Nonlinear mobility continuity equations and generalized displacement convexity

We consider the geometry of the space of Borel measures endowed with a distance that is defined by generalizing the dynamical formulation of the Wasserstein distance to concave, nonlinear mobilities. We investigate the energy landscape of internal, potential, and interaction energies. For the internal energy, we give an explicit sufficient condition for geodesic convexity which generalizes the condition of McCann. We take an eulerian approach that does not require global information on the geodesics. As by-product, we obtain existence, stability, and contraction results for the semigroup obtained by solving the homogeneous Neumann boundary value problem for a nonlinear diffusion equation in a convex bounded domain. For the potential energy and the interaction energy, we present a nonrigorous argument indicating that they are not displacement semiconvex.     

Generalized Solutions of Nonlinear Parabolic Equations and Diffusion Processes

We reduce the construction of a weak solution of the Cauchy problem for a quasilinear parabolic equation to the construction of a solution to a stochastic problem. Namely, we construct a diffusion process that allows us to obtain a probabilistic representation of a weak (in distributional sense) solution to the Cauchy problem for a nonlinear PDE.      

Isolated periodic minima are unstable

A classical result, studied, among others, by Carathéodory [C. Carathéodory, Calculus of Variations and Partial Differential Equations of the First Order, Chelsea, New York, 1989], says that, at least generically, periodic minimizers are hyperbolic, and consequently, unstable as solutions of the associated Euler–Lagrange equation. A new version of this fact, also valid in the nonhyperbolic case, is given.     

The J-flow and stability

We study the J-flow from the point of view of an algebro-geometric stability condition. In terms of this we give a lower bound for the natural associated energy functional, and we show that the blowup behavior found by Fang and Lai [11] is reflected by the optimal destabilizer. Finally we prove a general existence result on complex tori.     

Existence of nontrivial nonnegative periodic solutions for a class of doubly degenerate parabolic equation with nonlocal terms

In this paper, the authors establish the existence of nontrivial nonnegative periodic solutions for a class of doubly degenerate parabolic equation with nonlocal terms by using the theory of Leray-Schauder's degree.     

Backward Euler Scheme,Singular Hölder Norms,and Maximal Regularity for Parabolic Difference Equations

We show finite difference analogues of maximal regularity results for discretizations of abstract linear parabolic problems. The involved spaces are discrete versions of spaces of Hölder continuous functions, which can be singular in 0. The main tools are real interpolation and Da Prato–Grisvard's theory of the sum of linear operators.     

Global-in-time Uniform Convergence for Linear Hyperbolic-Parabolic Singular Perturbations

We consider the Cauchy problem εu^″ε ＋ δu′ε ＋ Auε = 0, uε（0） = uo, u′ε（0） = ul, where ε 〉 0, δ 〉 0, H is a Hilbert space, and A is a self-adjoint linear non-negative operator on H with dense domain D（A）. We study the convergence of （uε） to the solution of the limit problem ,δu＇ ＋ Au = 0, u（0） = u0. For initial data （u0, u1） ∈ D（A1/2）× H, we prove global-in-time convergence with respect to strong topologies. Moreover, we estimate the convergence rate in the case where （u0, u1）∈ D（A3/2） ∈ D（A1/2）, and we show that this regularity requirement is sharp for our estimates. We give also an upper bound for ｜u′ε（t）｜ which does not depend on ε.     

Lateral boundary differentiability of solutions of parabolic equations in nondivergence form

The lateral boundary differentiability is shown for solutions of parabolic differential equations in nondivergence form under the assumptions that the parabolic boundary satisfies the exterior Dini condition and is punctually C¹$C^1$ differentiable one-sided in t-direction. The classical barrier technique, the maximum principle, the interior Harnack inequality and an iteration procedure are the main analytical tools.     

Carleman Estimates for Parabolic Equations with Nonhomogeneous Boundary Conditions

The authors prove a new Carleman estimate for general linear second order parabolic equation with nonhomogeneous boundary conditions. On the basis of this estimate, improved Carleman estimates for the Stokes system and for a system of parabolic equations with a penalty term are obtained. This system can be viewed as an approximation of the Stokes system.     

Γ-CONVERGENCE OF INTEGRAL FUNCTIONALS DEPENDING ON VECTOR-VALUED FUNCTIONS OVER PARABOLIC DOMAINS

51.IntroductionWebeginwiththecharacterizationofr--convergencein[1,2].Definition1.1.Let(X,T)beafirstcountabletOPologicalspaceand{F'}7=,bease-quenceOjfunctionalshemXtoR=RU{--co,co},u6X,AER.Wecallifandonlyifforeverysequence{u'}concealingtouin(X,T)andthereedestsasequence{u'}conveneingtouin(X,T)suchthatWecallA~r(T)timF"(u)ifandonlyifjoreveryah-- a(h-co)Throughoutthispaper,weassumethatfiisaboundedopensetinR".Letp>1,T>0,andmbeapositiveinteger.Denoteforavectorvaluedfunctionu.ConsiderthefUc…     

The blow-up rate for semilinear parabolic problems on general domains

We derive results on blow-up rates for parabolic equations and systems from Fujita-type theorems. We complement a previous study by allowing (possibly unbounded) domains with boundary. Received May 2000     

Fourier expansion based recursive algorithms for periodic Riccati and Lyapunov matrix differential equations

Combining Fourier series expansion with recursive matrix formulas, new reliable algorithms to compute the periodic, non-negative, definite stabilizing solutions of the periodic Riccati and Lyapunov matrix differential equations are proposed in this paper. First, periodic coefficients are expanded in terms of Fourier series to solve the time-varying periodic Riccati differential equation, and the state transition matrix of the associated Hamiltonian system is evaluated precisely with sine and cosine series. By introducing the Riccati transformation method, recursive matrix formulas are derived to solve the periodic Riccati differential equation, which is composed of four blocks of the state transition matrix. Second, two numerical sub-methods for solving Lyapunov differential equations with time-varying periodic coefficients are proposed, both based on Fourier series expansion and the recursive matrix formulas. The former algorithm is a dimension expanding method, and the latter one uses the solutions of the homogeneous periodic Riccati differential equations. Finally, the efficiency and reliability of the proposed algorithms are demonstrated by four numerical examples.