Showing papers in "Journal of Evolution Equations in 2007"
TL;DR: Using an integral formula of Droniou and Imbert (2005) for the fractional Laplacian, the authors define an entropy formulation for fractal conservation laws with pure fractional diffusion of order λ ∈]0, 1].
Abstract: Using an integral formula of Droniou and Imbert (2005) for the fractional Laplacian, we define an entropy formulation for fractal conservation laws with pure fractional diffusion of order λ ∈]0, 1]. This allows to show the existence and the uniqueness of a solution in the L∞ framework. We also establish a result of controled speed of propagation that generalizes the finite propagation speed result of scalar conservation laws. We finally let the non-local term vanish to approximate solutions of scalar conservation laws, with optimal error estimates for BV initial conditions as Kuznecov (1976) for λ = 2 and Droniou (2003) for λ ∈]1, 2].
97 citations
TL;DR: In this paper, the second part of a series of four articles on weighted norm inequalities, off-diagonal estimates and elliptic operators is presented, which is a substitute to the notion of pointwise bounds for kernels of operators which usually is a measure of decay.
Abstract: This is the second part of a series of four articles on weighted norm inequalities, off-diagonal estimates and elliptic operators. We consider a substitute to the notion of pointwise bounds for kernels of operators which usually is a measure of decay. This substitute is that of off-diagonal estimates expressed in terms of local and scale invariant L
p
− L
q
estimates. We propose a definition in spaces of homogeneous type that is stable under composition. It is particularly well suited to semigroups. We study the case of semigroups generated by elliptic operators.
77 citations
TL;DR: The model considered in this article consists of an ordinary differential equation coupled with an integro-partial differential equation and describes the interaction between non-infectious and infectious prion proteins.
Abstract: The model considered consists of an ordinary differential equation coupled with an integro-partial differential equation and describes the interaction between non-infectious and infectious prion proteins. We provide sufficient conditions for uniqueness of monomer-preserving weak solutions. In addition, we also prove existence of weak solutions under rather general assumptions on the involved degradation rates.
46 citations
TL;DR: In this paper, the realization of the differential operator in the space of continuous time periodic functions was studied in L2 with respect to its unique invariant measure. But the authors focused on the special case of the Ornstein-Uhlenbeck operator.
Abstract: We study the realization of the differential operator \(u \mapsto u_t - L(t)u\) in the space of continuous time periodic functions, and in L2 with respect to its (unique) invariant measure. Here L(t) is an Ornstein-Uhlenbeck operator in \({\mathbb{R}}^n\), such that L(t + T) = L(t) for each \(t \in {\mathbb{R}}\).
41 citations
TL;DR: In this article, a special case of the well-known Lojasiewicz gradient inequality is shown to give a unified background for many convergence results in gradient or gradient-like systems appearing previously in the Literature.
Abstract: It is shown that a special case of the well-known Lojasiewicz gradient inequality is sufficient to give a unified background for many convergence results in gradient or gradient-like systems appearing previously in the Literature. Besides as an illustration we give a direct proof of convergence in the case of 1D wave equations by a suitable adaptation of Zelenyak’s method.
36 citations
TL;DR: In this paper, the authors considered weak solutions of parabolic equations in divergence form with discontinuous coefficients on nonsmooth domains and obtained an optimal condition on the coefficients under which the global W1,p regularity theory holds.
Abstract: This work treats Lp regularity theory for weak solutions of parabolic equations in divergence form with discontinuous coefficients on nonsmooth domains. We essentially obtain an optimal condition on the coefficients under which the global W1,p regularity theory holds.
34 citations
TL;DR: In this article, a transference principle was used to obtain results of the form ==================\/\/\/\/\/\/£££/$££ £££€££$££\/\/£ £ ££ £€£ £/$£ £$£ £•££•£ £|££ • ££• ££|£•• £• £ £ £| ££ |££
Abstract: Let –iA be the generator of a C0-group U on a Banach space X. Via a transference principle we obtain results of the form
$$\mathop {sup}\limits_{t \in {\mathbb{R}}} \parallel f(A + t ) \parallel < \infty$$
for certain functions f, provided that X is a UMD space. Special examples are
$$f(z) = (1 + e^z)^{-1} \,\,\,{\rm{or}}\,\,\, f(z) = {\rm arctan} \,z.$$
The first choice leads to easy proofs of the theorems of Monniaux and Dore–Venni, the second is related to a new proof of Fattorini’s theorem on cosine functions.
33 citations
TL;DR: In this paper, the homogeneous relativistic heat equation (HRHE) obtained as the asymptotic limit of the so-called RHE when the kinematic viscosity ν → ∞ was studied.
Abstract: In this paper we study the homogeneous relativistic heat equation (HRHE) obtained as asymptotic limit of the so-called relativistic heat equation (RHE) when the kinematic viscosity ν → ∞. These equations were introduced in the theory of radiation hydrodynamics to guarantee a bounded speed of propagation of radiating energy. We shall prove that this is indeed true, and we shall construct some explicit solutions of the HRHE exhibiting fronts propagating at light speed.
27 citations
TL;DR: In this paper, the authors show that the solutions of nonlinear diffusion equations of the form ut = ΔΦ(u) appearing in filtration theory may present complicated asymptotics as t → ∞ whenever we alternate infinitely many times in a suitable manner the behavior of the nonlinearity Φ.
Abstract: We show that the solutions of nonlinear diffusion equations of the form ut = ΔΦ(u) appearing in filtration theory may present complicated asymptotics as t → ∞ whenever we alternate infinitely many times in a suitable manner the behavior of the nonlinearity Φ. Oscillatory behaviour is demonstrated for finite-mass solutions defined in the whole space when they are renormalized at each time t > 0 with respect to their own second moment, as proposed in [Tos05, CDT05]; they are measured in the L1 norm and also in the Euclidean Wasserstein distance W2. This complicated asymptotic pattern formation can be constructed in such a way that even a chaotic behavior may arise depending on the form of Φ.
26 citations
TL;DR: In this paper, sufficient conditions in terms of resolvents implying the stability of the essential or critical spectra of perturbed C0-semigroups on Hilbert spaces are given.
Abstract: We give sufficient conditions in terms of resolvents implying the stability of the essential or critical spectra of perturbed C0-semigroups on Hilbert spaces. We also show how these results apply to transport theory.
17 citations
TL;DR: In this article, the authors established the short-time asymptotic behavior of strongly local Dirichlet semigroups under very general hypotheses and applied it to a wide class of strongly elliptic, subelliptic and degenerate elliptic operators.
Abstract: We establish the short-time asymptotic behaviour of the Markovian semigroups associated with strongly local Dirichlet forms under very general hypotheses. Our results apply to a wide class of strongly elliptic, subelliptic and degenerate elliptic operators. In the degenerate case the asymptotics incorporate possible non-ergodicity.
TL;DR: In this article, stochastic Volterra equations driven by cylindrical Wiener process in Hilbert space are investigated and sufficient conditions for existence of strong solutions are given, where the key role is played by convergence of α-times resolvent families.
Abstract: In this paper, stochastic Volterra equations driven by cylindrical Wiener process in Hilbert space are investigated. Sufficient conditions for existence of strong solutions are given. The key role is played by convergence of α-times resolvent families.
TL;DR: In this paper, the exterior problem of the Newtonian filtration equation with nonlinear boundary sources is dealt with and the large time behavior of solutions including the critical Fujita exponent are determined or estimated.
Abstract: This paper deals with the exterior problem of the Newtonian filtration equation with nonlinear boundary sources. The large time behavior of solutions including the critical Fujita exponent are determined or estimated. An interesting phenomenon is illustrated that there exists a threshold value for the coefficient of the lower order term, which depends on the spacial dimension. Exactly speaking, the critical global exponent is strictly less than the critical Fujita exponent when the coefficient is under this threshold, while these two exponents are identically equal when the coefficient is over this threshold.
TL;DR: In this paper, the authors consider a class of structural acoustics models with thermoelastic flexible wall and show that under three different sets of coupled (mechanical/thermal) boundary conditions, the overall coupled system inherits some specific regularity properties of its flexible wall component.
Abstract: We consider a class of structural acoustics models with thermoelastic flexible wall. More precisely, the PDE system consists of a wave equation (within an acoustic chamber) which is coupled to a system of thermoelastic plate equations with rotational inertia; the coupling is strong as it is accomplished via boundary terms. Moreover, the system is subject to boundary thermal control. We show that—under three different sets of coupled (mechanical/thermal) boundary conditions—the overall coupled system inherits some specific regularity properties of its thermoelastic component, as it satisfies the same singular estimates recently established for the thermoelastic system alone. These regularity estimates are of central importance for (i) well-posedness of Differential and Algebraic Riccati equations arising in the associated optimal control problems, and (ii) existence of solutions to the semilinear initial/boundary value problem under nonlinear boundary conditions. The proof given uses as a critical ingredient a sharp trace theorem pertaining to second-order hyperbolic equations with Neumann boundary data.
TL;DR: In this article, a topological degree for the class of maps of the form − A + F where M is a closed neighborhood retract in a Banach space is constructed and applied to studying the existence and branching of periodic points of differential inclusions.
Abstract: In the paper a topological degree is constructed for the class of maps of the form − A + F where M is a closed neighborhood retract in a Banach space
$$E, A : D(A) \multimap E$$
is a m-accretive map such that − A generates a compact semigroup and F : M→ E is a locally Lipschitz map. The obtained degree is applied to studying the existence and branching of periodic points of differential inclusions of the type
$$ \left\{ \begin{aligned} &\dot{u} \in - \lambda Au + \lambda F(t,u),\lambda > 0\\
& u(t) \in M\\
& u(0) = u(T).\\
\end{aligned} \right. $$
TL;DR: In this article, the authors give necessary and sufficient conditions on the generator (A, D(A)) of T which ensure that eventual or immediate differentiability of T is inherited by S petertodd B� for all ��B \in {\mathcal{L}}(X)$$======.
Abstract: If (A, D(A)) generates a C
0-semigroup T on a Banach space X and
$$B \in {\mathcal{L}}(X)$$
then (A + B, D(A)) is also the generator of a C
0-semigroup, S
B
. There are easy examples to show that if T is eventually differentiable then S
B
need not be eventually differentiable. In 1995 an example was constructed to show that if T is immediately differentiable then S
B
need not be immediately differentiable. In this paper we establish necessary and sufficient conditions on the generator (A, D(A)) of T which ensure that eventual or immediate differentiability of T is inherited by S
B
for all
$$B \in {\mathcal{L}}(X)$$
. We are therefore able to give a characterization of the immediately and eventually differentiable C
0-semigroups for which differentiability is a stable property under bounded perturbations of the generator. We also prove a characterization of the C
0-semigroups for which the norm of the resolvent of the generator decays on vertical lines and a new characterization of the Crandall-Pazy class of semigroups.
TL;DR: In this article, the authors consider a reaction-diffusion system subject to homogeneous Neumann boundary conditions on a given bounded domain and prove the existence of an exponential attractor, provided that a bounded invariant region exists and the past history decays exponentially fast.
Abstract: We consider a reaction-diffusion system subject to homogeneous Neumann boundary conditions on a given bounded domain. The reaction term depends on the population densities as well as on their past histories in a very general way. This class of systems is widely used in population dynamics modelling. Due to its generality, the longtime behavior of the solutions can display a certain complexity. Here we prove a qualitative result which can be considered as a common denominator of a large family of specific models. More precisely, we demonstrate the existence of an exponential attractor, provided that a bounded invariant region exists and the past history decays exponentially fast. This result will be achieved by means of a suitable adaptation of the l-trajectory method coming back to the seminal paper of Malek and Necas.
TL;DR: In this article, short time estimates for the heat kernels of certain Schrodinger operators with unbounded potentials were proved for the asymptotic distribution of the eigenvalues.
Abstract: We prove short time estimates for the heat kernels of certain Schrodinger operators with unbounded potentials in
$${\mathbb{R}}^N$$
. The asymptotic distribution of the eigenvalues is also considered.
TL;DR: In this article, the authors improved the existing results of the long time asymptotical behavior of some basic semilinear diffusion equations (with space variable in the whole space) using explicit calculations, weighted sup-norm estimates and a fixed point argument.
Abstract: We improve the existing results of the long time asymptotical behaviour of some basic semilinear diffusion equations (with space variable in the whole space \(\mathbb{R}\)). Our method is elementary: it is based on explicit calculations, weighted sup–norm estimates and a fixed point argument.
TL;DR: In this paper, an approximation of a related Feller process (XC(t), t ≥ 0) via a sequence of semigroups acting in a locally compact Hausdorff space is provided.
Abstract: Let \({\mathcal{S}}\) be a locally compact Hausdorff space. Let A and B be two generators of Feller semigroups in \({C_0(\mathcal{S})}\) with related Feller processes {XA(t), t ≥ 0} and {XB(t), t ≥ 0} and let α and β be two non-negative continuous functions on \({\mathcal{S}}\) with α + β = 1. Assume that the closure C of C0 = αA + βB with \({\mathcal{D}(C_0) = \mathcal{D}(A) \cap \mathcal{D}(B)}\) generates a Feller semigroup {TC(t), t ≥ 0} in \({C_0(\mathcal{S})}\) . It is natural to think of a related Feller process {XC(t), t ≥ 0} as that evolving according to the following heuristic rules. Conditional on being at a point \({p \in \mathcal{S}}\) , with probability α(p) the process behaves like {XA(t), t ≥ 0} and with probability β(p) it behaves like {XB(t), t ≥ 0}. We provide an approximation of {TC(t), t ≥ 0} via a sequence of semigroups acting in \({C_0(\mathcal{S}) \times C_0(\mathcal{S})}\) that supports this interpretation. This work is motivated by the recent model of stochastic gene expression due to Lipniacki et al. [17].
TL;DR: In this article, it was proved that the Stokes operator in Lq -space on an infinite cylindrical domain of \({mathbb{R}^{n}}}, \({n \geq 3}}\), with several exits to infinity, generates a bounded and exponentially decaying analytic semigroup and admits a bounded H^{H^{\infty}}} -calculus.
Abstract: It is proved that the Stokes operator in Lq -space on an infinite cylindrical domain of \({{\mathbb{R}^{n}}}\) , \({{n \geq 3}}\) , with several exits to infinity generates a bounded and exponentially decaying analytic semigroup and admits a bounded \({{H^{\infty}}}\) -calculus. For the resolvent estimates, the Stokes resolvent system with a prescribed divergence in an infinite straight cylinder with bounded cross-section \({{\Sigma}}\) is studied in Lq\({{(\mathbb{R}; L^{r}_{\omega} (\Sigma))}}\) where \({{1 < q,r < \infty}}\) and \({{\omega \, \epsilon \, A_{r}(\mathbb{R}^{n-1})}}\) is an arbitrary Muckenhoupt weight. The proofs use cut-off techniques and the theory of Schauder decomposition of UMD spaces based on \({{\mathcal{R}}}\) -boundedness of operator families and on square function estimates involving Muckenhoupt weights.
TL;DR: In this paper, the existence and uniqueness of a viscosity-measure solution (S, ρ) of the WKB method is established. But the measure solution is defined by means of the Filippov flow of ∇S.
Abstract: Let H (t, x, p) be a Hamiltonian function that is convex in p. Let the associated Lagrangian satisfy the nonstandard minorization condition \(L(t, x, v) \geq \frac{1}{2} m({\mid v \mid}^2 - \omega^{2}{\mid x \mid}^{2})-C\) where m > 0, ω > 0, and C ≥ 0 are constants. Under some additional conditions, we prove that the associated value function is the unique viscosity solution of St + H(t, x, ∇S) = 0 in \(Q_T = (0, T)\times {\mathbb{R}}^{n}, S\mid_{t=0} = S_0\), without any conditions at infinity on the solution. Here ωT < π/2. To the Hamilton–Jacobi equation corresponding to the classical action integrand in mechanics, we adjoin the continuity equation and establish the existence and uniqueness of a viscosity–measure solution (S, ρ) of
$$ S_t + \frac{1}{2}{\mid
abla S\mid}^2 + U(x) = 0 \quad {\rm in}\, Q_T, \quad S(0,x) = S_0(x) \quad {\rm in} \, {\mathbb{R}}^n; $$
$$ \rho_t + {\rm div}(\rho
abla S) = 0 \quad {\rm in}\, Q_{T}, \quad \rho(0,x) = \rho_0(x) \quad {\rm in} \, {\mathbb{R}}^n. $$
This system arises in the WKB method. The measure solution is defined by means of the Filippov flow of ∇S.
TL;DR: In this article, a conserved phase-field system coupling two nonlinear hyperbolic integro-differential equations was considered and the existence of the solution to the corresponding initial-boundary value problem associated with the resulting PDE system was proved.
Abstract: We consider a conserved phase-field system coupling two nonlinear hyperbolic integro-differential equations. The model results from the assumption that the material undergoing phase transition exhibits some thermal memory effects (cf. [15]) and that the response of the order parameter to the variation of the free-energy functional is delayed (cf. [10, 23]). We prove the existence of the solution to the corresponding initial-boundary value problem associated with the resulting PDE system and a (conditioned) continuous dependence estimate of the solution with respect to the data of the problem.
TL;DR: In this article, the Cauchy problem for singularly perturbed parabolic equations is considered, and weighted L2-estimates as well as certain decay properties of bounded classical solutions to it are established.
Abstract: The Cauchy problem for singularly perturbed parabolic equations is considered, and weighted L2-estimates as well as certain decay properties of bounded classical solutions to it are established. These do not depend on the value of the small perturbation parameter, and allow to prove global in time existence of strong solutions to certain boundary-value problems for ultraparabolic equations with unbounded coefficients. Optimal decay estimates are proved for such solutions. All results concerning ultraparabolic equations apply, in particular, to the Kolmogorov equation for diffusion with inertia, to the (linear) Fokker-Planck equation, to the linearized Boltzmann equation, and to some nonlinear integro-differential ultraparabolic equations of the Fokker-Planck type, arising from biophysics. Optimal decay estimates are derived for global in time strong solutions to such equations.
TL;DR: In this article, the authors studied the motion of an incompressible homogeneous Newtonian fluid in a rigid porous medium of infinite extent and proved that the stability properties of this equilibrium are determined by the slope of b in c : the equilibrium is unstable if b′(c) 0 implies exponential stability.
Abstract: In this article, we study the motion of an incompressible homogeneous Newtonian fluid in a rigid porous medium of infinite extent. The fluid is bounded below by a fixed layer having an external source (with an injection rate b), and above by a free surface moving under the influence of gravity. The flow is governed by Darcy’s law. If b(c) = 0 for some c > 0 then the system admits (u, f) ≡ (c, c) as an equilibrium solution. We shall prove that the stability properties of this equilibrium are determined by the slope of b in c : The equilibrium is unstable if b′(c) 0 implies exponential stability.
TL;DR: In this paper, a degenerate nonlinear variational inequality which can be reduced to a multivalued inclusion by an appropriate change of the unknown function is studied and its existence, uniqueness and regularity results are established.
Abstract: We study a degenerate nonlinear variational inequality which can be reduced to a multivalued inclusion by an appropriate change of the unknown function. We establish existence, uniqueness and regularity results. An application arising in the theory of water diffusion in porous media is discussed as an example.
TL;DR: In this article, an abstract second order nonlinear evolution equation in a real Hilbert space is studied, where time-dependent convex functions and their subdifferentials operating on the first derivative of the unknown function are considered.
Abstract: We study an abstract second order nonlinear evolution equation in a real Hilbert space. We consider time-dependent convex functions and their subdifferentials operating on the first derivative of the unknown function. Introducing appropriate assumptions on the convex functions and other data, we prove the existence and uniqueness of a strong solution, and give some applications of the abstract theorem to hyperbolic variational inequalities with time-dependent constraints.
TL;DR: In this article, the authors prove the existence of a finite-dimensional attractor for the Stefan problem without heat losses in the weighted Sobolev space, and show that the Hausdorff dimension of the attractor is finite.
Abstract: We study a two-phase Stefan problem with kinetics. Here we prove existence of a finite-dimensional attractor for the problem without heat losses. Fot the most part we use a more elegant technique of energetic type estimates in appropriately defined weighted Sobolev spaces as opposite to the parabolic potentials of [9]. We demonstrate existence of compact attractors in the Sobolev spaces and prove that the attractor consists of sufficiently regular functions. This allows us to show that the Hausdorff dimension of the attractor is finite.
TL;DR: In this article, the existence of weak entropy solutions for the Riemann problem with respect to x/t has been shown for a general class of data, thanks to the convergence proof of a coupled finite volume scheme.
Abstract: The system of equations (f (u))t − (a(u)v + b(u))x = 0 and ut − (c(u)v + d(u))x = 0, where the unknowns u and v are functions depending on \({(x, t) \, \in \, \mathbb{R} \times \mathbb{R}_+}\), arises within the study of some physical model of the flow of miscible fluids in a porous medium. We give a definition for a weak entropy solution (u, v), inspired by the Liu condition for admissible shocks and by Krushkov entropy pairs. We then prove, in the case of a natural generalization of the Riemann problem, the existence of a weak entropy solution only depending on x/t. This property results from the proof of the existence, by passing to the limit on some approximations, of a function g such that u is the classical entropy solution of ut − ((cg + d)(u))x = 0 and simultaneously w = f (u) is the entropy solution of wt − ((ag + b)(f(−1)(w)))x = 0. We then take v = g(u), and the proof that (u, v) is a weak entropy solution of the coupled problem follows from a linear combination of the weak entropy inequalities satisfied by u and f (u). We then show the existence of an entropy weak solution for a general class of data, thanks to the convergence proof of a coupled finite volume scheme. The principle of this scheme is to compute the Godunov numerical flux with some interface functions ensuring the symmetry of the finite volume scheme with respect to both conservation equations.