L2 Initial Data Research Articles

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.

Weak and classical solutions to an asymptotic model for atmospheric flows

In this paper we study a recently derived mathematical model for nonlinear propagation of waves in the atmosphere, for which we establish the local well-posedness in the setting of classical solutions. This is achieved by formulating the model as a quasilinear parabolic evolution problem in an appropriate functional analytic framework and by using abstract theory for such problems. Moreover, for L2-initial data, we construct global weak solutions by employing a two-step approximation strategy based on a Galerkin scheme, where an equivalent formulation of the problem in terms of a new variable is used. Compared to the original model, the latter has the advantage that the L2-norm is a Liapunov functional.

Uniform Error Estimates of the Finite Element Method for the Navier-Stokes Equations in R2 with L2 Initial Data.

In this paper, we study the finite element method of the Navier-Stokes equations with the initial data belonging to the L2 space for all time t>0. Due to the poor smoothness of the initial data, the solution of the problem is singular, although in the H1-norm, when t∈[0,1). Under the uniqueness condition, by applying the integral technique and the estimates in the negative norm, we deduce the uniform-in-time optimal error bounds for the velocity in H1-norm and the pressure in L2-norm.

One-sided Hölder regularity of global weak solutions of negative order dispersive equations

We prove global existence, uniqueness and stability of entropy solutions with L2 initial data for a general family of negative order dispersive equations. These weak solutions are found to satisfy one-sided Hölder conditions whose coefficients decay in time. The latter result controls the height of solutions and further provides a way to bound the maximal lifespan of classical solutions from their initial data.

Analysis of fully discrete finite element methods for 2D Navier–Stokes equations with critical initial data

First-order convergence in time and space is proved for a fully discrete semi-implicit finite element method for the two-dimensional Navier–Stokes equations with L2 initial data in convex polygonal domains, without extra regularity assumptions or grid-ratio conditions. The proof utilises the smoothing properties of the Navier–Stokes equations in the analysis of the consistency errors, an appropriate duality argument, and the smallness of the numerical solution in the discrete L2(0, tm; H1) norm when tm is smaller than some constant. Numerical examples are provided to support the theoretical analysis.

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.

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.

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.

Almost sure existence of global solutions for supercritical semilinear wave equations

We prove that for almost every initial data (u0,u1)∈Hs×Hs−1 with s>p−3p−1 there exists a global weak solution to the supercritical semilinear wave equation ∂t2u−Δu+|u|p−1u=0 where p>5, in both R3 and T3. This improves in a probabilistic framework the classical result of Strauss [20] who proved global existence of weak solutions associated to H1×L2 initial data. The proof relies on techniques introduced by Oh and Pocovnicu in [16] based on the pioneer work of Burq and Tzvetkov in [7]. We also improve the global well-posedness result in [21] for the subcritical regime p<5 to the endpoint s=p−3p−1.

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.

Large time decay of weak solutions for the 2D Oldroyd-B model of non-Newtonian flows

This paper is dedicated to investigating large time decay of weak solutions for the 2D Oldroyd-B model of non-Newtonian flows. Due to the inconsistencies of the dissipative effect between Laplacian velocity viscosity and linear stress damping, either the classic Fourier splitting methods or Kato’s methods for both velocity u and stress tensor τ are not available directly. Based on a new observation on the structure of Oldroyd-B model instead of the complex spectral analysis of linear problem, we explore the low frequency effect for weak solutions which allows us to develop the generalized Fourier splitting technique. More precisely, we first examine that the weak solutions with L2 initial data have the non-uniformly decay ‖u(t)‖L2(R2)+‖τ(t)‖L2(R2)→0,t→∞. Furthermore, we obtain the optimal decay rates of the weak solutions ‖τ‖L2(R2)+‖u‖L2(R2)≤C(1+t)−12 when (u0,τ0)∈L2(R2)∩L1(R2). Our methods here are also available for the time decay issue of the complex fluid flows with partial dissipation.

Well-posedness of semilinear heat equations in L1

The problem of obtaining necessary and sufficient conditions for local existence of non-negative solutions in Lebesgue spaces for semilinear heat equations having monotonically increasing source term f has only recently been resolved (Laister et al. (2016)). There, for the more difficult case of initial data in L1, a necessary and sufficient integral condition on f emerged. Here, subject to this integral condition, we consider other fundamental properties of solutions with L1 initial data of indefinite sign, namely: uniqueness, regularity, continuous dependence and comparison. We also establish sufficient conditions for the global-in-time continuation of solutions for small initial data in L1.

The role of superlinear damping in the construction of solutions to drift-diffusion problems with initial data in L1

AbstractIn boundedn-dimensional domainsΩ, the Neumann problem for the parabolic equation$$\begin{array}{} \displaystyle u_t = \nabla \cdot \Big( A(x,t)\cdot\nabla u\Big) + \nabla \cdot \Big(b(x,t)u\Big) - f(x,t,u)+g(x,t) \end{array}$$(*)is considered for sufficiently regular matrix-valuedA, vector-valuedband real valuedg, and withfrepresenting superlinear absorption in generalizing the prototypical choice given byf(⋅, ⋅,s) =sαwithα&gt; 1. Problems of this form arise in a natural manner as sub-problems in several applications such as cross-diffusion systems either of Keller-Segel or of Shigesada-Kawasaki-Teramoto type in mathematical biology, and accordingly a natural space for initial data appears to beL1(Ω).The main objective thus consists in examining how far solutions can be constructed for initial data merely assumed to be integrable, with major challenges potentially resulting from the interplay between nonlinear degradation on the one hand, and the possibly destabilizing drift-type action on the other in such contexts. Especially, the applicability of well-established methods such as techniques relying on entropy-like structures available in some particular cases, for instance, seems quite limited in the present setting, as these typically rely on higher initial regularity properties.The first of the main results shows that in the general framework of (*), nevertheless certain global very weak solutions can be constructed through a limit process involving smooth solutions to approximate variants thereof, provided that the ingredients of the latter satisfy appropriate assumptions with regard to their stabilization behavior.The second and seemingly most substantial part of the paper develops a method by which it can be shown, under suitably stregthened hypotheses on the integrability ofband the degradation parameterα, that the solutions obtained above in fact form genuine weak solutions in a naturally defined sense. This is achieved by properly exploiting a weak integral inequality, as satisfied by the very weak solution at hand, through a testing procedure that appears to be novel and of potentially independent interest.To underline the strength of this approach, both these general results are thereafter applied to two specific cross-diffusion systems. Inter alia, this leads to a statement on global solvability in a logistic Keller-Segel system under the assumptionα&gt;$\begin{array}{} \frac{2n+4}{n+4} \end{array}$on the respective degradation rate which seems substantially milder than any previously found condition in the literature. Apart from that, for a Shigesada-Kawasaki-Teramoto system some apparently first results on global solvability forL1initial data are derived.

Global regularity for the 3D micropolar equations

In this paper a global weak solution of the 3D generalized micropolar equations with any α≥1 and β>0 corresponding to any L2 initial data are considered. Actually, weak solutions associated with α≥54 and β≥1 are global classical solutions when their initial data are sufficiently smooth.

Perturbations of superstable linear hyperbolic systems

The paper deals with initial-boundary value problems for linear non-autonomous first order hyperbolic systems whose solutions stabilize to zero in a finite time. We prove that problems in this class remain exponentially stable in L2 as well as in C1 under small bounded perturbations. To show this for C1, we prove a general smoothing result implying that the solutions to the perturbed problems become eventually C1-smooth for any L2-initial data.

Global well-posedness and blow-up for the 2-D Patlak–Keller–Segel equation

In this paper, we prove that for every nonnegative initial data in L1(R2), the Patlak–Keller–Segel equation is globally well-posed if and only if the total mass M≤8π. Our proof is based on some monotonicity formulas of nonnegative mild solutions.

Two timescales asymptotes of the weakly compressible Stokes system

We establish a two timescale asymptotics of the weakly compressible Stokes system which has dissipation of order ε>0. For any L2 initial data, over time scale of order 1, the solutions of the weakly compressible Stokes system converge strongly to those of the acoustic system as ε→0. Over time scale of order 1/ε, the limit system is the incompressible Stokes system with the initial data projected on the incompressible mode. For the periodic domain, the convergence is weak due to the fast oscillation generated by acoustic waves. For the Navier-slip boundary condition with the reciprocal of slip length being square of the Knudsen number, the acoustic waves are damped by the viscous boundary layer, and consequently the strong convergence is justified.

Asymptotic behavior of solutions to perturbed superstable wave equations

The paper deals with initial-boundary value problems for the linear wave equation whose solutions stabilize to zero in a finite time. We prove that problems in this class remain exponentially stable in L2 as well as in C2 under small bounded perturbations of the wave operator. To show this for C2, we prove a smoothing result implying that the solutions to the perturbed problems become eventually C2-smooth for any H1 × L2-initial data.

Space-time asymptotics of the two dimensional Navier–Stokes flow in the whole plane

We consider the space-time behavior of the two dimensional Navier–Stokes flow. Introducing some qualitative structure of initial data, we succeed to derive the first order asymptotic expansion of the Navier–Stokes flow without moment condition on initial data in L1(R2)∩Lσ2(R2). Moreover, we characterize the necessary and sufficient condition for the rapid energy decay ‖u(t)‖2=o(t−1) as t→∞ motivated by Miyakawa–Schonbek [21]. By weighted estimated in Hardy spaces, we discuss the possibility of the second order asymptotic expansion of the Navier–Stokes flow assuming the first order moment condition on initial data. Moreover, observing that the Navier–Stokes flow u(t) lies in the Hardy space H1(R2) for t>0, we consider the asymptotic expansions in terms of Hardy-norm. Finally we consider the rapid time decay ‖u(t)‖2=o(t−32) as t→∞ with cyclic symmetry introduced by Brandolese [2].