Models issued from ecology, chemical reactions and several other application fields lead to semi-linear parabolic equations with super-linear growth. Even if, in general, blow-up can occur, these models share the property that mass control is essential. In many circumstances, it is known that this $L^1$ control is enough to prove the global existence of weak solutions. The theory is based on basic estimates initiated by M. Pierre and collaborators, who have introduced methods to prove $L^2$ a priori estimates for the solution. Here, we establish such a key estimate with initial data in $L^1$ while the usual theory uses $L^2$. This allows us to greatly simplify the proof of some results. We also establish new existence results of semilinearity which are super-quadratic as they occur in complex chemical reactions. Our method can be extended to semi-linear porous medium equations.
Boundary characteristic point regularity is studied for a class of semilinear heat equations and an ODE criterion of regularity is obtained. Extensions to higher-order semilinear parabolic problems are discussed.
We consider semi-stable, radially symmetric, and decreasing solutions of a reaction equation involving the p-Laplacian, where the reaction term is a locally Lipschitz function, and the domain is the unit ball. For this class of radial solutions, which includes local minimizers, we establish pointwise and Sobolev estimates which are optimal and do not depend on the specific nonlinear reaction term. Under standard assumptions we also prove the regularity of the corresponding extremal solution.
We prove an $L^p(I,C^α(Ω))$ regularity result for a reaction-diffusion equation with mixed boundary conditions, symmetric $L^\infty$ coefficients and an $L^\infty$ initial condition. We provide explicit control of the $L^p(I,C^α(Ω))$ norm with respect to the data. To prove our result, we first establish $C^α(Ω)$ control of the stationary equation, extending a result by Haller-Dintelmann et al. (2009).
The close-to-equilibrium regularity of solutions to a class of reaction-diffusion systems is investigated. The considered systems typically arise from chemical reaction networks and satisfy a complex balanced condition. Under some restrictions on spatial dimensions ($d\leq 4$) and order of nonlinearities ($μ= 1 + 4/d$), we show that if the initial data is close to a complex balanced equilibrium in $L^2$-norm, then classical solutions are shown global and converging exponentially to equilibrium in $L^{\infty}$-norm. Possible extensions to higher dimensions and order of nonlinearities are also d…
The boundedness of stable solutions to semilinear (or reaction-diffusion) elliptic PDEs has been studied since the 1970's. In dimensions 10 and higher, there exist stable energy solutions which are unbounded (or singular). This note describes, for non-expert readers, a recent work in collaboration with Figalli, Ros-Oton, and Serra, where we prove that stable solutions are smooth up to the optimal dimension 9. This answers to an open problem posed by Brezis in the mid-nineties concerning the regularity of extremal solutions to Gelfand-type problems. We also describe, briefly, a famous analogue …
H{ö}lder regularity of parabolic equations with Dirichlet boundary conditions and application to reaction-diffusion and reaction-cross-diffusion systems
We investigate the behavior of the time derivatives of the solution to a linear time-fractional, advection-diffusion-reaction equation, allowing space- and time-dependent coefficients as well as initial data that may have low regularity. Our focus is on proving estimates that are needed for the error analysis of numerical methods. The nonlocal nature of the fractional derivative creates substantial difficulties compared with the case of a classical parabolic PDE. In our analysis, we rely on novel energy methods in combination with a fractional Gronwall inequality and certain properties of frac…
We derive some regularity estimates of the solution to a time fractional diffusion equation, that are useful for numerical analysis, and partially unravel the singularity structure of the solution with respect to the time variable.
In this paper we address the regularity issues of drift-diffusion equation with nonlocal diffusion, where the diffusion operator is in the realm of stable-type Lévy operator and the velocity field is defined from the considered quantity by a zero-order pseudo-differential operator. Through using the method of nonlocal maximum principle in a unified way, we prove the eventual regularity result in the supercritical type cases and the global regularity at some logarithmically supercritical cases. The feature of these results is that the time after which the solution is smoothly regular in the sup…
In this article we investigate the existence and regularity of 1-d steady state fractional order diffusion equations. Two models are investigated:the Riemann-Liouville fractional diffusion equation, and the Riemann-Liouville-Caputo fractional diffusion equation. For these models we explicitly show how the regularity of the solution depends upon the right hand side (rhs) function. We also establish for which Dirichlet and Neumann boundary conditions the models are well posed.
Whether or not classical solutions of the 2D incompressible MHD equations without full dissipation and magnetic diffusion can develop finite-time singularities is a difficult issue. A major result of this paper establishes the global regularity of classical solutions for the MHD equations with mixed partial dissipation and magnetic diffusion. In addition, the global existence, conditional regularity and uniqueness of a weak solution is obtained for the 2D MHD equations with only magnetic diffusion.
We prove global Hölder gradient estimates for bounded positive weak solutions of fast diffusion equations in smooth bounded domains with the homogeneous Dirichlet boundary condition, which then lead us to establish their optimal global regularity. This solves a problem raised by Berryman and Holland in 1980.
In this paper, logarithmically improved regularity criteria for the Navier--Stokes/Poisson--Nernst--Planck system are established in terms of both the pressure and the gradient of pressure in the homogeneous Besov space.
This paper is concerned with the global regularity of the 2D (two-dimensional) generalized magnetohydrodynamic equations with only magnetic diffusion $Λ^{2β} b$. It is proved that when $β>1 $ there exists a unique global regular solution for this equations. The obtained result improves the previous known one which requires that $β>\frac32$.
We consider the Cauchy problem on nonlinear scalar conservation laws with a diffusion-type source term related to an index $s\in \R$ over the whole space $\R^n$ for any spatial dimension $n\geq 1$. Here, the diffusion-type source term behaves as the usual diffusion term over the low frequency domain while it admits on the high frequency part a feature of regularity-gain and regularity-loss for $s< 1$ and $s>1$, respectively. For all $s\in \R$, we not only obtain the $L^p$-$L^q$ time-decay estimates on the linear solution semigroup but also establish the global existence and optimal time-decay …
In this paper, a standard about the existence and upper semi-continuity of pullback attractors in the non-initial space is established for some classes of non-autonomous SPDE. This pullback attractor, which is the omega-limit set of the absorbing set constructed in the initial space, is completely determined by the asymptotic compactness of solutions in both the initial and non-initial spaces. As applications, the existences and upper semi-continuity of pullback attractors in $H^1(\mathbb{R}^N)$ are proved for stochastic non-autonomous reaction-diffusion equation driven by a multiplicative noi…
The global existence of classical solutions to cross diffusion systems of more than 2 equations given on a planar domain is established. The results can apply to generalized Shigesada-Kawasaki-Teramoto (SKT) and food pyramid models whose diffusion and reaction can have polynomial growth of any order.
In this paper, we present an approach to characterising fast-reaction limits of systems with nonlinear diffusion, when there are either two reaction-diffusion equations, or one reaction-diffusion equation and one ordinary differential equation, on unbounded domains. Here, we replace the terms of the form uxx in usual reaction-diffusion equation, which represent linear diffusion, by terms of form phi(u)xx, representing nonlinear diffusion. We prove the convergence as k tends to infinity to a limit that is determined by the unique solution of a certain scalar nonlinear diffusion limit problem.
We study the free boundary regularity of the traveling wave solutions to a degenerate advection-diffusion problem of Porous Medium type, whose existence was proved in \cite{MonsaingonNovikovRoquejoffre}. We set up a finite difference scheme allowing to compute approximate solutions and capture the free boundaries, and we carry out a numerical investigation of their regularity. Based on some nondegeneracy assumptions supported by solid numerical evidence, we prove the Lipschitz regularity of the free boundaries. Our simulations indicate that this regularity is optimal, and the free boundaries s…
We study the boundedness and convergence to equilibrium of weak solutions to reaction-diffusion systems with nonlinear diffusion. The nonlinear diffusion is of porous medium type and the nonlinear reaction terms are assumed to grow polynomially and to dissipate (or conserve) the total mass. By utilising duality estimates, the dissipation of the total mass and the smoothing effect of the porous medium equation, we prove that if the exponents of the nonlinear diffusion terms are high enough, then weak solutions are bounded, locally Hölder continuous and their $L^{\infty}(Ω)$-norm grows in time a…
We study a nonlocal Venttsel' problem in a non-convex bounded domain with a Koch-type boundary. Regularity results of the strict solution are proved in weighted Sobolev spaces. The numerical approximation of the problem is carried out and optimal a priori error estimates are obtained.
We study the existence of global weak solutions of a nonlinear transport-diffusion equation with a fractional derivative in the time variable and under some extra hypotheses, we also study some regularity properties for this type of solutions. In the system considered here, the diffusion operator is given by a fractional Laplacian and the nonlinear drift is assumed to be divergence free and it is assumed to satisfy some general stability and boundedness properties in Lebesgue spaces.
This paper considers a class of nonlinear, degenerate drift- diffusion equations. We study well-posedness and regularity properties of the solutions, with the goal to achieve uniform Hölder regularity in terms of $L^p$-bound on the drift vector field. A formal scaling argument yields that the threshold for such estimates is $p=d$, while our estimates are for $p>d+\frac{4}{d+2}$. On the other hand we are able to show by a series of examples that one needs $p>d$ for such estimates, even for divergence free drift.
This article studies the global regularity problem of the two-dimensional zero thermal diffusion tropical climate model with fractional dissipation, given by $(-Δ)^αu$ in the barotropic mode equation and by $(-Δ)^βv$ in the first baroclinic mode of the vector velocity equation. More precisely, we show that the global regularity result holds true as long as $α+β\geq2$ with $1<α<2$. In addition, with no dissipation from both the temperature and the first baroclinic mode of the vector velocity, we also establish the global regularity result with the dissipation strength at the logarithmically sup…
We establish the uniqueness and regularity of weak (and very weak) solutions to a class of cross diffusion systems which is inspired by models in mathematical biology/ecology, in particular the Shigesada-Kawasaki-Teramoto (SKT) model in population biology. No boundedness assumption on these solutions is supposed here as known techniques for scalar equations such as maximum/comparison principles are generally unavailable for systems. Furthermore, for planar domains we show that unbounded weak solutions satisfying mild integrability conditions are in fact smooth.
We provide an asymptotic analysis of linear transport problems in the diffusion limit under minimal regularity assumptions on the domain, the coefficients, and the data. The weak form of the limit equation is derived and the convergence of the solution in the $L^2$ norm is established without artificial regularity requirements. This is important to be able to deal with problems involving realistic geometries and heterogeneous media. In a second step we prove the usual $O(\varepsilon)$ convergence rates under very mild additional assumptions. The generalization of the results to convergence in …
This work addresses the regularity of solutions for a nonlocal diffusion equation over the space of periodic distributions. The spatial operator for the nonlocal diffusion equation is given by a nonlocal Laplace operator with a compactly supported integral kernel. We follow a unified approach based on the Fourier multipliers of the nonlocal Laplace operator, which allows the study of regular as well as distributional solutions of the nonlocal diffusion equation, integrable as well as singular kernels, in any spatial dimension. In addition, the results extend beyond operators with singular kern…
A spatio-temporal evolution of chemicals appearing in a reversible enzyme reaction and modelled by a four component reaction-diffusion system with the reaction terms obtained by the law of mass action is considered. The large time behaviour of the system is studied by means of entropy methods.
In this work we study global well-posedness and large time behaviour for a typical reaction--diffusion system, which include degenerate diffusion, and whose non-linearities arise from chemical reactions. We show that there is an {\it indirect diffusion effect}, i.e. an effective diffusion for the non-diffusive species which is incurred by a combination of diffusion from diffusive species and reversible reactions between the species. Deriving new estimates for such degenerate reaction-diffusion system, we show, by applying the entropy method, that the solution converges exponentially to equilib…
This paper is dedicated to the application of the DeGiorgi-Nash-Moser regularity theory to the kinetic Fokker-Planck equation. This equation is hypoelliptic. It is parabolic only in the velocity variable, while the Liouville transport operator has a mixing effect in the position/velocity phase space. The mixing effect is incorporated in the classical DeGiorgi method via the averaging lemmas. The result can be seen as a Hölder regularity version of the classical averaging lemmas.
Whether or not the classical solutions of the two-dimensional (2D) incompressible magnetohydrodynamics (MHD) equations with only Laplacian magnetic diffusion (without velocity dissipation) are globally well-posed is a difficult problem and remains completely open. In this paper, we establish the global regularity of solutions to the 2D incompressible MHD equations with almost Laplacian magnetic diffusion in the whole space. This result can be regarded as a further improvement and generalization of the previous works. Consequently, our result is more closer to the resolution of the global regul…
This paper establishes the global regularity of classical solution to the 2D MHD system with only horizontal dissipation and horizontal magnetic diffusion in a strip domain $\mathbb{T}\times\mathbb{R}$ when the initial data is suitable small. To prove this, we combine the Littlewood-Paley decomposition with anisotropic inequalities to establish a crucial commutator estimate. We also analysis the asymptotic behavior of the solution. In addition, the global existence and uniqueness of classical solution is obtained for the 2D simplified tropical climate model with only horizontal dissipations.
This paper is concerned with the local and global existence of solutions for a generalized $m$-component reaction--diffusion system with a tridiagonal $2$--Toeplitz diffusion matrix and polynomial growth. We derive the eigenvalues and eigenvectors and determine the parabolicity conditions in order to diagonalize the proposed system. We, then,determine the invariant regions and utilize a Lyapunov functional to establish the global existence of solutions for the proposed system. A numerical example is used to illustrate and confirm the findings of the study.
Singular limit problems of reaction-diffusion systems have been studied in cases where the effects of the reaction terms are very large compared with those of the other terms. Such problems appear in literature in various fields such as chemistry, ecology, biology, geology and approximation theory. In this paper, we deal with the singular limit of a general reaction-diffusion system including many problems in the literature. We formulate the problem, derive the limit equation and establish a rigorous mathematical theory.
In this paper we study the local instability to the boundary equilibria and the local stability to the positive equilibria for some chemical reaction-diffusion systems. We first analyze a three-species system with boundary equilibria in some stoichiometric classes. Then We prove the convergence to the positive equilibria for a general reversible reaction-diffusion network as long as the initial data is closed enough to the the positive equilibria.
We prove existence and uniqueness of travelling waves for a reaction-diffusion system coupling a classical reaction-diffusion equation in a strip with a diffusion equation on a line. To do this we use a continuation method which leads to further insight into the system. In particular, the transition occurs through a singular perturbation which seems new in this context, connecting the system with a Wentzell type boundary value problem.
We study in this paper the periodic homogenization problem related to a strongly nonlinear reaction-diffusion equation. Owing to the large reaction term, the homogenized equation has a rather quite different form which puts together both the reaction and convection effects. We show in a special case that, the homogenized equation is exactly of a convection-diffusion type. The study relies on a suitable version of the well-known two-scale convergence method.
For a class of competition-diffusion nonlinear systems involving fractional powers of the Laplacian, including as a special case the fractional Gross-Pitaevskii system, we prove that uniform boundedness implies Hölder boundedness for sufficiently small positive exponents, uniformly as the interspecific competition parameter diverges. This implies strong convergence of the family of solutions as the segregation of their support occurs.
In this paper we consider a final value problem for a diffusion equation with time-space fractional differentiation on a bounded domain $D$ of $ \mathbb{R}^{k}$, $k\ge 1$, which includes the fractional power $\mathcal L^β$, $0<β\le 1$, of a symmetric uniformly elliptic operator $\mathcal L$ defined on $L^2(D)$. A representation of solutions is given by using the Laplace transform and the spectrum of $\mathcal L^β$. We establish some existence and regularity results for our problem in both the linear and nonlinear case.
We prove optimal boundary regularity for bounded positive weak solutions of fast diffusion equations in smooth bounded domains. This solves a problem raised by Berryman and Holland in 1980 for these equations in the subcritical and critical regimes. Our proof of the a priori estimates uses a geometric type structure of the fast diffusion equations, where an important ingredient is an evolution equation for a curvature-like quantity.
This paper considers the temperature patch problem for the incompressible Boussinesq system with no diffusion and viscosity in the whole space $\mathbb{R}^2$. We prove that for initial patches with $W^{2,\infty}$ boundary the curvature remains bounded for all time. The proof explores new cancellations that allow us to bound $\nabla^2u$, even for those components given by time dependent singular integrals with kernels with nonzero mean on circles. In addition, we give a different proof of the $C^{1+γ}$ regularity result in [23], $0<γ<1$, using the scale of Sobolev spaces for the velocity. Furth…
We study the regularity properties of the solutions to the nonlinear equation with fractional diffusion $$ \partial_tu+(-Δ)^{σ/2}φ(u)=0, $$ posed for $x\in \mathbb{R}^N$, $t>0$, with $0<σ<2$, $N\ge1$. If the nonlinearity satisfies some not very restrictive conditions: $φ\in C^{1,γ}(\mathbb{R})$, $1+γ>σ$, and $φ'(u)>0$ for every $u\in\mathbb{R}$, we prove that bounded weak solutions are classical solutions for all positive times. We also explore sufficient conditions on the non-linearity to obtain higher regularity for the solutions, even $C^\infty$ regularity. Degenerate and singular cases, in…
We study quasilinear reaction diffusion systems relative to the Shigesada-Kawasaki-Teramoto model. Nonlinearity standing for the external force is provided with mass dissipation. Estimate in several norms of the solution is provided under the restriction of diffusion coefficients, growth rate of reaction, and space dimension.
In this paper we consider a multidimensional semilinear reaction-diffusion equation and we obtain at any arbitrary time an approximate controllability result between nonnegative states using as control term the reaction coefficient, that is via multiplicative controls.
The aim of this paper is to construct invariant regions of a generalized m-component reaction-diffusion system with a tri-diagonal Toeplitz matrix of diffusion coefficients and prove the global existence of solutions using Lyapunov functional. The paper assumes nonhomogeneous boundary conditions and polynomial growth for the non-linear reaction term.
In this paper we study the structure of the global attractor for a reaction- di§usion equation in which uniqueness of the Cauchy problem is not guarantied. We prove that the global attractor can be characterized using either the unstable manifold of the set of stationary points or the stable one but considering in this last case only solutions in the set of bounded complete trajectories.
