We consider a class of fully nonlinear nonlocal degenerate elliptic operators which are modeled on the fractional Laplacian and converge to the truncated Laplacians. We investigate the validity of (strong) maximum and minimum principles, and their relation with suitably defined principal eigenvalues. We also show a Hopf type Lemma, the existence of solutions for the corresponding Dirichlet problem, and representation formulas in some particular cases.
There has been recently an important interest in deriving rigorously the Cahn-Hilliard equation from the nonlocal equation, also called aggregation equation. So far, only non-degenerate mobilities were treated. Since we are motivated by models for the biomechanics of living tissues, it is useful to include degenerate motilities. In this framework, we present a new method to show the convergence of the nonlocal to the local degenerate Cahn-Hilliard equation. The method includes the use of nonlocal Poincaré and compactness inequalities.
Conditions for the existence and uniqueness of weak solutions for a class of nonlinear nonlocal degenerate parabolic equations are established. The asymptotic behaviour of the solutions as time tends to infinity are also studied. In particular, the finite time extinction and polynomial decay properties are proved.
We provide a rigorous mathematical framework to establish the limit of a nonlocal model of cell-cell adhesion system to a local model. When the parameter of the nonlocality goes to 0, the system tends to a Cahn-Hilliard system with degenerate mobility and cross interaction forces. Our analysis relies on a priori estimates and compactness properties.
This work is concerned with new results on long-time dynamics of a class of hyperbolic evolution equations related to extensible beams with three distinguished nonlocal nonlinear damping terms. In the first possibly degenerate case, the results feature the existence of a family of compact global attractors and a thickness estimate for their Kolmogorov's $\varepsilon$-entropy. Then, in the non-degenerate context, the structure of the helpful nonlocal damping leads to the existence of finite-dimensional compact global and exponential attractors. Lastly, in a degenerate and critical framework, it…
We study a Dirichlet problem in the entire space for some nonlocal degenerate elliptic operators with internal nonlinearities. With very mild assumptions on the boundary datum, we prove existence and uniqueness of the solution in the viscosity sense. If we further assume uniform ellipticity then the solution is shown to be classical, and even smooth if both the operator and the boundary datum are smooth.
In this paper, for a degenerate higher order equation with a fractional derivative in the sense of Caputo, a nonlocal problem with conjugation conditions in a rectangular domain is studied. The solution is constructed in the form of a Fourier series in the eigenfunctions of a one-dimensional problem. A criterion for the uniqueness of a solution is given.
We show that weak solutions of degenerate Navier-Stokes equations converge to the strong solutions of the pressureless Euler system with linear drag term, Newtonian repulsion and quadratic confinement. The proof is based on the relative entropy method using the artificial velocity formulation for the one-dimensional Navier-Stokes system.
Existence and uniqueness of solutions for nonlocal Cahn-Hilliard equations with degenerate potential is shown. The nonlocality is described by means of a symmetric singular kernel not falling within the framework of any previous existence theory. A convection term is also taken into account. Building upon this novel existence result, we prove convergence of solutions for this class of nonlocal Cahn-Hilliard equations to their local counterparts, as the nonlocal convolution kernels approximate a Dirac delta. Eventually, we show that, under suitable assumptions on the data, the solutions to the …
We consider equations involving a combination of local and nonlocal degenerate $p$-Laplace operators. The main contribution of the paper is almost Lipschitz regularity for the homogeneous equation and Hölder continuity with an explicit Hölder exponent in the general case. For certain parameters, our results also imply Hölder continuity of the gradient. In addition, we establish existence, uniqueness and local boundedness. The approach is based on an iteration in the spirit of Moser combined with an approximation method.
We investigate a class of systems of partial differential equations with nonlinear cross-diffusion and nonlocal interactions, which are of interest in several contexts in social sciences, finance, biology, and real world applications. Assuming a uniform "coerciveness" assumption on the diffusion part, which allows to consider a large class of systems with degenerate cross-diffusion (i.e. of porous medium type) and relaxes sets of assumptions previously considered in the literature, we prove global-in-time existence of weak solutions by means of a semi-implicit version of the Jordan-Kinderlehre…
We establish a local null controllability result for following the nonlinear parabolic equation: $$u_t-\left(b\left(x,\int_0^1u \ \right)u_x \right)_x+f(t,x,u)=hχ_ω,\ (t,x)\in (0,T)\times (0,1) $$ where $b(x,r)=\ell(r)a(x)$ is a function with separated variables that defines an operator which degenerates at $x=0$ and has a nonlocal term. Our approach relies on an application of Liusternik's inverse mapping theorem that demands the proof of a suitable Carleman estimate.
We study the existence of non-trivial, non-negative periodic solutions for systems of singular-degenerate parabolic equations with nonlocal terms and satisfying Dirichlet boundary conditions. The method employed in this paper is based on the Leray-Schauder topological degree theory. However, verifying the conditions under which such a theory applies is more involved due to the presence of the singularity. The system can be regarded as a possible model of the interactions of two biological species sharing the same isolated territory, and our results give conditions that ensure the coexistence o…
In this paper, we study a nonlocal degenerate parabolic equation of order α + 2 for 0<α<2. The equation is a generalization of the one arising in the modeling of hydraulic fractures studied by Imbert and Mellet in 2011. Using the same approach, we prove the existence of solutions for this equation for 0<α<2 and for nonnegative initial data satisfying appropriate assumptions. The main difference is the compactness results due to different Sobolev embeddings. Furthermore, for α > 1, we construct a nonnegative solution for nonnegative initial data under weaker assumptions.
Even without a variational background, a multiplicity result of positive solutions with ordered $L^{p}(Ω)$-norms is provided to the following boundary value problem \begin{equation*} \left \{ \begin{array}{ll} -a(\int_Ωu^{p}dx)Δu = f(u) & \mbox{in $Ω$,}\\ u=0 & \mbox{on $\partialΩ$,} \end{array}\right. \end{equation*} where $Ω$ is a bounded domain and $a$, $f$ are continuous real functions with $a$ vanishing in many positive points.
A unified treatment of degenerate nonlocal elliptic problems
University of the National Education Commission（国家教育委员会大学）；CIEM-Conicet, Universidad Nacional de Córdoba（科尔多瓦国立大学CONICET-CEM研究所）；Universidade Federal do Pará（帕拉联邦大学）；Universidade Federal de Goiás（戈亚斯联邦大学）
In this paper, we are concerned with the internal control of a class of one-dimensional nonlinear parabolic systems with nonlocal and weakly degenerate diffusion coefficients. Our main theorem establishes a local null controllability result with only one internal control for a system of two equations. The proof, based on the ideias developed by Fursikov and Imanivilov, is obtained from the global null controllability of the linearized system provided by \textit{Lyusternik's Inverse Mapping Theorem}. This work extends the results previously treated by the authors for just one equation. For the …
We prove convergence of positive solutions to \[ u_t = uΔu + u\int_Ω |\nabla u|^2, \qquad u\rvert_{\partialΩ} =0, \qquad u(\cdot,0)=u_0 \] in a bounded domain $Ω\subset \mathbb{R}^n$, $n\ge 1$, with smooth boundary in the case of $\int_Ωu_0=1$ and identify the $W_0^{1,2}(Ω)$-limit of $u(t)$ as $t\to \infty$ as the solution of the corresponding stationary problem. This behaviour is different from the cases of $\int_Ωu_0<1$ and $\int_Ωu_0>1$ which are known to result in convergence to zero or blow-up in finite time, respectively. The proof is based on a monotonicity property of $\int_Ω |\nabla u…
We consider an epidemic model with nonlocal diffusion and free boundaries, which describes the evolution of an infectious agents with nonlocal diffusion and the infected humans without diffusion, where humans get infected by the agents, and infected humans in return contribute to the growth of the agents. The model can be viewed as a nonlocal version of the free boundary model studied by Ahn, Beak and Lin \cite{ABL2016}, with its origin tracing back to Capasso et al. \cite{CP1979, CM1981}. We prove that the problem has a unique solution defined for all $t>0$, and its long-time dynamical behavi…
We study a Cahn-Hilliard-Hele-Shaw (or Cahn-Hilliard-Darcy) system for an incompressible mixture of two fluids. The relative concentration difference $φ$ is governed by a convective nonlocal Cahn-Hilliard equation with degenerate mobility and logarithmic potential. The volume averaged fluid velocity $\mathbf{u}$ obeys a Darcy's law depending on the so-called Korteweg force $μ\nabla φ$, where $μ$ is the nonlocal chemical potential. In addition, the kinematic viscosity $η$ may depend on $φ$. We establish first the existence of a global weak solution which satisfies the energy identity. Then we p…
In this paper we study the geometry of certain functionals associated to quasilinear elliptic boundary value problems with a degenerate nonlocal term of Kirchhoff type. Due to the degeneration of the nonlocal term it is not possible to directly use classical results such as uniform a-priori estimates and "Sobolev versus Hölder local minimizers" type of results. We prove that results similar to these hold true or not, depending on how degenerate the problem is. We apply our findings in order to show existence and multiplicity of solutions for the associated quasilinear equations, considering se…
In recent years, there has been a spike in the interest in multi-phase tissue growth models. Depending on the type of tissue, the velocity is linked to the pressure through Stoke's law, Brinkman's law or Darcy's law. While each of these velocity-pressure relations has been studied in the literature, little emphasis has been placed on the fine relationship between them. In this paper, we want to address this dearth in the literature, providing a rigorous argument that bridges the gap between a viscoelastic tumour model (of Brinkman type) and an inviscid tumour model (of Darcy type).
We consider a diffuse interface model for incompressible isothermal mixtures of two immiscible fluids with matched constant densities. This model consists of the Navier-Stokes system coupled with a convective nonlocal Cahn-Hilliard equation with non-constant mobility. We first prove the existence of a global weak solution in the case of non-degenerate mobilities and regular potentials of polynomial growth. Then we extend the result to degenerate mobilities and singular (e.g. logarithmic) potentials. In the latter case we also establish the existence of the global attractor in dimension two. Us…
Existence and uniqueness are established for a degenerate regularization of the well-known Perona-Malik equation proposed by the first author for non-smooth initial data. The results heavily rely on the choice of appropriate functional setting inspired by a recent approach to degenerate parabolic equations via so-called singular Riemannian manifolds introduced by Herbert Amann.
We consider a family of non-local problems that model the effects of transport and vortex stretching in the incompressible Euler equations. Using modulation techniques, we establish stable self-similar blow-up near a family of known self-similar blow-up solutions.
We prove the comparison principle for viscosity sub and super solutions of degenerate nonlocal operators with general nonlocal gradient nonlinearities. The proofs apply to purely Hamilton-Jacobi equations of order $0<s<1$.
We state and prove a general Harnack inequality for minimizers of nonlocal, possibly degenerate, integro-differential operators, whose model is the fractional p-Laplacian.
This paper is devoted to the analysis of non-negative solutions for a generalisation of the parabolic equation with porous medium like nonlinear diffusion and nonlinear nonlocal reaction. We investigate under which conditions equilibration between two competing effects, repulsion modelled by nonlinear diffusion and aggregation modelled by nonlinear reaction, occurs. Precisely, we exhibit that the qualitative behavior of solutions is decided by the nonlinear diffusion which is chosen in such a way that its scaling and the reaction term coincide, i.e. that there is a critical exponent $m+2/n$ fo…
We introduce a new class of quasilinear nonlocal operators and study equations involving these operators. The operators are degenerate elliptic and may have arbitrary growth in the gradient. Included are new nonlocal versions of p-Laplace, $\infty$-Laplace, mean curvature of graph, and even strongly degenerate operators, in addition to some nonlocal quasilinear operators appearing in the existing literature. Our main results are comparison, uniqueness, and existence results for viscosity solutions of linear and fully nonlinear equations involving these operators. Because of the structure of ou…
In this paper we obtain interior regularity estimates for viscosity solutions of nonlocal Dirichlet problems that degenerate when the gradient of the solution vanishes. Interior Hölder estimates are obtained when the order of the fractional diffusion is less or equal than one, and Lipschitz estimates when it is bigger than one. In the latter case, the estimates are robust enough to conclude interior $C^{1, α}$ regularity by an improvement of the flatness procedure, which is possible when the nonlocal term is close enough to a second-order diffusion.
In this work, Holder continuity is obtained for solutions to the nonlocal kinetic Fokker-Planck Equation, and to a family of related equations with general integro-differential operators. These equations can be seen as a generalization of the Fokker-Planck Equation, or as a linearization of non-cutoff Boltzmann. Difficulties arise because our equations are hypoelliptic, so we utilize the theory of averaging lemmas. Regularity is obtained using De Giorgi's method, so it does not depend on the regularity of initial conditions or coefficients. This work assumes stronger constraints on the nonloca…
In this paper, we consider a two-dimensional diffuse interface model for the phase separation of an incompressible and isothermal binary fluid mixture with matched densities. This model consists of the Navier--Stokes equations, nonlinearly coupled with a convective nonlocal Cahn--Hilliard equation. The system rules the evolution of the volume-averaged velocity $\uvec$ of the mixture and the (relative) concentration difference $φ$ of the two phases. The aim of this work is to study an optimal control problem for such a system, the control being a time-dependent external force acting on the flui…
We prove existence, uniqueness, regularity and separation properties for a nonlocal Cahn-Hilliard equation with a reaction term. We deal here with the case of logarithmic potential and degenerate mobility as well an uniformly lipschitz in $u$ reaction term $g(x,t,u).$
We study the validity of the comparison and maximum principles, and their relation with principal eigenvalues, for a class of degenerate nonlinear operators that are extremal among operators with one dimensional fractional diffusion.
Variational characterizations of weighted eigenvalue and basic reproduction ratio for nonlocal dispersal systems and application
School of Mathematics(Zhuhai), Sun Yat-sen University（中山大学数学学院（珠海））；School of Mathematics and Systems Science, Guangdong Polytechnic Normal University（广东技术师范大学数学与系统科学学院）
We provide a rigorous mathematical framework to establish the hydrodynamic limit of the Vlasov model introduced by Noguchi and Takata in order to describe phase transition of fluids by kinetic equations. We prove that, when the scale parameter tends to 0, this model converges to a nonlocal Cahn-Hilliard equation with degenerate mobility. For our analysis, we introduce apropriate forms of the short and long range potentials which allow us to derive Helmhotlz free energy estimates. Several compactness properties follow from the energy, the energy dissipation and kinetic averaging lemmas. In part…
