Energy Initial Data Research Articles

The Landau--Lifshitz Equation of the Ferromagnetic Spin Chain and Oseen--Frank Flow

We investigate the general Oseen--Frank energy for the Landau--Lifshitz equation of the ferromagnetic spin chain from $\Bbb R^2$ to the unit sphere $S^2$. Our main results are global existence and uniqueness of weak solutions for large energy initial data. Moreover, the number of singular points for weak solutions is proved to be finite.

Limiting Motion for the Parabolic Ginzburg–Landau Equation with Infinite Energy Data

We study a class of solutions to the parabolic Ginzburg-Landau equation in dimension 2 or higher, with ill-prepared infinite energy initial data. We show that, asymptotically, vorticity evolves according to motion by mean curvature in Brakke's weak formulation. Then, we prove that in the plane, point vortices do not move in the original time scale. These results extend the work of Bethuel, Orlandi and Smets [8, 9] for infinite energy data, they allow to consider the point vortices on a lattice (in dimension 2), or filament vortices of infinite length (in dimension 3).

Local Well-Posedness of the (4 + 1)-Dimensional Maxwell–Klein–Gordon Equation at Energy Regularity

This paper is the first part of a trilogy [22, 23] dedicated to a proof of global well-posedness and scattering of the \((4+1)\)-dimensional mass-less Maxwell–Klein–Gordon equation (MKG) for any finite energy initial data. The main result of the present paper is a large energy local well-posedness theorem for MKG in the global Coulomb gauge, where the lifespan is bounded from below by the energy concentration scale of the data. Hence the proof of global well-posedness is reduced to establishing non-concentration of energy. To deal with non-local features of MKG we develop initial data excision and gluing techniques at critical regularity, which might be of independent interest.

Global well-posedness and scattering of the (4+1)-dimensional Maxwell-Klein-Gordon equation

This article constitutes the final and main part of a three-paper sequence, whose goal is to prove global well-posedness and scattering of the energy critical Maxwell-Klein-Gordon equation (MKG) on $\mathbb{R}^{1+4}$ for arbitrary finite energy initial data. Using the successively stronger continuation/scattering criteria established in the previous two papers, we carry out a blow-up analysis and deduce that the failure of global well-posedness and scattering implies the existence of a nontrivial stationary or self-similar solution to MKG. Then, by establishing that such solutions do not exist, we complete the proof.

Existence of global strong solution for Korteweg system with large infinite energy initial data

This work is devoted to the study of the initial boundary value problem for a general isothermal model of capillary fluids derived by J.E. Dunn and J. Serrin (1985) (see [18]), which can be used as a phase transition model. We aim at proving the existence of local and global (under a condition of smallness on the initial data) strong solutions with initial density ln⁡ρ0 belonging to the Besov space B2,∞N2. It implies in particular that some classes of discontinuous initial density generate strong solutions. The proof relies on the fact that the density can be written as the sum of the solution ρL of the associated linear system and a remainder term ρ¯; this last term is more regular than ρL provided that we have regularizing effects induced on the bilinear convection term. The main difficulty consists in obtaining new estimates of maximum principle type for the associated linear system; this is based on a characterization of the Besov space in terms of the semi-group associated with this linear system. We show in particular the existence of global strong solution for small initial data in (B˜2,∞N2−1,N2∩L∞)×B2,∞N2−1; it allows us to exhibit a family of large energy initial data when N=2 providing global strong solution. In conclusion we introduce the notion of quasi-solutions for the Korteweg's system (a tool which has been developed in the framework of the compressible Navier–Stokes equations [31,30,32,26,27]) which enables to obtain the existence of global strong solution with a smallness condition which is subcritical. Indeed we can deal with large initial velocity in B2,1N2−1. As a corollary, we get global strong solution for highly compressible Korteweg system when N≥2. It means that for any large initial data (under an irrotational condition on the initial velocity) we have the existence of global strong solution provided that the Mach number is sufficiently large.

Maximal dissipation in Hunter–Saxton equation for bounded energy initial data

In [13] it was conjectured by Zhang and Zheng that dissipative solutions of the Hunter–Saxton equation, which are known to be unique in the class of weak solutions, dissipate the energy at the highest possible rate. The conjecture of Zhang and Zheng was proven in [4] by Dafermos for monotone increasing initial data with bounded energy. In this note we prove the conjecture in [13] in full generality. To this end we examine the evolution of the energy of any weak solution of the Hunter–Saxton equation. Our proof shows in fact that for every time t>0 the energy of the dissipative solution is not greater than the energy of any weak solution with the same initial data.

On weak solutions to the 2D Savage–Hutter model of the motion of a gravity-driven avalanche flow

ABSTRACTWe consider the Savage–Hutter system consisting of two-dimensional depth-integrated shallow water equations for the incompressible fluid with the Coulomb-type friction term. Using the method of convex integration we show that the associated initial value problem possesses infinitely many weak solutions for any finite energy initial data. On the other hand, the problem enjoys the weak-strong uniqueness property provided the system of equations is supplemented with the energy inequality.

On weak solutions to a diffuse interface model of a binary mixture of compressible fluids

We consider the Euler-Cahn-Hilliard systemproposed by Lowengrub and Truskinovsky describingthe motion of a binary mixture of compressible fluids. We show that the associated initial-valueproblem possesses infinitely many global-in-time weak solutions for any finite energy initial data. A modification of the method of convex integration is used to prove the result.

Profile decompositions for wave equations on hyperbolic space with applications

The goal for this paper is twofold. Our first main objective is to develop Bahouri–Gerard type profile decompositions for waves on hyperbolic space. Recently, such profile decompositions have proved to be a versatile tool in the study of the asymptotic dynamics of solutions to nonlinear wave equations with large energy. With an eye towards further applications, we develop this theory in a fairly general framework, which includes the case of waves on hyperbolic space perturbed by a time-independent potential. Our second objective is to use the profile decomposition to address a specific nonlinear problem, namely the question of global well-posedness and scattering for the defocusing, energy critical, semi-linear wave equation on three-dimensional hyperbolic space, possibly perturbed by a repulsive time-independent potential. Using the concentration compactness/rigidity method introduced by Kenig and Merle, we prove that all finite energy initial data lead to a global evolution that scatters to linear waves as $$t \rightarrow \pm \infty $$ . This proof serves as a blueprint for the arguments in the work [45] by the authors, where we study the asymptotic behavior of large energy equivariant wave maps on the hyperbolic plane.

Blow‐up phenomena for a class of fourth‐order nonlinear wave equations with a viscous damping term

This paper deals with the blow‐up phenomena for a class of fourth‐order nonlinear wave equations with a viscous damping term urn:x-wiley:mma:media:mma3623:mma3623-math-0046 with Ω = (0,1) and α > 0. Here, f(s) is a given nonlinear smooth function.For 0 < α < p – 1, we prove that the blow‐up occurs in finite time for arbitrary positive initial energy and suitable initial data. This result extends the recent results obtained by Xu et al. (Applicable Analysis)(2013) and Chen and Lu (J. Math. Anal. Appl.)(2009). Copyright © 2015 John Wiley & Sons, Ltd.

Local Equilibrium of the Carleman Equation

UDC 517.9 We consider the Cauchy problem for the one-dimensional Carleman equation with bounded energy and periodic initial data and obtain the local equilibrium conditions. We prove exponential stabilization to the equilibrium state. Bibliography :1 0titles. Illustrations :3 figures. Dedicated to N. N. Uraltseva

Going Beyond the Threshold: Scattering and Blow-up in the Focusing NLS Equation

We study the focusing nonlinear Schr\"odinger equation in the $L^2$-supercritical regime with finite energy and finite variance initial data. We investigate solutions above the energy (or mass-energy) threshold. In our first result, we extend the known scattering versus blow-up dichotomy above that threshold for finite variance solutions in the energy-subcritical and energy-critical regimes, obtaining scattering and blow-up criteria for solutions with arbitrarily large mass and energy. As a consequence, we characterize the behavior of the ground state initial data modulated by a quadratic phase. Our second result gives two blow up criteria, which are also applicable in the energy-supercritical NLS setting. We finish with various examples illustrating our results.

A New Continuation Criterion for the Relativistic Vlasov–Maxwell System

The global existence of solutions to the relativistic Vlasov-Maxwell system given sufficiently regular finite energy initial data is a longstanding open problem. The main result of Glassey-Strauss (1986) shows that a solution $(f, E, B)$ remains $C^1$ as long as the momentum support of $f$ remains bounded. Alternate proofs were later given by Bouchut-Golse-Pallard (2003) and Klainerman-Staffilani (2002). We show that only the boundedness of the momentum support of $f$ after projecting to any two dimensional plane is needed for $(f, E, B)$ to remain $C^1$.

On the existence of global solutions for a nonlinear Klein-Gordon equation

The aim of this work is to study the global existence of solutions for the Cauchy problem of a Klein-Gordon equation with high energy initial data. The proof relies on constructing a new functional, which includes both the initial displacement and the initial velocity: with sign preserving property of the new functional we show the existence of global weak solutions.

The Cauchy problem of a focusing energy-critical nonlinear Schrödinger equation

In this paper, we combine variational methods and harmonic analysis to discuss the Cauchy problem of a focusing nonlinear Schrödinger equation. We study the global well-posedness, finite time blowup and asymptotic behavior of this problem. By Hamiltonian property, we establish two types of invariant evolution flows. Then from one flow and the stability of classical energy-critical nonlinear Schrödinger equation, we find that the solution exists globally and scattering occurs. Finally, we get a precise blowup criterion of this problem for positive energy initial data via the other flow.

Equivariant wave maps exterior to a ball

We consider the exterior Cauchy–Dirichlet problem for equivariant wave maps from 3 + 1 dimensional Minkowski spacetime into the three sphere. Using mixed analytical and numerical methods we show that, for a given topological degree of the map, all solutions starting from smooth finite energy initial data converge to the unique static solution (harmonic map). The asymptotics of this relaxation process is described in detail. We hope that our model will provide an attractive mathematical setting for gaining insight into dissipation-by-dispersion phenomena, in particular the soliton resolution conjecture.

Saddle-point dynamics of a Yang–Mills field on the exterior Schwarzschild spacetime

We consider the Cauchy problem for a spherically symmetric SU(2) Yang–Mills field propagating outside the Schwarzschild black hole. Although solutions starting from smooth finite energy initial data remain smooth for all times, not all of them scatter since there are non-generic solutions which asymptotically tend towards unstable static solutions. We show that a static solution with one unstable mode appears as an intermediate attractor in the evolution of initial data near a border between basins of attraction of two different vacuum states. We study the saddle-point dynamics near this attractor; in particular, we identify the universal phases of evolution: the ringdown approach, the exponential departure and the eventual decay to one of the vacuum states.

Blowup and global existence of the nonlinear Schrödinger equations with multiple potentials

This paper is concerned with the Cauchy problem for the nonlinearSchrödinger equations with multiple potentials. Under someassumptions on these potentials, we first obtain some sufficientconditions of blowup according to the initial energy and theCauchy initial data directly. We next establish a sufficientcondition of global existence by using potential well method andthe relation between the Cauchy data and the ground state. Wefinally answer the question that how small the Cauchy initial dataneed to be for global existence by using scaling argument.

Global infinite energy solutions of the critical semilinear wave equation

We consider the critical semilinear wave equation (NLW)_{2^*-1} \;\;\; \left\{ \begin{aligned} \square u + |u|^{2^*-2} u & = 0 \\ u_{|t=0} & = u_0 \\ \partial_t u_{|t=0} & = u_1 \, \,, \end{aligned} \right. set in \mathbb{R}^d , d \geq 3 , with 2^* = \frac{2d}{d-2} \,\cdotp Shatah and Struwe [Shatah, J. and Struwe, M.: Geometric wave equations. Courant Lecture Notes in Mathematics 2. New York University, Courant Institute of Mathematical Sciences. American Mathematical Society, RI, 1998] proved that, for finite energy initial data (ie if (u_0,u_1) \in \dot{H}^1 \times L^2 ), there exists a global solution such that (u,\partial_t u)\in \mathcal{C}(\mathbb{R},\dot{H}^1 \times L^2) . Planchon [Planchon, F.: Self-similar solutions and semi-linear wave equations in Besov spaces. J. Math. Pures Appl. (9) 79 (2000), no. 8, 809-820] showed that there also exists a global solution for certain infinite energy initial data, namely, if the norm of (u_0,u_1) in \dot{B}^1_{2,\infty} \times \dot{B}^0_{2,\infty} is small enough. In this article, we build up global solutions of (NLW)_{2^*-1} for arbitrarily big initial data of infinite energy, by using two methods which enable to interpolate between finite and infinite energy initial data: the method of Calderón, and the method of Bourgain. These two methods give complementary results.

Global solutions and finite time blow up for damped semilinear wave equations

A class of damped wave equations with superlinear source term is considered. It is shown that every global solution is uniformly bounded in the natural phase space. Global existence of solutions with initial data in the potential well is obtained. Finally, not only finite time blow up for solutions starting in the unstable set is proved, but also high energy initial data for which the solution blows up are constructed.