L2 Initial Data Research Articles

On the Burgers–Poisson equation

In this paper, we prove the existence and uniqueness of weak entropy solutions to the Burgers–Poisson equation for initial data in L1(R). In addition an Oleinik type estimate is established and some criteria on local smoothness and wave breaking for weak entropy solutions are provided.

Are the incompressible 3d Navier–Stokes equations locally ill-posed in the natural energy space?

An important open problem in the theory of the Navier–Stokes equations is the uniqueness of the Leray–Hopf weak solutions with L2 initial data. In this paper we give sufficient conditions for non-uniqueness in terms of spectral properties of a natural linear operator associated to scale-invariant solutions recently constructed in [8]. If the spectral conditions are satisfied, non-uniqueness and ill-posedness can appear for quite benign compactly supported data, just at the borderline of applicability of the classical perturbation theory. The verification of the spectral conditions seems to be approachable by numerical simulations which involve only smooth functions.

Global existence and uniqueness for the inhomogeneous 1-Laplace evolution equation

In this paper we introduce a new approach to the Dirichlet problem for the total variation flow in a bounded domain and analyze the associated inhomogeneous problem. We prove global existence and uniqueness for source data belonging to \({L^{1}_{loc}(0,+ \infty; L^2(\Omega))}\) and L2-initial data. We compare solutions corresponding to different data as well as study the long-term behaviour of the solutions. We also show explicit examples of radial solutions.

Estimates on fractional higher derivatives of weak solutions for the Navier–Stokes equations

We study weak solutions of the 3D Navier–Stokes equations with L2 initial data. We prove that ∇αu is locally integrable in space–time for any real α such that 1<α<3. Up to now, only the second derivative ∇2u was known to be locally integrable by standard parabolic regularization. We also present sharp estimates of those quantities in weak-Lloc4/(α+1). These estimates depend only on the L2-norm of the initial data and on the domain of integration. Moreover, they are valid even for α⩾3 as long as u is smooth. The proof uses a standard approximation of Navier–Stokes from Leray and blow-up techniques. The local study is based on De Giorgi techniques with a new pressure decomposition. To handle the non-locality of fractional Laplacians, Hardy space and Maximal functions are introduced.

On the existence and weak stability of solutions to the compressible semigeostrophic equations

In this article, existence and weak stability of solutions to the compressible semigeostrophic equations are studied both in physical and in dual spaces. We present a proof of the existence of a weak solution to these equations in dual space assuming initial data in L1(R3), extending previous results (Cullen and Maroofi (2003) [8]). An extension to the concept of Lagrangian solutions due to Cullen and Feldman (2006) [6] is presented to the compressible semigeostrophic system, and it is shown the existence of Lagrangian solutions in physical space, with initial potential vorticities in L1, as well a weak stability result with respect to perturbations of the potential vorticity in L1.

Weak solutions for the Navier-Stokes equations with $B^{-1(\ln )}_{\infty \infty }+B_{X_r}^{-1+r,\frac{2}{1-r}}+L^2$B∞∞−1(ln)+BXr−1+r,21−r+L2 initial data

One of the major topics in the study of nonlinear partial differential equations of the evolutionary type is to look for as large as possible initial value spaces so that as many as possible solutions of such equations can be obtained. In the book “Recent Developments in the Navier-Stokes Problems,” Lemarié-Rieusset proved that the Navier-Stokes equations have global weak solutions for initial data in the space \documentclass[12pt]{minimal}\begin{document}$B_{\widetilde{X}_r}^{-1+r,\frac{2}{1-r}} (\mathbb {R}^N)+L^2(\mathbb {R}^N)$\end{document}BX̃r−1+r,21−r(RN)+L2(RN) (0 &amp;lt; r &amp;lt; 1), where Xr is the space of functions whose pointwise products with Hr functions belong to L2, \documentclass[12pt]{minimal}\begin{document}$\widetilde{X}_r$\end{document}X̃r denotes the closure of \documentclass[12pt]{minimal}\begin{document}$C_0^\infty (\mathbb {R}^N)$\end{document}C0∞(RN) in Xr, and \documentclass[12pt]{minimal}\begin{document}$B_{\widetilde{X}_r}^{-1+r,\frac{2}{1-r}}(\mathbb {R}^N)$\end{document}BX̃r−1+r,21−r(RN) is the Besov space over \documentclass[12pt]{minimal}\begin{document}$\widetilde{X}_r$\end{document}X̃r. In this paper we partially extend this result of Lemarié-Rieusset to the larger initial value space \documentclass[12pt]{minimal}\begin{document}$B^{-1(\ln )}_{\infty \infty } (\mathbb {R}^N)+B_{\widetilde{X}_r}^{-1+r,\frac{2}{1-r}}(\mathbb {R}^N)+ L^2(\mathbb {R}^N)$\end{document}B∞∞−1(ln)(RN)+BX̃r−1+r,21−r(RN)+L2(RN) (0 &amp;lt; r &amp;lt; 1), where \documentclass[12pt]{minimal}\begin{document}$B^{-1(\ln )}_{\infty \infty }(\mathbb {R}^N)$\end{document}B∞∞−1(ln)(RN) is a logarithmically modified version of the usual Besov space \documentclass[12pt]{minimal}\begin{document}$B^{-1}_{\infty \infty } (\mathbb {R}^N)$\end{document}B∞∞−1(RN).

Pointwise asymptotic behavior of modulated periodic reaction–diffusion waves

By working with the periodic resolvent kernel and the Bloch-decomposition, we establish pointwise bounds for the Green function of the linearized equation associated with spatially periodic traveling waves of a system of reaction–diffusion equations. With our linearized estimates together with a nonlinear iteration scheme developed by Johnson–Zumbrun, we obtain Lp-behavior (p⩾1) of a nonlinear solution to a perturbation equation of a reaction–diffusion equation with respect to initial data in L1∩H2 recovering and slightly sharpening results obtained by Schneider using weighted energy and renormalization techniques. We obtain also pointwise nonlinear estimates with respect to two different initial perturbations |u0|⩽E0e−|x|2/M, |u0|H2⩽E0 and |u0|⩽E0(1+|x|)−r, r>2, |u0|H2⩽E0 respectively, E0>0 sufficiently small and M>1 sufficiently large, showing that behavior is that of a heat kernel. These pointwise bounds have not been obtained elsewhere, and do not appear to be accessible by previous techniques.

Random attractors for stochastic porous media equations perturbed by space–time linear multiplicative noise

Unique existence of solutions to porous media equations driven by continuous linear multiplicative space–time rough signals is proven for initial data in L1(O). The generation of a continuous, order-preserving random dynamical system (RDS) on L1(O) and the existence of a “small” random attractor for stochastic porous media equations perturbed by linear multiplicative noise in space and time is obtained. Uniform L∞ bounds and uniform space–time continuity of solutions is shown. General noise including fractional Brownian Motion for all Hurst parameters is contained. A pathwise Wong–Zakai result for driving noise given by a continuous semimartingale is obtained.

Uniqueness of weak solutions to the Ginzburg-Landau model for superconductivity

We prove the uniqueness for weak solutions of the time-dependent 2-D Ginzburg-Landau model for superconductivity with L 2 initial data in the case of Coulomb gauge. This question was left open in Tang and Wang (Physica D, 88:139–166, 1995). We also prove the uniqueness of the 3-D radially symmetric solution in bounded annular domain with the choice of Lorentz gauge and L 2 initial data.

Weak-Strong Uniqueness for Measure-Valued Solutions

We prove the weak-strong uniqueness for measure-valued solutions of the incompressible Euler equations. These were introduced by DiPerna and Majda in their landmark paper (Commun Math Phys 108(4):667–689, 1987), where in particular global existence to any L2 initial data was proven. Whether measure-valued solutions agree with classical solutions if the latter exist has apparently remained open.

Global well-posedness for an advection–diffusion equation arising in magneto-geostrophic dynamics

We use De Giorgi techniques to prove Hölder continuity of weak solutions to a class of drift-diffusion equations, with L2 initial data and divergence free drift velocity that lies in Lt∞BMOx−1. We apply this result to prove global regularity for a family of active scalar equations which includes the advection–diffusion equation that has been proposed by Moffatt in the context of magnetostrophic turbulence in the Earthʼs fluid core.

THE MASS-CRITICAL FOURTH-ORDER SCHRÖDINGER EQUATION IN HIGH DIMENSIONS

We prove global wellposedness and scattering for the Mass-critical homogeneous fourth-order Schrödinger equation in high dimensions n ≥ 5, for general L2initial data in the defocusing case, and for general initial data with Mass less than certain fraction of the Mass of the ground state in the focusing case.

A strongly degenerate quasilinear equation: the elliptic case

We prove existence and uniqueness of entropy solutions for the Neumann problem for the quasilinear elliptic equation u − div a(u, Du) = v, where v ∈ L1, a(z, ξ) = ∇ξ f (z, ξ), and f is a convex function of ξ with linear growth as ‖ξ‖ → ∞, satisfying other additional assumptions. In particular, this class includes the case where f (z, ξ) = φ(z)ψ(ξ), φ > 0, ψ being a convex function with linear growth as ‖ξ‖ → ∞. In the second part of this work, using Crandall-Ligget’s iteration scheme, this result will permit us to prove existence and uniqueness of entropy solutions for the corresponding parabolic problem with initial data in L1. Mathematics Subject Classification (2000): 35J60 (primary); 47H06, 47H20 (secondary).

Well-posedness of an inverse problem of Navier–Stokes equations with the final overdetermination

This paper proves the existence, uniqueness and stability of solutions of an inverse problem for the 2-D Navier–Stokes equations with the final overdetermination with L2 initial data and sufficiently large viscosity.

Quantum diffusion of the random Schrödinger evolution in the scaling limit

We consider random Schrödinger equations on Rd for d ≽ 3 with a homogeneous Anderson–Poisson type random potential. Denote by λ the coupling constant and $\psi_t$ the solution with initial data $\psi_0$. The space and time variables scale as $ x\sim\lambda ^{{ - 2 - \varkappa/2}} {\text{ and }}t\sim\lambda ^{{ - 2 - \varkappa}} {\text{ with }}0 < \varkappa < \varkappa_{0} {\left( d \right)} $. We prove that, in the limit λ → 0, the expectation of the Wigner distribution of $\psi_t$ converges weakly to the solution of a heat equation in the space variable x for arbitrary L2 initial data. The proof is based on analyzing the phase cancellations of multiple scatterings on the random potential by expanding the propagator into a sum of Feynman graphs. In this paper we consider the non-recollision graphs and prove that the amplitude of the non-ladder diagrams is smaller than their “naive size” by an extra λc factor per non-(anti)ladder vertex for some c > 0. This is the first rigorous result showing that the improvement over the naive estimates on the Feynman graphs grows as a power of the small parameter with the exponent depending linearly on the number of vertices. This estimate allows us to prove the convergence of the perturbation series.

Quantum Diffusion for the Anderson Model in the Scaling Limit

We consider random Schrodinger equations on \({\mathbb{Z}}^d\) for d ≥ 3 with identically distributed random potential. Denote by λ the coupling constant and ψt the solution with initial data ψ0. The space and time variables scale as \(x \sim \lambda^{-2-\kappa/2},t \sim \lambda^{-2-\kappa}\) with 0 < κ < κ0(d). We prove that, in the limit λ → 0, the expectation of the Wigner distribution of ψt converges weakly to a solution of a heat equation in the space variable x for arbitrary L2 initial data. The diffusion coefficient is uniquely determined by the kinetic energy associated to the momentum υ.

Stability and error analysis for spectral Galerkin method for the Navier–Stokes equations with L2 initial data

AbstractIn this article we consider the spectral Galerkin method with the implicit/explicit Euler scheme for the two‐dimensional Navier–Stokes equations with the L2 initial data. Due to the poor smoothness of the solution on [0,1), we use the the spectral Galerkin method based on high‐dimensional spectral space HM and small time step Δt2 on this interval. While on [1,∞), we use the spectral Galerkin method based on low‐dimensional spectral space Hm(m = O(M1/2)) and large time step Δt. For the spectral Galerkin method, we provide the standard H2‐stability and the L2‐error analysis. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2007

Quantum Diffusion of the Random Schrödinger Evolution in the Scaling Limit II. The Recollision Diagrams

We consider random Schrodinger equations on \({\mathbb{R}^{d}}\) for d≥ 3 with a homogeneous Anderson-Poisson type random potential. Denote by λ the coupling constant and ψt the solution with initial data ψ0. The space and time variables scale as \({x\sim \lambda^{-2 -\kappa/2}, t \sim \lambda^{-2 -\kappa}}\) with 0 < κ < κ0(d). We prove that, in the limit λ → 0, the expectation of the Wigner distribution of ψt converges weakly to the solution of a heat equation in the space variable x for arbitrary L2 initial data. The proof is based on a rigorous analysis of Feynman diagrams. In the companion paper [10] the analysis of the non-repetition diagrams was presented. In this paper we complete the proof by estimating the recollision diagrams and showing that the main terms, i.e. the ladder diagrams with renormalized propagator, converge to the heat equation.

Global Existence of Solutions for the Cauchy Problem of the Kawahara Equation with L 2 Initial Data

In this paper we study solvability of the Cauchy problem of the Kawahara equation \( \partial _{t} u + au\partial _{x} u + \beta \partial ^{3}_{x} u + \gamma \partial ^{5}_{x} u = 0 \) with L2 initial data. By working on the Bourgain space Xr,s(R2) associated with this equation, we prove that the Cauchy problem of the Kawahara equation is locally solvable if initial data belong to Hr(R) and −1 < r ≤ 0. This result combined with the energy conservation law of the Kawahara equation yields that global solutions exist if initial data belong to L2(R).

EXISTENCE RESULTS FOR L1 DATA OF SOME QUASI-LINEAR PARABOLIC PROBLEMS WITH A QUADRATIC GRADIENT TERM AND SOURCE

In this paper we deal with a Cauchy–Dirichlet quasilinear parabolic problem containing a gradient lower order term; namely, ut - Δu + |u|2 γ-2u |∇u|2 = |u|p-2u. We prove that if p ≥ 1, γ ≥ ½ and p &lt; 2 γ + 2, then there exists a global weak solution for all initial data in L1 (Ω). We also see that there exists a non-negative solution if the initial datum is non-negative.