat the degenerate case of $2 < p < \infty$ and allow also for the nontrivial convection velocity $b\colon [-1,1]\to \mathbb{R}$ in the underlying domain $(-1,1)$. We establish the weak comparison principle under a rather general, natural sufficient condition on the convection velocity, $b(x)$, and the reaction function, $φ(x,u)$. Furthermore, we establish also the strong comparison principle under a number of various ad…
We consider three-dimensional (3D) Boussinesq convection system of an incompressible fluid in a closed sample of a porous medium. Specifically, we introduce and analyze a 3D Brinkman-Forchheimer-Bénard convection problem describing the behavior of an incompressible fluid in a porous medium between two plates heated from the bottom and cooled from the top. We show the existence and uniqueness of global in-time solutions, and the existence of absorbing balls in $L^2$ and $H^1$. Eventually, we comment on the applicability of a data assimilation algorithm to our system.
This paper is concerned with theoretical analysis of a heat and moisture transfer model arising from textile industries, which is described by a degenerate and strongly coupled parabolic system. We prove the global (in time) existence of weak solution by constructing an approximate solution with some standard smoothing. The proof is based on the physcial nature of gas convection, in which the heat (energy) flux in convection is determined by the mass (vapor) flux in convection.
In this contribution we prove the existence of weak solutions to degenerate parabolic systems arising from the coupled moisture movement, transport of dissolved species and heat transfer through partially saturated porous materials. Physically motivated mixed Dirichlet-Neumann boundary conditions and initial conditions are considered. Existence of a global weak solution of the problem is proved by means of semidiscretization in time and by passing to the limit from discrete approximations. Degeneration occurs in the nonlinear transport coefficients which are not assumed to be bounded below and…
In this paper we consider quasilinear elliptic equations with double phase phenomena and a reaction term depending on the gradient. Under quite general assumptions on the convection term we prove the existence of a weak solution by applying the theory of pseudomonotone operators. Imposing some linear conditions on the gradient variable the uniqueness of the solution is obtained.
In this paper, we consider the existence and uniqueness of weak solutions of a nonlinear elliptic equation with a variable exponent, a monotonic type operator and a convection term. With the topological degree theory, we prove the existence of at least one weak solution under some Leray-Lions and growth conditions. Moreover, we obtain the uniqueness of the solution of the problem under some additional assumptions. Our results generalize and improve existing results with another approach.
The existence of a positive entire weak solution to a singular quasi-linear elliptic system with convection terms is established, chiefly through perturbation techniques, fixed point arguments, and a priori estimates. Some regularity results are then employed to show that the obtained solution is actually strong.
In this survey we report on some recent results related to various singular phenomena arising in the study of some classes of nonlinear elliptic equations. We establish qualitative results on the existence, nonexistence or the uniqueness of solutions and we focus on the following types of problems: (i) blow-up boundary solutions of logistic equations; (ii) Lane-Emden-Fowler equations with singular nonlinearities and subquadratic convection term. We study the combined effects of various terms involved in these problems: sublinear or superlinear nonlinearities, singular nonlinear terms, convecti…
We study existence and Lorentz regularity of distributional solutions to elliptic equations with either a convection or a drift first order term. The presence of such a term makes the problem not coercive. The main tools are pointwise estimates of the rearrangements of both the solution and its gradient.
We are concerned with singular elliptic equations of the form $-Δu= p(x)(g(u)+ f(u)+|\nabla u|^a)$ in $\RR^N$ ($N\geq 3$), where $p$ is a positive weight and $0< a <1$. Under the hypothesis that $f$ is a nondecreasing function with sublinear growth and $g$ is decreasing and unbounded around the origin, we establish the existence of a ground state solution vanishing at infinity. Our arguments rely essentially on the maximum principle.
The existence of entire solutions to quasilinear elliptic systems exhibiting both singular and convective reaction terms is discussed. An auxiliary problem, obtained by `freezing' the convection terms and `shifting' the singular ones, is first solved. Then, a priori estimates, fixed point arguments, nonlinear regularity, compactness results concerning the gradient terms, besides a regularization-localization procedure, yield the existence of a weak solution to the problem.
We establish some existence and regularity results to the Dirichlet problem, for a class of quasilinear elliptic equations involving a partial differential operator, depending on the gradient of the solution. Our results are formulated in the Orlicz Sobolev spaces and under general growth conditions on the convection term. The sub and supersolutions method is a key tool in the proof of the existence results.
In this paper we study the existence and partial regularity of weak solutions to an elliptic-parabolic system that models the single-phase miscible displacement of one incompressible fluid by another in a porous media. The system is singular and involves quadratic nonlinearity. A partial regularity result is obtained via the Fefferman-Stein inequality.
We study an elliptic equation with measurable coefficients arising from photo-acoustic imaging in inhomogeneous media. We establish Holder continuity of weak solutions and obtain pointwise bounds for Green's functions subject to Dirichlet or Neumann condition.
We report on some recent existence and uniqueness results for elliptic equations subject to Dirichlet boundary condition and involving a singular nonlinearity. We take into account the following types of problems: (i) singular problems with sublinear nonlinearity and two parameters; (ii) combined effects of asymptotically linear and singular nonlinearities in bifurcation problems; (iii) bifurcation for a class of singular elliptic problems with subquadratic convection term. In some concrete situations we also establish the asymptotic behaviour of the solution around the bifurcation point. Our …
In this paper we propose a continuous data assimilation (downscaling) algorithm for the Bénard convection in porous media using only coarse mesh measurements of the temperature. In this algorithm, we incorporate the observables as a feedback (nudging) term in the evolution equation of the temperature. We show that under an appropriate choice of the nudging parameter and the size of the mesh, and under the assumption that the observed data is error free, the solution of the proposed algorithm converges at an exponential rate, asymptotically in time, to the unique exact unknown reference solutio…
In this note we study periodic homogenization of Dirichlet problem for divergence type elliptic systems when both the coefficients and the boundary data are oscillating. One of the key difficulties here is the determination of the fixed boundary data corresponding to the limiting (homogenized) problem. This issue has been addressed in recent papers by D. Gérard-Varet and N. Masmoudi, and by C. Prange, however, not much is known about the regularity of this fixed data. The main objective of this note is to initiate a study of this problem, and to prove several regularity results in this connect…
We analyze a second order, linear, elliptic PDE with mixed boundary conditions. This problem arose as a limiting case of a Markov-modulated queueing model for data handling switches in communications networks. We use singular perturbation methods to analyze the problem. In particular we use the ray method to solve the PDE in the limit where convection dominates diffusion. We show that there are both interior and boundary caustics, as well as a cusp point where two caustics meet, an internal layer, boundary layers and a corner layer. Our analysis leads to approximate formulas for the queue leng…
The paper deals with the homogenization of reaction-diffusion equations with large reaction terms in a multi-scale porous medium. We assume that the fractures and pores are equidistributed and that the coefficients of the equations are periodic. Using the multi-scale convergence method, we derive a homogenization result whose limit problem is defined on a fixed domain and is of convection-diffusion-reaction type.
The paper deals with the existence of solutions for quasilinear elliptic systems involving singular and convection terms with variable exponents. Our approach combines the sub-supersolutions method and Schauder's fixed point theorem.
We consider the homogenization of a model of reactive flows through periodic porous media involving a single solute which can be absorbed and desorbed on the pore boundaries. This is a system of two convection-diffusion equations, one in the bulk and one on the pore boundaries, coupled by an exchange reaction term. The novelty of our work is to consider a nonlinear reaction term, a so-called Langmuir isotherm, in an asymptotic regime of strong convection. We therefore generalize previous works on a similar linear model [G. Allaire et al, Chemical Engineering Science, 65 (2010), pp.2292-2300], …
We consider an incompressible non-isothermal fluid flow with non-linear slip boundary conditions governed by Tresca's friction law. We assume that the stress tensor is given as $σ= 2 μ\bigl( θ, u, | D(u) |) |D(u) |^{p-2} D(u) - π{\rm Id}$ where $θ$ is the temperature, $π$ is the pressure, $u$ is the velocity and $D(u)$ is the strain rate tensor of the fluid while $p$ is a real parameter. The problem is thus given by the $p$-Laplacian Stokes system with subdifferential type boundary conditions coupled to a $L^1$ elliptic equation describing the heat conduction in the fluid. We establish first a…
We consider the quadratic Schrödinger system $$iu_t+Δ_{γ_1}u+\overline{u}v=0$$ $$2iv_t+Δ_{γ_2}v-βv+\frac 12 u^2=0,$$ where $t\in\mathbf{R},\,x\in \mathbf{R}^d\times \mathbf{R}$, in dimensions $1\leq d\leq 4$ and for $γ_1,γ_2>0$, the so-called elliptic-elliptic case. We show the formation of singularities and blow-up in the $L^2$-(super)critical case. Furthermore, we derive several stability results concerning the ground state solutions of this system.
We solve time-harmonic Maxwell's equations in anisotropic, spatially homogeneous media in intersections of $L^p$-spaces. The material laws are time-independent. The analysis requires Fourier restriction-extension estimates for perturbations of Fresnel's wave surface. This surface can be decomposed into finitely many components of the following three types: smooth surfaces with non-vanishing Gaussian curvature, smooth surfaces with Gaussian curvature vanishing along one-dimensional submanifolds, but without flat points, and surfaces with conical singularities. Our estimates are based on new Boc…
We consider a class of $L^1$ critical nonlocal aggregation equations with linear or nonlinear porous media-type diffusion which are characterized by a long-range interaction potential that decays faster than the Newtonian potential at infinity. The fast decay breaks the $L^1$ scaling symmetry and we prove that `sufficiently spread out' initial data, regardless of the mass, result in global spreading solutions. This is in contrast to the classical parabolic-elliptic PKS for which essentially all solutions with more than critical mass are known to blow up in finite time. In all cases, the long-t…
In this paper we study optimal control problems governed by a semilinear elliptic equation. The equation is nonmonotone due to the presence of a convection term, despite the monotonocity of the nonlinear term. The resulting operator is neither monotone nor coervive. However, by using conveniently a comparison principle we prove existence and uniqueness of solution for the state equation. In addition, we prove some regularity of the solution and differentiability of the relation control-to-state. This allows us to derive first and second order conditions for local optimality.
We study the nonlinear elliptic problem $-Δu=ρ(x)f(u)$ in $\RR^N$ ($N\geq 3$), $\lim\_{|x|\ri\infty}u(x)=\ell$, where $\ell\geq 0$ is a real number, $ρ(x)$ is a nonnegative potential belonging to a certain Kato class, and $f(u)$ has a sublinear growth. We distinguish the cases $\ell>0$ and $\ell=0$ and we prove existence and uniqueness results if the potential $ρ(x)$ decays fast enough at infinity. Our arguments rely on comparison techniques and on a theorem of Brezis and Oswald for sublinear elliptic equations.
The paper concerns the discreteness of the eigenvalues and the solvability of the interior transmission problem for anisotropic media. Conditions for the ellipticity of the problem are written explicitly, and it is shown that they do not guarantee the discreteness of the eigenvalues. Some simple sufficient conditions for the discreteness and solvability are found. They are expressed in terms of the values of the anisotropy matrix and the refraction index at the boundary of the domain. The discreteness of the eigenvalues and the solvability of the interior transmission problem are shown if a sm…
A finite volume method on general meshes for a degenerate parabolic convection-reaction-diffusion equation
EDF R&D（法国电力集团研发部）；Laboratoire de Mathématiques, Université de Paris-Sud 11（巴黎第十一大学数学实验室）；CNRS and Laboratoire de Mathématiques, Université de Paris-Sud 11（法国国家科学研究中心及巴黎第十一大学数学实验室）
We present a complete analysis of the problem of convection-diffusion in low Re, 2-dimensional flows with distributions of singularities, such as those found in open-space microfluidics and in groundwater flows. Using Boussinesq transformations and solving the problem in streamline coordinates, we obtain concentration profiles in flows with complex arrangements of sources and sinks for both high and low Pe. These yield the complete analytical concentration profile at every point in applications that previously relied on material surface tracking, local lump models or numerical analysis such as…
