Data In L1 Research Articles

Nonsmooth data error estimates for fully discrete finite element approximations of semilinear parabolic equations in Banach space

Implicit–explicit (IMEX) Euler formula together with finite element methods is proposed to solve numerically rather general semilinear parabolic differential equations with initial data in Banach space Xα, 0<α≤1. The time-space regularity of solutions to this class of equations is first investigated. Stability and error estimates are provided for this fully discrete scheme by discrete semigroup method. In the special case when X is the Hilbert space L2(Ω), these error estimates bridge the gap between the existing results on this scheme for semilinear parabolic problems with initial data in L2(Ω) and in H1(Ω) under general nonlinearity. Numerical experiments for nonlinear Allen–Cahn equations verify and complement our theoretical results.

Optimal decay rates and space–time analyticity of solutions to the Patlak-Keller–Segel equations

Based on some new elementary estimates for the space–time derivatives of the heat kernel, we use a bootstrapping approach to establish quantitative estimates on the optimal decay rates for the Lq(Rd) (1≤q≤∞, d∈N) norm of the space–time derivatives of solutions to the (modified) Patlak-Keller–Segel equations with initial data in L1(Rd), which implies the joint space–time analyticity of solutions. When the L1(Rd) norm of the initial datum is small, the upper bound for the decay estimates is global in time, which yields a lower bound on the growth rate of the radius of space–time analyticity in time. As a byproduct, the space analyticity is obtained for any initial data in L1(Rd). The decay estimates and space–time analyticity are also established for solutions bounded in both space and time variables. The results can be extended to a more general class of equations.

Global well-posedness of the 1d compressible Navier–Stokes system with rough data

In this paper, we study the global well-posedness problem for the 1d compressible Navier–Stokes systems (cNSE) in gas dynamics with rough initial data. First, Liu and Yu (2022) [30] established the global well-posedness theory for the 1d isentropic cNSE with initial velocity data in BV space. Then, it was extended to the 1d full cNSE with initial velocity and temperature data in BV space by Wang et al. (2022) [31]. We improve the global well-posedness result of Liu and Yu with initial velocity data in W2γ,1 space; and of Wang–Yu–Zhang with initial velocity data in L2∩W2γ,1 space and initial data of temperature in W˙−23,65∩W˙2γ−1,1 for any γ>0arbitrarily small. Our essential ideas are based on establishing various “end-point” smoothing estimates for the 1d parabolic equation.

Global existence and uniqueness of weak solutions of a Stokes-Magneto system with fractional diffusions

We consider a Stokes-Magneto system in Rd (d≥2) with fractional diffusions Λ2αu and Λ2βb for the velocity u and the magnetic field b, respectively. Here α,β are positive constants and Λs=(−Δ)s/2 is the fractional Laplacian of order s. We establish global existence of weak solutions of the Stokes-Magneto system for any initial data in L2 when α, β satisfy 1/2<α<(d+1)/2, β>0, and min⁡{α+β,2α+β−1}>d/2. It is also shown that weak solutions are unique if β≥1 and min⁡{α+β,2α+β−1}≥d/2+1, in addition.

Pointwise decay of solutions to the energy critical nonlinear Schrödinger equations

In this note, we prove pointwise decay in time of solutions to the 3D energy-critical nonlinear Schrödinger equations assuming data in L1∩H3. The main ingredients are the boundness of the Schrödinger propagators in Hardy space due to Miyachi [9] and a fractional Leibniz rule in the Hardy space. We also extend the fractional chain rule to the Hardy space.

Dirichlet and Neumann problems for elliptic equations with singular drifts on Lipschitz domains

We consider the Dirichlet and Neumann problems for second-order linear elliptic equations: \[ − △ u + div ⁡ ( u b ) = f and − △ v − b ⋅ ∇ v = g -\triangle u +\operatorname {div}(u\mathbf {b}) =f \quad \text { and }\quad -\triangle v -\mathbf {b} \cdot \nabla v =g \] in a bounded Lipschitz domain Ω \Omega in R n \mathbb {R}^n ( n ≥ 3 ) (n\geq 3) , where b : Ω → R n \mathbf {b}:\Omega \rightarrow \mathbb {R}^n is a given vector field. Under the assumption that b ∈ L n ( Ω ) n \mathbf {b} \in L^{n}(\Omega )^n , we first establish existence and uniqueness of solutions in L α p ( Ω ) L_{\alpha }^{p}(\Omega ) for the Dirichlet and Neumann problems. Here L α p ( Ω ) L_{\alpha }^{p}(\Omega ) denotes the Sobolev space (or Bessel potential space) with the pair ( α , p ) (\alpha ,p) satisfying certain conditions. These results extend the classical works of Jerison-Kenig [J. Funct. Anal. 130 (1995), pp. 161–219] and Fabes-Mendez-Mitrea [J. Funct. Anal. 159 (1998), pp. 323–368] for the Poisson equation. We also prove existence and uniqueness of solutions of the Dirichlet problem with boundary data in L 2 ( ∂ Ω ) L^{2}(\partial \Omega ) . Our results for the Dirichlet problems hold even for the case n = 2 n=2 .

Continuous data assimilation for the 3D Ladyzhenskaya model: analysis and computations

We analyze continuous data assimilation by nudging for the 3D Ladyzhenskaya equations. The analysis provides conditions on the spatial resolution of the observed data that guarantee synchronization to the reference solution associated with the observed, spatially coarse data. This synchronization holds even though it is not known whether the reference solution, with initial data in L2, is unique; any particular reference solution is determined by the observed, coarse data. The efficacy of the algorithm in both 2D and 3D is demonstrated by numerical computations.

Pointwise convergence problem of the Korteweg–de Vries–Benjamin–Ono equation

In this paper we investigate the pointwise convergence problem for the Korteweg–de Vries–Benjamin–Ono equation ut+γH(∂x2u)−β∂x3u=0,(x,t)∈R×R,u(x,0)=f(x)∈Hs(R),where γβ>0. We prove that the solution u(x,t)=Utf(x) converges pointwisely to the initial data f(x) for a.e. x∈R when f∈Hs(R) with s≥14, and that the Hausdorff dimension of the divergence set of points of the solution is αU(s)=1−2s when 14≤s≤12. We also obtain the stochastic continuity for the initial data with much less regularity, i.e. for a large class of the initial data in L2(R), via the randomization technique.

Quantitative analysis of fundamental frequency in Spanish (L2) and Brazilian Portuguese (L1)

This paper analyzes the intonation of Spanish and Brazilian Portuguese (BP) produced by monolingual speakers of both languages and bilingual BP speakers that lived in Spain on average for 6 years. Bilinguals produced data in both Spanish L2 (BL2) and BP L1 (BL1). Speech materials are sentences in different modalities (declaratives, yes-no and wh-questions) and reading styles (isolated sentences and storytelling). Fundamental frequency (f0) contours were analyzed to assess learning in Spanish L2 and language attrition in the L1 production of bilinguals. Variability in the f0 contours of the four language conditions was gauged by means of three indices (peak rate, peak range and global standard deviation). Dynamic time warping (DTW) distances between pairs of f0 contours were also measured as a way to measure within- and between-language differences in intonation patterns. The main results are: 1) BL2 and BL1 contours are significantly more variable than the monolingual ones both quantitatively and qualitatively; 2) BL2 contours partially converge towards the patterns of Spanish monolinguals, showing that there is learning; 3) there is evidence for language attrition in the form of transfer of Spanish patterns to BP contours produced by bilinguals; 4) Learning and attrition levels are different depending on sentence modality, such that learning is greater in modalities that differ less between BP and Spanish and attrition is greater in modalities that differ the most.

Entropy solution for a nonlinear parabolic problem with homogeneous Neumann boundary condition involving variable exponents

In this paper we prove the existence and uniqueness of an entropy solution for a non-linear parabolic equation with homogeneous Neumann boundary condition and initial data in L1. By a time discretization technique we analyze the existence, uniqueness and stability questions. The functional setting involves Lebesgue and Sobolev spaces with variable exponents.

Probabilistic pointwise convergence problem of Schrödinger equations on manifolds

In this paper, we investigate the probabilistic pointwise convergence problem of Schrödinger equation on the manifolds. We prove probabilistic pointwise convergence of the solutions to Schrödinger equations with the initial data in L 2 ( T n ) L^{2}(\mathrm {\mathbf {T}}^{n}) , where T = [ 0 , 2 π ) \mathrm {\mathbf {T}}=[0,2\pi ) , which require much less regularity for the initial data than the rough data case. We also prove probabilistic pointwise convergence of the solutions to Schrödinger equation with Dirichlet boundary condition for a large set of random initial data in ∩ s &gt; 1 2 H s ( Θ ) \cap _{s&gt;\frac {1}{2}}H^{s}(\Theta ) , where Θ \Theta is three dimensional unit ball, which require much less regularity for the initial data than the rough data case.

Unconditional well-posedness for the Kawahara equation

This article is concerned with the unconditional well-posedness for the Kawahara equation on the real line and shows that this holds true for initial data in L2(R). This is achieved by applying an infinite iteration scheme of normal form reductions.

SBV regularity for Burgers-Poisson equation

The SBV regularity of weak entropy solutions to the Burgers-Poisson equation for initial data in L1(R) is considered. We show that the derivative of a solution consists of only the absolutely continuous part and the jump part.

Global endpoint Strichartz estimates for Schrödinger equations on the cylinder [formula omitted

We prove global in time Strichartz estimates on the semi-periodic cylinder with initial data in L2. This extends the local results of Takaoka-Tzvetkov [13] and the global estimates with loss of derivatives of Hani-Pausader [9] and Barron [1].

Approximations of Lyapunov functionals for ISS analysis of a class of higher dimensional nonlinear parabolic PDEs

This paper introduces a novel method, namely the approximations of Lyapunov functionals, for input-to-state stability (ISS) analysis of a class of higher dimensional nonlinear parabolic partial differential equations (PDEs) with variable coefficients. Specifically, for any q∈[1,+∞] and the considered nonlinear parabolic PDEs with different types of boundary disturbances in Llocq(R+;L1(∂Ω)) and initial data in L1(Ω), we show that ISS-like estimates in L1-norm (or weighted L1-norm) can be established by constructing approximations of (coercive and non-coercive) Lyapunov functionals. Some examples are provided to illustrate the application of the proposed method.

Global analytic solutions for the nonlinear Schrödinger equation

Abstract We prove the existence of global analytic solutions to the nonlinear Schrödinger equation in one dimension for a certain type of analytic initial data in L 2 {L^{2}} .

Approximation by group invariant subspaces

In this article we study the structure of Γ-invariant spaces of L2(S). Here S is a second countable LCA group. The invariance is with respect to the action of Γ, a non commutative group in the form of a semidirect product of a discrete cocompact subgroup of S and a group of automorphisms. This class includes in particular most of the crystallographic groups. We obtain a complete characterization of Γ-invariant subspaces in terms of range functions associated to shift-invariant spaces. We also define a new notion of range function adapted to the Γ-invariance and construct Parseval frames of orbits of some elements in the subspace, under the group action. These results are then applied to prove the existence and construction of a Γ-invariant subspace that best approximates a set of functional data in L2(S). This is very relevant in applications since in the euclidean case, Γ-invariant subspaces are invariant under rigid movements, a very sought feature in models for signal processing.

The Euler equations in a critical case of the generalized Campanato space

In this paper we prove local in time well-posedness for the incompressible Euler equations in Rn for the initial data in L1(1)1(Rn), which corresponds to a critical case of the generalized Campanato spaces Lq(N)s(Rn). The space is studied extensively in our companion paper [9], and in the critical case we have embeddings B∞,11(Rn)↪L1(1)1(Rn)↪C0,1(Rn), where B∞,11(Rn) and C0,1(Rn) are the Besov space and the Lipschitz space respectively. In particular L1(1)1(Rn) contains non-C1(Rn) functions as well as linearly growing functions at spatial infinity. We can also construct a class of simple initial velocity belonging to L1(1)1(Rn), for which the solution to the Euler equations blows up in finite time.

Reaction-diffusion systems with initial data of low regularity

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 L1 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 L2 a priori estimates for the solution.Here, we establish such a key estimate with initial data in L1 while the usual theory uses L2. 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.

A structurally damped σ‐evolution equation with nonlinear memory

In this paper, we investigate the global (in time) existence of small data solutions to the Cauchy problem for the following structurally damped σ‐evolution model with nonlinear memory term: with σ&gt;0. In particular, for γ∈((n−σ)/n,1), we find the sharp critical exponent, under the assumption of small data in L1. Dropping the L1 smallness assumption of initial data, we show how the critical exponent is consequently modified for the problem. In particular, we obtain a new interplay between the fractional order of integration 1−γ in the nonlinear memory term and the assumption that initial data are small in Lm, for some m&gt;1.