Gradient estimates below the duality exponent

We show sharp local a priori estimates and regularity results for possibly degenerate non-linear elliptic problems, with data not lying in the natural dual space. We provide a precise non-linear potential theoretic analog of classical potential theory results due to Adams (Duke Math J 42:765–778, 1975) and Adams and Lewis (Studia Math 74:169–182, 1982), concerning Morrey spaces imbedding/regularity properties. For this we introduce a technique allowing for a “non-local representation” of solutions via Riesz potentials, in turn yielding optimal local estimates simultaneously in both rearrangement and non-rearrangement invariant function spaces. In fact we also derive sharp estimates in Lorentz spaces, covering borderline cases which remained open for some while.     

Well-posedness for the fractional Landau–Lifshitz equation without Gilbert damping

The main purpose is to consider the well-posedness of the fractional Landau– Lifshitz equation without Gilbert damping. The local existence of classical solutions is obtained by combining Kato’s method and vanishing viscosity method, by carefully choosing the working space. Since this equation is strongly degenerate and nonlocal and no regularizing effect is available, it is a challenging problem to extend this smooth solution to global. Instead, we give some regularity criteria to show that the solution is global if some additional regularity is assumed, which seems minimal in the sense of dimensional analysis. Finally, we introduce the commutator and show the global existence of weak solutions by vanishing viscosity method.     

A nonconvex dissipative system and its applications (I)

In order to study the uniformly translating solution of some non-linear evolution equations such as the complex Ginzburg–Landau equation, this paper presents a qualitative analysis to a Duffing–van der Pol non-linear oscillator. Monotonic property of the bounded exact solution is established based on the construction of a convex domain. Under certain parametric choices, one first integral to the Duffing–van der Pol non-linear system is obtained by using the Lie symmetry analysis, which constitutes one of the bases for further work of obtaining uniformly translating solutions of the complex Ginzburg–Landau equation. Dedicated to Professor G. Strang on the occasion of his 70th birthday     

Logarithmic Decay of Hyperbolic Equations with Arbitrary Small Boundary Damping

This paper addresses an analysis on the longtime behavior of the hyperbolic equations with a partially boundary damping, under sharp regularity assumptions on the coefficients appeared in the equation. Based on a global Carleman estimate, we establish an estimate on the underlying resolvent operator of the equation, via which we show the logarithmic decay rate for solutions of the hyperbolic equations without any geometric assumption on the subboundary in which the damping is effective.     

Kirchhoff equations with strong damping

We consider Kirchhoff equations with strong damping, namely with a friction term which depends on a power of the “elastic” operator. We address local and global existence of solutions in two different regimes depending on the exponent in the friction term. When the exponent is greater than 1/2, the dissipation prevails, and we obtain global existence in the energy space, assuming only degenerate hyperbolicity and continuity of the nonlinear term. When the exponent is less than 1/2, we assume strict hyperbolicity and we consider a phase space depending on the continuity modulus of the nonlinear term and on the exponent in the damping. In this phase space, we prove local existence and global existence if initial data are small enough. The regularity we assume both on initial data and on the nonlinear term is weaker than in the classical results for Kirchhoff equations with standard damping. Proofs exploit some recent sharp results for the linearized equation and suitably defined interpolation spaces.     

On a phase field model of Cahn–Hilliard type for tumour growth with mechanical effects

Mechanical effects have mostly been neglected so far in phase field tumour models that are based on a Cahn–Hilliard approach. In this paper we study a macroscopic mechanical model for tumour growth in which cell–cell adhesion effects are taken into account with the help of a Ginzburg–Landau type energy. In the overall model an equation of Cahn–Hilliard type is coupled to the system of linear elasticity and a reaction–diffusion equation for a nutrient concentration. The highly non-linear coupling between a fourth-order Cahn–Hilliard equation and the quasi-static elasticity system lead to new challenges which cannot be dealt within a gradient flow setting which was the method of choice for other elastic Cahn–Hilliard systems. We show existence, uniqueness and regularity results. In addition, several continuous dependence results with respect to different topologies are shown. Some of these results give uniqueness for weak solutions and other results will be helpful for optimal control problems.     

Finite-Element Methods for a Strongly Damped Wave Equation

Error estimates of optimal order are proved for semidiscreteand completely discrete finite-element methods for a linearwave equation with strong damping, arising in viscoelastic theory.It is demonstrated that the exact solution may be interpretedin terms of an analytic semigroup, and as a result that, althoughthe solution has essentially the spatial regularity of its initialdata, it is infinitely differentiable in time for t>0. Theestimates for the spatially discrete method are derived by energyarguments. Rational approximation of analytic semigroups isdiscussed in a general setting, by means of spectral representation,and the results are used to analyse the completely discreteschemes. Both smooth (and compatible) and less smooth data areconsidered.     

Newton’s method for generalized equations: a sequential implicit function theorem

In an extension of Newton’s method to generalized equations, we carry further the implicit function theorem paradigm and place it in the framework of a mapping acting from the parameter and the starting point to the set of all associated sequences of Newton’s iterates as elements of a sequence space. An inverse function version of this result shows that the strong regularity of the mapping associated with the Newton sequences is equivalent to the strong regularity of the generalized equation mapping.     

Landau Damping in Finite Regularity for Unconfined Systems with Screened Interactions

We prove Landau damping for the collisionless Vlasov equation with a class of L¹ interaction potentials (including the physical case of screened Coulomb interactions) on for localized disturbances of an infinite, homogeneous background. Unlike the confined case , results are obtained for initial data in Sobolev spaces (as well as Gevrey and analytic classes). For spatial frequencies bounded away from 0, the Landau damping of the density is similar to the confined case. The finite regularity is possible due to an additional dispersive mechanism available on that reduces the strength of the plasma echo resonance.© 2017 Wiley Periodicals, Inc.     

一个三阶牛顿变形方法

基于反函数建立的积分方程,结合Simpson公式,给出了一个非线性方程求根的新方法,即为牛顿变形方法.证明了它至少三次收敛到单根,与牛顿法相比,提高了收敛阶和效率指数.文末给出数值试验,且与牛顿法和同类型牛顿变形法做了比较.结果表明方法具有较好的优越性,它丰富了非线性方程求根的方法.     

Propagation d'ondes en milieu ferromagnétique 1D : existence et unicité des solutions

In order to model the electromagnetic behaviour of ferromagnetic materials, we consider Maxwell's equations together with a non-linear vectorial differential equation called Landau—Lifschitz—Gilbert's law. Using the theory of semi-linear equations, we show that the associated Cauchy problem admits a unique global weak solution in the monodimensional case.     

Longtime behavior of the hyperbolic equations with an arbitrary internal damping

This paper is devoted to a study of the longtime behavior of the hyperbolic equations with an arbitrary internal damping, under sharp regularity assumptions that both the principal part coefficients and the boundary of the space domain (in which the system evolves) are continuously differentiable. For this purpose, we derive a new point-wise inequality for second differential operators with symmetric coefficients. Then, based on a global Carleman estimate, we establish an estimate on the underlying resolvent operator of the equation, via which, we show the logarithmic decay rate for solutions of the hyperbolic equations.     

Longtime behavior of the hyperbolic equations with an arbitrary internal damping

This paper is devoted to a study of the longtime behavior of the hyperbolic equations with an arbitrary internal damping, under sharp regularity assumptions that both the principal part coefficients and the boundary of the space domain (in which the system evolves) are continuously differentiable. For this purpose, we derive a new point-wise inequality for second differential operators with symmetric coefficients. Then, based on a global Carleman estimate, we establish an estimate on the underlying resolvent operator of the equation, via which, we show the logarithmic decay rate for solutions of the hyperbolic equations.     

A singular control approach to highly damped second-order abstract equations and applications

In this paper we restudy, by a radically different approach, the optimal quadratic cost problem for an abstract dynamics, which models a special class of second-order partial differential equations subject to high internal damping and acted upon by boundary control. A theory for this problem was recently derived in [LLP] and [T1] (see also [T2]) by a change of variable method and by a direct approach, respectively. Unlike [LLP] and [T1], the approach of the present paper is based on singular control theory, combined with regularity theory of the optimal pair from [T1]. This way, not only do we rederive the basic control-theoretic results of [LLP] and [T1]—the (first) synthesis of the optimal pair, and the (first) nonstandard algebraic Riccati equation for the (unique) Riccati operator which enters into the gain operator of the synthesis—but in addition, this method also yields new results—a second form of the synthesis of the optimal pair, and a second (still nonstandard) algebraic Riccati equation for the (still unique) Riccati operator of the synthesis. These results, which show new pathologies in the solution of the problem, are new even in the finite-dimensional case. This research was made possible by NATO Collaborative Research Grant SA.5-2-05 (CRG.940161) 274/94/JARC-501, whose support is gratefully acknowledged. The research of I. Lasiecka and R. Triggiani was supported also by the National Science Foundation under Grant NSF-DMS-92-04338. The research of L. Pandolfi was written with the programs of GNAFA-CNR. The main results of the present paper were announced in [LPT].     

C ∞ regularity of solutions of an equation of Levi’s type in R 2 n +1

We study the regularity of the solutions u of a class of P.D.E., whose prototype is the prescribed Levi curvature equation in ℝ^{2 n +1}. It is a second-order quasilinear equation whose characteristic matrix is positive semidefinite and has vanishing determinant at every point and for every function u∈C ². If the Levi curvature never vanishes, we represent the operator ℒ associated with the Levi equation as a sum of squares of non-linear vector fields which are linearly independent at every point. By using a freezing method we first study the regularity properties of the solutions of a linear operator, which has the same structure as ℒ. Then we apply these results to the classical solutions of the equation, and prove their C ^∞ regularity. Received: October 10, 1998; in final form: March 5, 1999?Published online: May 10, 2001     

Martingale weak solutions of the stochastic Landau–Lifshitz–Bloch equation

The present paper studies the stochastic Landau–Lifshitz–Bloch equation which is recommended as the only valid model at temperature around the Curie temperature and is especially important for the simulation of heat-assisted magnetic recording. We study the stochastic Landau–Lifshitz–Bloch equation in the case that the temperature is raised higher than the Curie temperature. The global existence of martingale weak solutions is proved by using a new argument and regularity properties of the weak solutions are discussed.     

Karush-Kuhn-Tucker systems: regularity conditions,error bounds and a class of Newton-type methods

 We consider optimality systems of Karush-Kuhn-Tucker (KKT) type, which arise, for example, as primal-dual conditions characterizing solutions of optimization problems or variational inequalities. In particular, we discuss error bounds and Newton-type methods for such systems. An exhaustive comparison of various regularity conditions which arise in this context is given. We obtain a new error bound under an assumption which we show to be strictly weaker than assumptions previously used for KKT systems, such as quasi-regularity or semistability (equivalently, the R ₀-property). Error bounds are useful, among other things, for identifying active constraints and developing efficient local algorithms. We propose a family of local Newton-type algorithms. This family contains some known active-set Newton methods, as well as some new methods. Regularity conditions required for local superlinear convergence compare favorably with convergence conditions of nonsmooth Newton methods and sequential quadratic programming methods. Received: December 10, 2001 / Accepted: July 28, 2002 Published online: February 14, 2003 Key words. KKT system – regularity – error bound – active constraints – Newton method Mathematics Subject Classification (1991): 90C30, 65K05     

Ultra-analytic effect of Cauchy problem for a class of kinetic equations

The smoothing effect of the Cauchy problem for a class of kinetic equations is studied. We firstly consider the spatially homogeneous nonlinear Landau equation with Maxwellian molecules and inhomogeneous linear Fokker-Planck equation to show the ultra-analytic effects of the Cauchy problem. Those smoothing effect results are optimal and similar to heat equation. In the second part, we study a model of spatially inhomogeneous linear Landau equation with Maxwellian molecules, and show the analytic effect of the Cauchy problem.     

L p -weak regularity and asymptotic behavior of solutions for critical equations with singular potentials on Carnot groups

In this paper, we consider a large class of critical problems with singular potentials on Carnot groups. In particular, we prove sharp regularity results in Marcinkiewicz spaces and the exact rate of decay of the solutions. Our general results apply, for instance, to the equation satisfied by the extremals of some relevant Hardy–Sobolev type inequalities on Stratified groups.     

A study of Landau damping with random initial inputs

For the Vlasov–Poisson equation with random uncertain initial data, we prove that the Landau damping solution given by the deterministic counterpart (Caglioti and Maffei (1998) [3]) depends smoothly on the random variable if the time asymptotic profile does, under the smoothness and smallness assumptions similar to the deterministic case. The main idea is to generalize the deterministic contraction argument to more complicated function spaces to estimate derivatives in space, velocity and random variables. This result suggests that the random space regularity can persist in long-time even in time-reversible nonlinear kinetic equations.