Regularizing effects for ut + Aϑ(u) = 0 in L1

Various initial-boundary value problems and Cauchy problems can be written in the form $\text{du}\text{dt}\text{+ A?(u) = 0}$, where ?:R→$\text{R}$ is nondecreasing and A is the linear generator of strongly continuous nonexpansive semigroup e^?tA in an L¹ space. For example, if A = ?Δ (subject, perhaps, to suitable boundary conditions) we obtain equations arising in flow in a porous medium or plasma physics (depending on the choice of ?) while if $\text{A =}\text{?}\text{?x}$ acting in L¹($\text{R}$) we have a scalar conservation law. In this paper we show that if M, m > 0 and m?′² ? ν??′' ? M?′², where ν ? {1,?1}, then (roughly speaking), the norm of $\text{t}\text{du}\text{dt}$ may be estimated in terms of the initial data u₀ in L¹. Such estimates give information about the regularity of solutions, asymptotic behaviour, etc., in applications. Side issues, such as the introduction of sufficiently regular approximate problems on which estimates can be made and the assignment of a precise meaning to the operator A?, are also dealt with. These considerations are of independent interest.     

Regularity properties of Schrödinger and Dirichlet semigroups

First we compute Brownian motion expectations of some Kac's functionals. This allows a complete study of the semigroups generated by the formal differential operator $\text{H = ?}\text{1}\text{2}\text{Δ + V}$ on the various Lebesgue's spaces L^q=L^q$\text{R}$ⁿ, dx, whenever the negative part of V is in L^∞ + L^p for some $\text{p >}\text{max}\text{{1,}\text{n}\text{2}\text{}}$. Our approach is probabilistic and some of the proofs are surprisingly elementary. The negative infinitesimal generators of our semigroups are shown to be reasonable self-adjoint extensions of H. Under mild assumptions on V, H is unitary equivalent to the Dirichlet operator, say D, associated to its groundstate measure. We study regularity of the semigroups generated by D. We concentrate on hyper and supercontractivity and we give, using probabilistic techniques, new examples of potential functions V which give rise to hyper and supercontractive Dirichlet semigroups.     

Contractive Korovkin approximations

Our results are related to $\text{L}$¹-shadows in L_p-spaces. For p = 1 we will complete the characterization of $\text{L}$¹-shadows and $\text{L}$^1,1-shadows. For 1 < p < ∞ S. J. Bernau has shown that the $\text{L}$¹-shadow of a set in L_p is the range of a contractive projection. We will show that the corresponding theorem is not true for all reflexive spaces, but is true for locally uniformly convex reflexive spaces.     

The automorphisms of convolution algebras of C∞ functions

We find the automorphisms and the spectra of several different topological convolution algebras of C^∞-functions on the real line. Starting with the convolution algebra of compactly supported C^∞-functions, equipped with the usual LF-topology, we define a corresponding convolution algebra of C^∞-functions of arbitrarily fast exponential decay at ∞; and convolution algebras of a given finite degree r of exponential decay at ∞. These algebras may be described topologically as “hyper Schwartz spaces.” With a natural Frechet topology, which we define, they get a structure as locally m-convex algebras. The continuous automorphisms and spectra of these algebras are described completely. We show that the algebra of C^∞-functions of infinitly fast exponential decay at ∞, $\text{H J}$, on the one hand, and the algebra of C^∞-functions of only a finite degree $\text{e}{}^{\mathrm{?r¦x¦}}$ decay at ∞, $\text{J}$_r⁰, on the other hand, have quite different automorphisms, although $\text{H J}$ = ∩_r$\text{J}$_r⁰. As an application, we show that the conformal group is canonically represented as the full group of automorphisms of $\text{J}$_r⁰, and that this representation does not extend to a representation on the Banach algebra L¹($\text{R}$).     

Compact Sobolev imbeddings on finite measure spaces

We provide conditions on a finite measure μ on $\text{R}$ⁿ which insure that the imbeddings W^{k, p}($\text{R}$ⁿdμ)?L^p($\text{R}$ⁿdμ) are compact, where 1 ? p < ∞ and k is a positive integer. The conditions involve uniform decay of the measure μ for large ¦x¦ and are satisfied, for example, by $\text{dμ = e}{}^{\mathrm{?¦x¦\alpha }}\text{dx,}\text{where}\text{α > 1}$.     

Functions and derivations of C1-algebras

It is shown that there is a closed symmetric derivation δ of a C¹-algebra with dense domain $\text{D}$(δ), an element A = A¹ ?$\text{D}$(δ), and a C¹-function $\text{f}$ such that $\text{f}$(A)?$\text{D}$(δ). Some estimates are derived for $\text{∥ δ(¦ A ¦)∥}\text{and}\text{∥ δ(A}{}_{\mathrm{+}}{}^{\mathrm{\alpha }}\text{)∥}$, where 0 < α < 1. It is shown that there exists a family of one-one self-adjoint operators S(t) in $\text{L}$(H) which depends linearly on t, while $\text{¦ S(t)¦}$ is not differentiable. It is also shown that there exists $\text{L}$(H) which is not C¹-self-adjoint even though it satisfies $\text{∥}\text{exp}\text{(itT)∥ ? C(1 + ¦ t ¦)}$ for all t ? $\text{R}$     

nth order Stieltjes differential boundary operators and Stieltjes differential boundary systems

This article discusses linear differential boundary systems, which include nth-order differential boundary relations as a special case, in $\text{L}$_n^p[0,1] × $\text{L}$_n^p[0,1], 1 ? p < ∞. The adjoint relation in $\text{L}$_n^q[0,1] × $\text{L}$_n^q[0,1], $\text{1}\text{p}\text{+}\text{1}\text{q}\text{= 1}$, is derived. Green's formula is also found. Self-adjoint relations are found in $\text{L}$_n²[0,1] × $\text{L}$_n²[0,1], and their connection with Coddington's extensions of symmetric operators on subspaces of $\text{L}$_n^p[0,1] × $\text{L}$_n²[0,1] is established.     

On convergence of LAD estimates in autoregression with infinite variance

The least absolute deviation estimates L(N), from N data points, of the autoregressive constants a = (a₁, …, a_q)′ for a stationary autoregressive model, are shown to have the property that N^σ(L(N) ? a) converge to zero in probability, for $\text{σ <}\text{1}\text{α}$, where the disturbances are i.i.d., attracted to a stable law of index α, 1 ≤ α < 2, and satisfy some other conditions.     

A duality principle for lattices and categories of modules

Let R be a ring with 1, R^op the opposite ring, and R-Mod the category of left unitary R-modules and R-linear maps. A characterization of well-powered abelian categories $\text{A}$ such that there exists an exact embedding functor $\text{A}$→R-Mod is given. Using this characterization and abelian category duality, the following duality principles can be established.Theorem. There exists an exact embedding functor $\text{A}$→R-Mod if and only if there exists an exact embedding functor $\text{A}$^op→R^op-Mod.Corollary. If R-Mod has a specified diagram-chasing property, then R^op-Mod has the dual property.A lattice L is representable by R-modules if it is embeddable in the lattice of submodules of some unitary left R-module; $\text{L}$(R) denotes the quasivariety of all lattices representable by R-modules.Theorem. A lattice L is representable by R-modules if and only if its order dual L¹ is representable by R^op-modules. That is, $\text{L}\text{(}\text{R}{}^{\mathrm{op}}\text{)={L}{}^1\text{:L?}\text{L}\text{(}\text{R}\text{)}}$.If $\text{R}$ is a commutative ring with 1 and a specified diagram-chasing result is satisfied in R-Mod, then the dual result is also satisfied in R-Mod. Furthermore, $\text{L}\text{(}\text{R}\text{)}$ is self-dual: $\text{L}\text{(}\text{R}\text{)= {L}{}^1\text{:L?}\text{L}\text{(}\text{R}\text{)}.}$     

Asymptotic stability of some quasilinear parabolic equations in divergence form. L∞ results

In this paper we study the behavior of solutions of some quasilinear parabolic equations of the form

$\text{(}\text{?u}\text{?t}\text{) ?}\text{∑}\text{i=1}\text{n}\text{(}\text{d}\text{dx}{}_{\mathrm{i}}\text{) a}{}_{\mathrm{i}}\text{(x, t, u, u}{}_{\mathrm{x}}\text{) + a(x, t, u, u}{}_{\mathrm{x}}\text{)u + f(x, t) = O,}$

as t → ∞. In particular, the solutions of these equations will decay to zero as t → ∞ in the L^∞ norm.     

Multipliers of the Hölder classes

Following the lines of [13] we introduce the classes of mixed smoothness $\text{L}$α_p(Rⁿ), $\text{L}$α_p(Zⁿ) for a multi-index α = (α₁,…, α_n). Such classes are naturally tied up with the study of semi-elliptic differential and difference equations.Besides a brief presentation of such classes, we concentrate our research on the study of mixed homogeneous multipliers with homogeneity β = (β₁, …, β_n) and their preservation of mixed homogeneous Hölder classes $\text{L}$_α^∞, for a different multi-index α.In the last paragraph we apply the results to produce various improvements of the classical Schauder's estimates, for differential and difference equations, in the parabolic and elliptic case.     

On moment-discretization and least-squares solutions of linear integral equations of the first kind

Let K(s, t) be a continuous function on [0, 1] × [0, 1], and let $\text{K}$ be the linear integral operator induced by the kernel K(s, t) on the space $\text{L}$₂[0, 1]. This note is concerned with moment-discretization of the problem of minimizing 6K_x?y6 in the $\text{L}$₂-norm, where y is a given continuous function. This is contrasted with the problem of least-squares solutions of the moment-discretized equation: ∝₀¹K(s_i, t) x(t) dt = y(s_i), i = 1, 2,h., n. A simple commutativity result between the operations of “moment-discretization” and “least-squares” is established. This suggests a procedure for approximating $\text{K2y}$ (where $\text{K}$² is the generalized inverse of $\text{K}$), without recourse to the normal equation $\text{K1Kx = K1y}$, that may be used in conjunction with simple numerical quadrature formulas plus collocation, or related numerical and regularization methods for least-squares solutions of linear integral equations of the first kind.     

Weak solutions for a nonlinear dispersive equation

In this paper we study the existence, uniqueness, and regularity of the solutions for the Cauchy problem for the evolution equation $\text{u}{}_{\mathrm{t}}\text{+ (f (u))}{}_{\mathrm{x}}\text{? u}{}_{\mathrm{xxt}}\text{= g(x, t), (}{}^1\text{)}\text{where}\text{u = u(x, t), x}$ is in (0, 1), 0 ? t ? T, T is an arbitrary positive real number,f(s)?C¹$\text{R}$, and g(x, t)?L^∞(0, T; L²(0, 1)). We prove the existence and uniqueness of the weak solutions for (¹) using the Galerkin method and a compactness argument such as that of J. L. Lions. We obtain regular solutions using eigenfunctions of the one-dimensional Laplace operator as a basis in the Galerkin method.     

Existence and bifurcation theorems for nonlinear elliptic eigenvalue problems on unbounded domains

We consider nonlinear elliptic eigenvalue problems on unbounded domains G?$\text{R}$ⁿ. Using an extended Ljusternik-Schnirelman theory we prove the existence of infinitely many eigenfunctions on every sphere in L²(G). Moreover, we establish that the infimum λ1 of the spectrum of the linearized problem L is always a bifurcation point. In addition, there is an infinity of branches emanating at λ1 from the trivial line of solutions if λ1 belongs to the essential spectrum of L.     

The Yang-Mills equations on the universal cosmos

Global existence and regularity of solutions for the Yang-Mills equations on the universal cosmos M?, which has the form R¹ × S³ for each of an 8-parameter continuum of factorizations of M? as time × space, are treated by general methods. The Cauchy problem in the temporal gauge is globally soluble in its abstract evolutionary form with arbitrary data for the field ⊕ potential in L_2,r(S³) ⊕ L_{2,r + 1}(S³), where r is an integer >1 and L_2,r denotes the class of sections whose first r derivatives are square-integrable; if r = 1, the problem is soluble locally in time. When r is 3 or more the solution is identifiable with a classical one; if infinite, the solution is in C^∞(M?). These results extend earlier work and approaches [1–5]. Solutions of the equations on Minkowski space-time M₀ extend canonically (modulo gauge transformations) to solutions on M? provided their Cauchy data are moderately smooth and small near spatial infinity. Precise asymptotic structures for solutions on M₀ follow, and in turn imply various decay estimates. Thus the energy in regions uniformly bounded in direction away from the light cone is $\text{O(¦x}{}_0\text{¦}{}^{\mathrm{?5}}\text{)}$, where x₀ is the Minkowski time coordinate; analysis solely in M₀ [8,9] earlier yielded the estimate $\text{O(¦x}{}_0\text{¦}{}^{\mathrm{?2}}\text{)}$ applicable to the region within the light cone. Similarly it follows that the action integral for a solution of the Yang-Mills equations in M₀ is finite, in fact absolutely convergent.     

A note on the similarity depth

In this note we demonstrate the existence of E0L forms F and G which are n-similar, i.e. $\text{L}$ⁿ(F) = $\text{L}$ⁿ(G) but $\text{L}$ⁿ⁺¹(F)≠$\text{L}$ⁿ⁺¹(G) for n ∈ {2, 3}. This partially solves an open problem from [3].     

On global discontinuous solutions of Hamilton–Jacobi equationsSur des solutions globales discontinues des équations d'Hamilton–Jacobi

The uniqueness of classical semicontinuous viscosity solutions of the Cauchy problem for Hamilton–Jacobi equations is established for globally Lipschitz continuous and convex Hamiltonian H=H(Du), provided the discontinuous initial value function ?(x) is continuous outside a set Γ of measure zero and satisfies

(1)$\text{?(x)??}{}_7\text{(x):=}\text{lim}\text{inf}\text{y→x,}\text{y∈}\text{R}{}^{\mathrm{d}}\text{?Γ}\text{?(y).}$

We prove that the discontinuous solutions with almost everywhere continuous initial data satisfying ($\text{1}$) become Lipschitz continuous after finite time for locally strictly convex Hamiltonians. The L¹-accessibility of initial data and a comparison principle for discontinuous solutions are shown for a general Hamiltonian. The equivalence of semicontinuous viscosity solutions, bi-lateral solutions, L-solutions, minimax solutions, and L^∞-solutions is clarified. To cite this article: G.-Q. Chen, B. Su, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 113–118     

On scattering and everywhere defined scattering operators for nonlinear Klein-Gordon equations

Asymptotic properties of solutions of the nonlinear Klein-Gordon equation ?_t²u ? Δu + m²u + f(u) = 0 (NLKG) $\text{u¦}{}_0\text{= θ}$, $\text{?}{}_{\mathrm{t}}\text{u¦}{}_0\text{= Ψ}$, are investigated, which are inherited from the corresponding solutions v of the (linear) Klein-Gordon equation ?_t²v ? Δv + m²v = 0$\text{v¦}{}_0\text{= θ}$, $\text{?}{}_{\mathrm{t}}\text{v¦}{}_0\text{= Ψ}$, (KG) In particular, the finiteness of time-integrals in L_q over R₊ of certain Sobolevnorms in space of the solution is proved to be such a hereditary property. Together with a device by W. A. Strauss and a weak decay result for the (KG) due to R. S. Strichartz, this is used to prove that under suitable restrictions on the nonlinearity, the scattering operator for the (NLKG) is defined on all of L₂¹ × L₂ for n = 3.     

Rate of decay for solutions of stochastic differential equations

A theorem is given which demonstrates that solutions of a stochastic differential equation decay to zero at least as fast as a function ψ(t) provided $\text{L[}\text{ln}\text{V] ?}\text{(d}\text{dt)}\text{(}\text{ln}\text{ψ)}$, where V is a Liapunov-like function and L is the operator associated with the stochastic equation. The result is an extension of a recent result of A. Friedman and M. Pinsky. A further generalization is discussed.     

Uniform estimates for solutions of nonlinear Klein-Gordon equations

Uniform estimates in $\text{H}{}_0{}^1\text{(Ω)}$ of global solutions to nonlinear Klein-Gordon equations of the form $\text{u}{}_{\mathrm{tt}}\text{? Δu + mu = g(u)}\text{in}\text{Ω, u = 0}\text{in}\text{?Ω}$, where Ω is an open subset of $\text{R}$^N, m > 0, and g satisfies some growth conditions are established.