Energy Type Estimates Research Articles

Weak solutions to the Cauchy problem of the time-dependent Thomas–Fermi equations

In this paper, we are concerned with the existence of weak solutions of the time-dependent Thomas–Fermi equations. We derive approximate solutions by the fractional step Lax–Friedrichs scheme and establish uniform boundedness of approximate solutions. Based on the uniform energy-type estimates, we establish that the entropy dissipation measures of the weak solution of the one-dimensional time-dependent Thomas–Fermi equations for weak entropy–entropy flux pairs, generated by compactly supported C0∞ test functions, are confined in a compact set in Hloc−1. We prove that the Young measure must be a Dirac measure by the Tartar–Murat commutator relation. The convergence of approximate solutions is established by using the compensated compactness method.

Systematic search for extreme and singular behaviour in some fundamental models of fluid mechanics.

This review article offers a survey of the research program focused on a systematic computational search for extreme and potentially singular behaviour in hydrodynamic models motivated by open questions concerning the possibility of a finite-time blow-up in the solutions of the Navier-Stokes system. Inspired by the seminal work of Lu & Doering (2008 Ind. Univ. Math. 57, 2693-2727), we sought such extreme behaviour by solving PDE optimization problems with objective functionals chosen based on certain conditional regularity results and a priori estimates available for different models. No evidence for singularity formation was found in extreme Navier-Stokes flows constructed in this manner in three dimensions. We also discuss the results obtained for one-dimensional Burgers and two-dimensional Navier-Stokes systems, and while singularities are ruled out in these flows, the results presented provide interesting insights about sharpness of different energy-type estimates known for these systems. Connections to other bounding techniques are also briefly discussed. This article is part of the theme issue 'Mathematical problems in physical fluid dynamics (part 1)'.

Scaling effects on the periodic homogenization of a reaction-diffusion-convection problem posed in homogeneous domains connected by a thin composite layer

We study the question of periodic homogenization of a variably scaled reaction-diffusion problem with non-linear drift posed for a domain crossed by a flat composite thin layer. The structure of the non-linearity in the drift was obtained in earlier works as hydrodynamic limit of a totally asymmetric simple exclusion process (TASEP) for a population of interacting particles crossing a domain with obstacle.Using energy-type estimates as well as concepts like thin-layer convergence and two-scale convergence, we derive the homogenized evolution equation and the corresponding effective model parameters for a regularized problem. Special attention is paid to the derivation of the effective transmission conditions across the separating limit interface in essentially two different situations: (i) finitely thin layer and (ii) infinitely thin layer.This study should be seen as a preliminary step needed for the investigation of averaging fast non-linear drifts across material interfaces—a topic with direct applications in the design of thin composite materials meant to be impenetrable to high-velocity impacts.

Global martingale solutions for a stochastic population cross-diffusion system

The existence of global nonnegative martingale solutions to a stochastic cross-diffusion system for an arbitrary but finite number of interacting population species is shown. The random influence of the environment is modeled by a multiplicative noise term. The diffusion matrix is generally neither symmetric nor positive definite, but it possesses a quadratic entropy structure. This structure allows us to work in a Hilbert space framework and to apply a stochastic Galerkin method. The existence proof is based on energy-type estimates, the tightness criterion of Brzeźniak and co-workers, and Jakubowski’s generalization of the Skorokhod theorem. The nonnegativity is proved by an extension of Stampacchia’s truncation method due to Chekroun, Park, and Temam.

Nonlinear Modulational Instability of Dispersive PDE Models

We prove nonlinear modulational instability for both periodic and localized perturbations of periodic traveling waves for several dispersive PDEs, including the KDV type equations (e.g. the Whitham equation, the generalized KDV equation, the Benjamin-Ono equation), the nonlinear Schr\"odinger equation and the BBM equation. First, the semigroup estimates required for the nonlinear proof are obtained by using the Hamiltonian structures of the linearized PDEs; Second, for KDV type equations the loss of derivative in the nonlinear term is overcome in two complementary cases: (1) for smooth nonlinear terms and general dispersive operators, we construct higher order approximation solutions and then use energy type estimates; (2) for nonlinear terms of low regularity, with some additional assumption on the dispersive operator, we use a bootstrap argument to overcome the loss of derivative.

Blowup solutions for a reaction–diffusion system with exponential nonlinearities

We consider the following parabolic system whose nonlinearity has no gradient structure:{∂tu=Δu+epv,∂tv=μΔv+equ,u(⋅,0)=u0,v(⋅,0)=v0,p,q,μ>0, in the whole space RN. We show the existence of a stable blowup solution and obtain a complete description of its singularity formation. The construction relies on the reduction of the problem to a finite dimensional one and a topological argument based on the index theory to conclude. In particular, our analysis uses neither the maximum principle nor the classical methods based on energy-type estimates which are not supported in this system. The stability is a consequence of the existence proof through a geometrical interpretation of the quantities of blowup parameters whose dimension is equal to the dimension of the finite dimensional problem.

On distributional solutions of local and nonlocal problems of porous medium type

We present a theory of well-posedness and a priori estimates for bounded distributional (or very weak) solutions of(0.1)∂tu−Lσ,μ[φ(u)]=g(x,t)inRN×(0,T), where φ is merely continuous and nondecreasing, and Lσ,μ is the generator of a general symmetric Lévy process. This means that Lσ,μ can have both local and nonlocal parts like, e.g., Lσ,μ=Δ−(−Δ)12. New uniqueness results for bounded distributional solutions to this problem and the corresponding elliptic equation are presented and proven. A key role is played by a new Liouville type result for Lσ,μ. Existence and a priori estimates are deduced from a numerical approximation, and energy-type estimates are also obtained.

Shock waves in conservation laws with physical viscosity

Introduction Preliminaries Green's functions for systems with constant coefficients Green's function for systems linearized along shock profiles Estimates on Green's function Estimates on crossing of initial layer Estimates on truncation error Energy type estimates Wave interaction Stability analysis Application to magnetohydrodynamics Bibliography

Energy-type estimates and global solvability in a two-dimensional chemotaxis–haptotaxis model with remodeling of non-diffusible attractant

This paper deals with the coupled chemotaxis–haptotaxis model{ut=Δu−χ∇⋅(u∇v)−ξ∇⋅(u∇w)+μu(1−u−w),x∈Ω,t>0,0=Δv+u−v,x∈Ω,t>0,wt=−vw+ηw(1−w−u),x∈Ω,t>0, which was initially proposed by Chaplain and Lolas (2006) [10] to describe the interactions between cancer cells, the matrix degrading enzyme and the host tissue in a process of cancer cell invasion of tissue (extracellular matrix). Here, Ω⊂R2 is a bounded domain with smooth boundary, and χ, ξ, μ and η are positive parameters. As compared to previous mathematical studies, the novelty here consists of allowing for positive values of η, reflecting processes of self-remodeling of the extracellular matrix.Under zero-flux boundary conditions, it is shown that for any sufficiently smooth initial data (u0,w0) satisfying first-order compatibility conditions, the model admits a unique global smooth solution. A crucial ingredient in the proof is an energy-like inequality which, given T>0, yields boundedness of u(⋅,t) in LlogL(Ω). This serves as a starting point for a bootstrap argument used to derive higher regularity estimates sufficient for global extensibility of solutions.

A Stochastic Weakly Damped, Forced KdV-BO Equation

We investigate the long time behavior of the damped, forced KdV-BO equation driven by white noise. We first show that the global solution generates a random dynamical system. By energy type estimates and dispersive properties, we then prove that this system possesses a weak random attractor in the spaceH1(ℝ).

Well-posedness and stabilization of a model system for long waves posed on a quarter plane

In this paper we are concerned with an initial--boundary-value problem for a coupled system of two KdV equations, posed on the positive half line, under the effect of a localized damping term. The model arises when modeling the propagation of long waves generated by a wave maker in a channel. It is shown that the solutions of the system are exponentially stable and globally well-posed in the weighted space $L^2(e^{2bx}\,dx)$ for $b>0$. The stabilization problem is studied using a Lyapunov approach, while the well-posedness result is obtained combining fixed-point arguments and energy-type estimates.

Solutions to the seepage face model for dual porosity flows with hysteresis

This paper is concerned with theoretical analysis of the initial-boundary value problem for the dual porosity model of Darcian flows with the Preisach hysteresis operator. Due to the presence of the so-called unilateral boundary condition, the problem comes as an evolution variational inequality. We apply the penalty method to prove the global-in-time existence of the variational solution by constructing approximate solutions satisfying suitable energy type estimates.

Shallow water equations: viscous solutions and inviscid limit

We establish the inviscid limit of the viscous shallow water equations to the Saint-Venant system. For the viscous equations, the viscosity terms are more degenerate when the shallow water is close to the bottom, in comparison with the classical Navier-Stokes equations for barotropic gases; thus, the analysis in our earlier work for the classical Navier-Stokes equations does not apply directly, which require new estimates to deal with the additional degeneracy. We first introduce a notion of entropy solutions to the viscous shallow water equations and develop an approach to establish the global existence of such solutions and their uniform energy-type estimates with respect to the viscosity coefficient. These uniform estimates yield the existence of measure-valued solutions to the Saint-Venant system generated by the viscous solutions. Based on the uniform energy-type estimates and the features of the Saint-Venant system, we further establish that the entropy dissipation measures of the viscous solutions for weak entropy-entropy flux pairs, generated by compactly supported C2 test-functions, are confined in a compact set in H−1, which yields that the measure-valued solutions are confined by the Tartar-Murat commutator relation. Then, the reduction theorem established in Chen and Perepelitsa [5] for the measure-valued solutions with unbounded support leads to the convergence of the viscous solutions to a finite-energy entropy solution of the Saint-Venant system with finite-energy initial data, which is relative with respect to the different end-states of the bottom topography of the shallow water at infinity. The analysis also applies to the inviscid limit problem for the Saint-Venant system in the presence of friction.

Global Existence for Two Dimensional Quadratic Derivative Nonlinear Schrödinger Equations

We study the global in time existence of small classical solutions to the nonlinear Schrödinger equation with quadratic interactions of derivative type in two space dimensions where λ jk ∈ C. We prove that if the initial data u 0 ∈ H 10 ∩ H 0, 10 satisfy smallness conditions in the weighted Sobolev norm, then the solution of the Cauchy problem (0.1) exists globally in time. Furthermore we prove the existence of the usual scattering states in homogeneous Sobolev space of order one. The proof depends on the energy type estimates, and smoothing property by Doi.

On the Local Existence for the Characteristic Initial Value Problem in General Relativity

Given a truncated incoming null cone and a truncated outgoing null cone intersecting at a two sphere $S$ with smooth characteristic initial data, a theorem of Rendall shows that the vacuum Einstein equations can be solved in a small neighborhood of $S$ in the future of $S$. We show that in fact the vacuum Einstein equations can be solved in a neighborhood in the future of the cones, as long as the constraint equations are initially satisfied on the null cones. The proof is based on energy type estimates and relies heavily on the null structure of the Einstein equations in the double null foliation.

Quadratic derivative nonlinear Schrödinger equations in two space dimensions

We study the global in time existence of small classical solutions to the nonlinear Schrödinger equation with quadratic interactions of derivative type in two space dimensions $$\left\{\begin{array}{l@{\quad}l}i \partial _{t} u+\frac{1}{2}\Delta u=\mathcal{N}\left( \nabla u,\nabla u\right),&t >0 ,\;x\in {\bf R}^{2},\\ u\left( 0,x\right) =u_{0} \left( x\right),&x\in {\bf R}^{2}, \end{array}\right.\quad\quad\quad\quad\quad\quad (0.1)$$ where the quadratic nonlinearity has the form $${\mathcal{N}( \nabla u,\nabla v) =\sum_{k,l=1,2}\lambda _{kl} (\partial _{k}u) ( \partial _{l}v) }$$ with $${\lambda \in \mathbf{C}}$$ . We prove that if the initial data $${u_{0}\in \mathbf{H}^{6}\cap \mathbf{H}^{3,3}}$$ satisfy smallness conditions in the weighted Sobolev norm, then the solution of the Cauchy problem (0.1) exists globally in time. Furthermore we prove the existence of the usual scattering states. The proof depends on the energy type estimates, smoothing property by Doi, and method of normal forms by Shatah.

Vanishing viscosity limit of the Navier‐Stokes equations to the euler equations for compressible fluid flow

AbstractWe establish the vanishing viscosity limit of the Navier‐Stokes equations to the isentropic Euler equations for one‐dimensional compressible fluid flow. For the Navier‐Stokes equations, there exist no natural invariant regions for the equations with the real physical viscosity term so that the uniform sup‐norm of solutions with respect to the physical viscosity coefficient may not be directly controllable. Furthermore, convex entropy‐entropy flux pairs may not produce signed entropy dissipation measures.To overcome these difficulties, we first develop uniform energy‐type estimates with respect to the viscosity coefficient for solutions of the Navier‐Stokes equations and establish the existence of measure‐valued solutions of the isentropic Euler equations generated by the Navier‐Stokes equations. Based on the uniform energy‐type estimates and the features of the isentropic Euler equations, we establish that the entropy dissipation measures of the solutions of the Navier‐Stokes equations for weak entropy‐entropy flux pairs, generated by compactly supported C2 test functions, are confined in a compact set in H−1, which leads to the existence of measure‐valued solutions that are confined by the Tartar‐Murat commutator relation.A careful characterization of the unbounded support of the measure‐valued solution confined by the commutator relation yields the reduction of the measurevalued solution to a Dirac mass, which leads to the convergence of solutions of the Navier‐Stokes equations to a finite‐energy entropy solution of the isentropic Euler equations with finite‐energy initial data, relative to the different end‐states at infinity. © 2010 Wiley Periodicals, Inc.

Oscillatory asymptotic expansion for semilinear wave equation

AbstractWe study oscillatory properties of the solution to semilinear wave equation, assuming oscillatory terms in initial data have sufficiently small amplitude. The main result gives an a priori estimate of the remainder in the approximation by means of the method of geometric optics. The method of establishing this estimate is based on a combination between energy type estimates for transport equation and Sobolev embedding. Copyright © 2001 John Wiley & Sons, Ltd.

Antiplane Shear Flows in Viscoplastic Solids Exhibiting Isotropic and Kinematic Hardening

The authors consider antiplane shearing motions of an incompressible viscoplastic solid. The particular constitutive equation employed assumes that the stress tensor has an "elastic" component and a component which can exhibit hysteresis. The model exhibits both "kinematic" and "isotropic" hardening. Our results consist of a set of energy type estimates for the resulting system, L2 contractivity estimates for the solution operator, and finally an analysis of the approach of our system to a "rate independent" model as a distinguished parameter describing our flow rule approaches zero. We also include some computational results for simple piecewise constant data.

A sharp interface limit of the phase field equations: one-dimensional and axisymmetric

We study the singular limit of the dimensionless phase-field equationsWe consider two cases: either the space dimension is 1 and then ɛ tends to zero; or the solutions are radially symmetric and then both ɛ and α tend to zero. It turns out that, in the first case, the limiting functions solve the Stefan problem with kinetic undercooling, provided the initial temperature is small compared to the surface tension and the latent heat. In the second case, the limiting functions satisfy the Stefan problem coupled with the Gibbs–Thomson law for the melting temperature. We show, in addition, that the multiplicity of the interface is always one, in a sense to be explained at the end of § 1. As main tool we use energy type estimates, and prove that the formal first-order asymptotic expansion with respect to ɛ in fact gives an approximation of the exact solution. Our results hold without smoothness assumptions on the limiting Stefan problem.