Showing papers in "Journal of Evolution Equations in 2003"

Asymptotic behaviour for the porous medium equation posed in the whole space

Abstract: This paper is devoted to present a detailed account of the asymptotic behaviour as t → ∞ of the solutions u(x, t) of the equation $$ {u_{{t = }}}\Delta ({u^{m}}) $$ (0.1) with exponent m > 1, a range in which it is known as the porous medium equation written here PME for short. The study extends the well-known theory of the classical heat equation (HE, the case m = 1) into a nonlinear situation, which needs a whole set of new tools. The space dimension can be any integer n ≥ 1. We will also present the extension of the results to exponents m < 1 (fast-diffusion equation, Fde). For definiteness we consider the Cauchy Problem posed in Q = ℝ n x ℝ+ with initial data $$ u(x,0) = {u_{0}}(x), x \in {\mathbb{R}^{n}} $$ (0.2) chosen in a suitable class of functions. In most of the paper we concentrate on the class X 0 of integrable and nonnegative data, $$ {u_{0}} \in {L^{1}}({\mathbb{R}^{n}}), {u_{0}} \geqslant 0, $$ (0.3) which is natural on physical grounds as the density or concentration of a diffusion process, the height of a ground-water mound, or the temperature of a hot medium (see a comment on the applications at the end). Consequently, we will deal mostly with nonnegative solutions u(x, t) ≥ 0 defined in Q. An existence and uniqueness theory exists for this problem so that for every data uo we can produce an orbit {u(·,t):t > 0} which lives in L 1 (ℝ n ) ∩ L ∞(ℝ n ) and describes the evolution of the process. The solution is not classical for m > 1, but it is proved that there exists a unique weak solution for all m > 0.

Nonlinear problems related to the Thomas-Fermi equation

Abstract: Most of the results in this work were obtained over the period 1975-77 and were announced at various meetings (see e.g. items [3], [4], [5] under Brezis [16]). This paper has a rather unusual history. Around 1972 I became interested in nonlinear elliptic equations of the form $$ - \Delta u + |u{|^{{p - 1}}}u = f in a domain \Omega \subset {\mathbb{R}^{N}}, $$ (P.1) with zero Dirichlet condition, where 0 < p < ∞ and f ∈ L 1. The motivation came from the study of the porous medium equation $$ \frac{{\partial \upsilon }}{{\partial t}} - \Delta (|\upsilon {|^{{m - 1}}}\upsilon ) = 0, $$ (P.2) with 0 < m < ∞.

Global solution and smoothing effect for a non-local regularization of a hyperbolic equation

Abstract: We study the problem $$ \left\{ \begin{gathered} {{\partial }_{t}}u\left( {t,x} \right) + {{\partial }_{x}}\left( {f\left( u \right)} \right)\left( {t,x} \right) + g\left[ {u\left( {t,\cdot } \right)} \right]\left( x \right) = 0 t \in ]0,\infty [,x \in \mathbb{R} \hfill \\ u\left( {0,x} \right) = {{u}_{0}}\left( x \right) x \in \mathbb{R}, \hfill \\ \end{gathered} \right. $$ (1.1) where \( f \in {C^\infty }\left( \mathbb{R} \right)\) is such that f (0) = 0(there is not loss of generality in assuming this), \( {u_0} \in {L^\infty }\left( \mathbb{R} \right)\) and gis the non-local (in general) operator defined through the Fourier transform by $$ \mathcal{F}\left( {g\left[ {u\left( {t, \cdot } \right)} \right]} \right)\left( \xi \right) = {\left| \xi \right|^\lambda }\mathcal{F}\left( {u\left( {t, \cdot } \right)} \right)\left( \xi \right), with \lambda \in \left] {1,2} \right]. $$ (1.1)

Minimization problems for eigenvalues of the Laplacian

Abstract: This paper is a survey on classical results and open questions about minimization problems concerning the lower eigenvalues of the Laplace operator. After recalling classical isoperimetric inequalities for the two first eigenvalues, we present recent advances on this topic. In particular, we study the minimization of the second eigenvalue among plane convex domains. We also discuss the minimization of the third eigenvalue. We prove existence of a minimizer. For others eigenvalues, we just give some conjectures. We also consider the case of Neumann, Robin and Stekloff boundary conditions together with various functions of the eigenvalues.

On the motion of rigid bodies in a viscous incompressible fluid

Abstract: The motion of one or several rigid bodies in a viscous incompressible fluid has been a topic of numerous theoretical studies. The time evolution of the fluid density ϱ f = ϱ f (t, x) and the velocity u f = u f (t, x) is governed by the Navier-Stokes system of equations $$ {\partial _{t}}{\varrho ^{f}} + {\text{div(}}{\varrho ^{f}}{u^{f}}{\text{)}} {\text{ = }} {\text{0,}} $$ (1.1) $$ {\partial _{t}}({\varrho ^{f}}{u^{f}}) + {\text{div(}}{\varrho ^{f}}{u^{f}} \oplus {u^{f}}{\text{)}} + abla p = {\text{div}} \mathbb{T} + {\varrho ^{f}}{g^{f}} $$ (1.2) satisfied in a region Q f of the space-time occupied by the fluid. We focus on linearly viscous (Newtonian) incompressible fluids where the stress tensor \( \mathbb{T} \) is determined through the constitutive relation $$ \mathbb{T} = \mathbb{T}(u) \equiv 2\mu \mathbb{D}(u), \mathbb{D}(u) \equiv \frac{1}{2}( abla u{\text{ + }} abla {u^{t}}),\mu > 0, $$ (1.3) and the velocity satisfies the incompressibility condition $$ {\text{div}} {u^{f}} = 0. $$ (1.4) .

Existence and uniqueness results for large solutions of general nonlinear elliptic equations

Abstract: We study under what condition there exists a solution of —Δu +f (u) =0 in a domain Ω which blows-up on the boundary, independently of the regularity of the boundary, and we provide criteria for uniqueness. We apply our results to the case f(u) = e au .

Oscillatory boundary conditions for acoustic wave equations

Abstract: In the textbook literature on theoretical acoustics, it was traditional to use the Robin boundary condition with the wave equation. But it was recognized that this was not the physically correct boundary condition. “Acoustic Boundary Conditions” (or ABC) were introduced in the monograph by Morse and Ingard [13, p. 263]. The presentation in [13] is not the usual approach to the wave equations, since the authors treat waves having definite frequency. The time dependent version of ABC was first formulated by Tom Beale and Steve Rosencrans [1] in a very interesting and original paper. ABC will be explained in detail in Section 2.

Dirichlet and Neumann boundary conditions: What is in between?

Abstract: Given an admissible measure µon oΩ where Ω ⊂ ℝ n is an open set, we define a realizationA µ of the Laplacian in L 2 (12) with general Robin boundary conditions and we show that Aµ generates a holomorphic C 0-semigroup on L2(Ω) which is sandwiched by the Dirichlet Laplacian and the Neumann Laplacian semigroups. Moreover, under a locality and a regularity assumption, the generator of each sandwiched semigroup is of the form Δµ. We also show that if D(Δµ) contains smooth functions, then µ is of the form dµ=βbσ(where σ is the (n — 1)-dimensional Hausdorff measure and β a positive measurable bounded function on ∂Ω); i.e. we have the classical Robin boundary conditions.

Rate of decay to equilibrium in some semilinear parabolic equations

Abstract: In this paper we prove, under various conditions, the so-called Lojasiewicz inequality $ \| E' (u + \varphi) \| \geq \gamma|E(u+\varphi) - E(\varphi)|^{1-\theta} $, where $ \theta \in (0,1/2] $, and γ > 0, while $ \| u \| $ is sufciently small and φ is a critical point of the energy functional E supposed to be only C², instead of analytic in the classical settings Here E can be for instance the energy associated to the semilinear heat equation $u_t = \Delta u - f(x,u) $ on a bounded domain $ \Omega \subset \mathbb{R}^N $ As a corollary of this inequality we give the rate of convergence of the solution u(t) to an equilibrium, and we exhibit examples showing that the given rate of convergence (which depends on the exponent θ and on the critical point through the nature of the kernel of the linear operator $ E'' (\varphi)) $ is optimal

Weak solutions and supersolutions in L 1 for reaction-diffusion systems

Abstract: We prove here that limits of nonnegative solutions to reaction-diffusion systems whose nonlinearities are bounded in \( L^1 \) always converge to supersolutions of the system. The motivation comes from the question of global existence in time of solutions for the wide class of systems preserving positivity and for which the total mass of the solution is uniformly bounded. We prove that, for a large subclass of these systems, weak solutions exist globally.

Some noncoercive parabolic equations with lower order terms in divergence form

Abstract: This paper deals with existence and regularity results for the problem $$ \left\{ \begin{gathered} {u_{t}} - {\text{div}}(a(x,t,u) abla u) = - {\text{div}}(u{\text{E}}), in \Omega x{\text{ }}(0,T), \hfill \\ u = 0 on \partial \Omega x (0,T), \hfill \\ u(0) = {u_{0}} in \Omega , \hfill \\ \end{gathered} \right. $$ , under various assumptions on E and u 0. The main difficulty in studying this problem is due to the presence of the term div(uE), which makes the differential operator non coercive on the “energy space” L2(0T; H 0 1 (Ω)).

On the motion of rigid bodies in a viscous incompressible fluid

Abstract: This paper is a survey on classical results and open questions about minimization problems concerning the lower eigenvalues of the Laplace operator. After recalling classical isoperimetric inequalities for the two first eigenvalues, we present recent advances on this topic. In particular, we study the minimization of the second eigenvalue among plane convex domains. We also discuss the minimization of the third eigenvalue. We prove existence of a minimizer. For others eigenvalues, we just give some conjectures. We also consider the case of Neumann, Robin and Stekloff boundary conditions together with various functions of the eigenvalues.

Uniqueness of entropy solutions for nonlinear degenerate parabolic problems

Abstract: We consider the general degenerate parabolic equation: $$ {u_t} - \Delta b(u) + div F(u) = f in Q \in \left] {0,T} \right[ \times {\mathbb{R}^N},T > 0. $$ We prove existence of Kruzkhov entropy solutions of the associated Cauchy problem for bounded data where the flux functionFis supposed to be continuous. Uniqueness is established under some additional assumptions on the modulus of continuity ofFandb.

Conservation laws with discontinuous flux functions and boundary condition

Abstract: In a bounded domain \( \Omega \) we study the existence and uniqueness of entropy solutions of \( \partial u/\partial t + \mathrm {div}\, \Phi (u) = f\quad \mathrm{with}\quad u(0)=u_0 \), where \( \Phi \) is allowed to have some discontinuities of first type.

Maximal LP-regularity for elliptic operators with VMO-coefficients

Abstract: In this paper we show that the parabolic problem u t Au = f on ℝ n associated with elliptic operators A having coefficients in VMO ∩L ∞ has the property of maximal L p -regularity.

Intrinsic metrics and Lipschitz functions

Abstract: We study the notions of measurable metric and Lipschitz function which were introduced by N. Weaver ([12]), in the framework of Dirichlet spaces. To this respect, we bring some precisions and complements to [15], notably concerning links with the notion of intrinsic metric ([2]). In the particular case of an abstract Wiener space, we establish the relationship between these notions and that of H-metric ([5]) and µ-a.e. H-Lipschitz continuous function ([4]).

On the regularizing effect of strongly increasing lower order terms

Abstract: We show an existence result of bounded weak solutions for some semilinear Dirichlet problems, even if the right hand side belongs only to L1(Ω). The model example is $$ \left\{ \begin{gathered} - \Delta u + h(u) = f(x) in \Omega , \hfill \\ u = 0 on \partial \Omega , \hfill \\ \end{gathered} \right. $$ where Ω is a bounded open set in ℝ N , h(s) is a continuous and increasing function such that \( \mathop{{\lim }}\limits_{{s \to \sigma }} h(s) = + \infty \), for some δ>0

Another way to say caloric

Abstract: This paper offers characterizations of subsolutions of the heat equation u t -Δu=0 (the subcaloric functions) and the infinity heat equation u t -Δ∞ u=0 (the infinity-subcaloric functions) by means of comparison properties with explicit families of solutions of the corresponding equations. The primary ingredients of functions in these families are translates of solutions which depend radially on the space variables. Results of independent interest include the presentation and study of the class of infinity-caloric functions employed in the characterization.

Analyticity of solutions to fully nonlinear parabolic evolution equations on symmetric spaces

Abstract: It is shown that solutions to fully nonlinear parabolic evolution equations on symmetric Riemannian manifolds are real analytic in space and time, provided the propagator is compatible with the underlying Lie structure. Applications to Bellman equations and to a class of mean curvature flows are also discussed.

Linearized stability for nonlinear evolution equations

Abstract: We present a general principle of linearized stability at an equilibrium point for the Cauchy problem \( \dot{u}(t) + Au(t) i 0,t \geqslant 0,u(0) = u0 \) for an ω-accretive, possibly multivalued, operator A ⊂ Xx X in a Banach space X that has a linear ‘resolvent-derivative’ A ⊂ X x X. The result is applied to derive linearized stability results for the case of A = (B + G) under ‘minimal’ differentiability assumptions on the operators B ⊂ X x X and G: cl D(B) → at the equilibrium point, as well as for partial differential delay equations.

On some singular limits of homogeneous semigroups

Abstract: This paper is based upon some handwritten notes Philippe sent us in late 1996, following his visit to UC Berkeley. He was interested in a scaling argument from our paper with Feldman [E-F-G], and in his notes extended this trick to cover general nonlinear evolutions governed by homogeneous accretive operators. We reproduce his proof in §2 below, and add some commentary and a few Pde examples

A new regularity result for Ornstein-Uhlenbeck generators and applications

Abstract: Let H be a separable real Hilbert space (norm \( \left|\cdot\right|\), inner product \( \left\langle{\cdot,\cdot}\right\rangle\)). We are given a linear operator \(A:D\left( A \right) \subset H \to H \) such that HYPOTHESIS 1.1. (i) A is self-adjoint and there exists ω > 0 such that $$ \left\langle {Ax,x} \right\rangle \leqslant - \omega {\left| x \right|^2}, x \in D(A). $$ (1.1) (ii) A −1 is of trace class.

Nonlinear evolutions with Carathéodory forcing

Abstract: Let A be an m-accretive operator in a real Banach space X and f : J x X → X a function of Caratheodory type, where J = [0,a] ⊂ ℝ. This paper investigates the existence of mild solutions of the evolution system $$ u' + Au i f(t,u) on J = [0,a]. $$ satisfying additional time-dependent constraints u(t)∈K(t) on J for a given tube K(·). Main emphasis is on existence results that are valid under minimal assumptions on f, K and X.

On the uniqueness of solutions for nonlinear elliptic‐parabolic equations

Abstract: We prove a priori estimates in L 2(0,T;W 1,2(Ω)) and L∞(Q) T existence and uniqueness of solutions to Cauchy-Dirichlet problems for elliptic-parabolic systems $$ \frac{{\partial \sigma (u)}}{{\partial t}} - \sum\limits_{{i = 1}}^{n} {\frac{\partial }{{\partial {x_{i}}}}\left\{ {\rho (u){b_{i}}\left( {t,x,\frac{{\partial (u - \upsilon )}}{{\partial x}}} \right)} \right\}} + a(t,x,\upsilon ,u) = 0, $$ $$ - \sum\limits_{{i = 1}}^{n} {\frac{\partial }{{\partial {x_{i}}}}\left[ {\kappa (x)\frac{{\partial \upsilon }}{{\partial {x_{i}}}}} \right]} + \sigma (u) = f(t,x),(t,x) \in {Q_{T}} = (0,T) x \Omega , $$ where \( \rho (u) = \frac{{\partial \sigma (u)}}{{\partial u}} \). Systems of such form arise as mathematical models of various applied problems, for instance, electron transport processes in semiconductors. Our basic assumption is that log ρ(u) is concave. Such assumption is natural in view of drift-diffusion models, whereahas to be specified as a probability distribution function like a Fermi integral and u resp. υ have to be interpreted as chemical resp. electrostatic potential.

Singular limit of changing sign solutions of the porous medium equation

Abstract: In this paper, we study the limit as m→ ∞ of changing sign solutions of the porous medium equation: u t = Δ:|u| m-1 u in a domain Ω of ℝ N with Dirichlet boundary condition.

Pointwise gradient estimates of solutions to onedimensional nonlinear parabolic equations

Abstract: We present here an improved version of the method introduced by the first author to derive pointwise gradient estimates for the solutions of one-dimensional parabolic problems. After considering a general qualinear equation in divergence form we apply the method to the case of a nonlinear diffusion-convection equation. The conclusions are stated first for classical solutions and then for generalized and mild solutions. In the case of unbounded initial datum we obtain several regularizing effects for t > 0. Some unilateral pointwise gradient estimates are also obtained. The case of the Dirichlet problem is also considered. Finally, we collect, in the last section, several comments showing the connections among these estimates and the study of the free boundaries associated to the solutions of the diffusion-convection equation.

Nonautonomous heat equations with generalized Wentzell boundary conditions

Abstract: In this paper, we study the nonautonomous heat equation in C[0, 1] with generalized Wentzell boundary conditions. It is shown, under appropriate assumptions, that there exists a unique evolution family for this problem and that the family satisfies various regularity properties. This enables us to obtain, for the corresponding inhomogeneous problem, classical and strict solutions having optimal regularity.

The Cauchy problem for linear growth functionals

Abstract: In this paper we are interested in the Cauchy problem $$ \left\{ \begin{gathered} \frac{{\partial u}}{{\partial t}} = div a (x, Du) in Q = (0,\infty ) x {\mathbb{R}^{{N }}} \hfill \\ u (0,x) = {u_{0}}(x) in x \in {\mathbb{R}^{N}}, \hfill \\ \end{gathered} \right. $$ (1.1) where \( {u_{0}} \in L_{{loc}}^{1}({\mathbb{R}^{N}}) \) and \( a(x,\xi ) = { abla _{\xi }}f(x,\xi ),f:{\mathbb{R}^{N}}x {\mathbb{R}^{N}} \to \mathbb{R} \)being a function with linear growth as ‖ξ‖ satisfying some additional assumptions we shall precise below. An example of function f(x, ξ) covered by our results is the nonparametric area integrand \( f(x,\xi ) = \sqrt {{1 + {{\left\| \xi \right\|}^{2}}}} \); in this case the right-hand side of the equation in (1.1) is the well-known mean-curvature operator. The case of the total variation, i.e., when f(ξ)= ‖ξ‖ is not covered by our results. This case has been recently studied by G. Bellettini, V. Caselles and M. Novaga in [8]. The case of a bounded domain for general equations of the form (1.1) has been studied in [3] and [4] (see also [18], [11] and [15]). Our aim here is to introduce a concept of solution of (1.1), for which existence and uniqueness for initial data in L loc 1 (ℝ N ) is proved.

Regularity of solutions of nonlinear Volterra equations

Abstract: We consider the regularity properties of solutions of the nonlinear Volterra equation¶¶\( \frac{d}{dt}\left(u(t)-u_0+\int_0^t k(t-s)\left(u(s)-u_0\right)ds\right) + Au(t) i f(t) \)¶¶in Banach spaces X without the Radon-Nikodym property. Existence of strong solutions for an m-completely accretive operator A in a normal Banach space \( X\subset L^1(\Omega;\mu) \) is shown for sufficiently smooth data.

The focusing problem for the Eikonal equation

Abstract: We study the focusing problem for the eikonal quation $$ {\partial _{t}}u = | abla u{|^{2}}, $$ i.e., the initial value problem in which the support of the initial datum is outside some compact set in R d . The hole in the support will be filled in finite time and we are interested in the asymptotics of the hole as it closes. We show that in the radially symmetric case there are self-similar asymptotics, while in the absence of radial symmetry essentially any convex final shape is possible. However in R 2 , for generic initial data the asymptotic shape will be either a vanishing triangle or the region between two parabolas moving in opposite directions (a closing eye). We compare these results with the known results for the porous medium pressure equation which approaches the eikonal equation in the limit as m → 1.