The notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems, existence theorems, and theorems about continuous dependence may now be proved by very efficient and striking arguments. The range of important applications of these results is enormous. This article is a self-contained exposition of the basic theory of viscosity solutions

https://www.ams.org/bull/1992-27-01/S0273-0979-1992-00266-5/S0273-0979-1992-00266-5.pdf

User’s guide to viscosity solutions of second order partial differential equations

This book is intended as an introduction to optimal stochastic control for continuous time Markov processes and to the theory of viscosity solutions. The authors approach stochastic control problems by the method of dynamic programming. The text provides an introduction to dynamic programming for deterministic optimal control problems, as well as to the corresponding theory of viscosity solutions. A new Chapter X gives an introduction to the role of stochastic optimal control in portfolio optimization and in pricing derivatives in incomplete markets. Chapter VI of the First Edition has been completely rewritten, to emphasize the relationships between logarithmic transformations and risk sensitivity. A new Chapter XI gives a concise introduction to two-controller, zero-sum differential games. Also covered are controlled Markov diffusions and viscosity solutions of Hamilton-Jacobi-Bellman equations. The authors have tried, through illustrative examples and selective material, to connect stochastic control theory with other mathematical areas (e.g. large deviations theory) and with applications to engineering, physics, management, and finance. In this Second Edition, new material on applications to mathematical finance has been added. Concise introductions to risk-sensitive control theory, nonlinear H-infinity control and differential games are also included.

/pdf/controlled-markov-processes-and-viscosity-solutions-4n8p6ywj5v.pdf

Controlled Markov processes and viscosity solutions

Ordinary differential equations Constant coefficient linear dispersive equations Semilinear dispersive equations The Korteweg de Vries equation Energy-critical semilinear dispersive equations Wave maps Tools from harmonic analysis Construction of ground states Bibliography.

/pdf/nonlinear-dispersive-equations-local-and-global-analysis-527whycdsg.pdf

Nonlinear dispersive equations : local and global analysis

The evolution of the state of many systems modeled by linear partial diﬁerentialequations (PDEs) or linear delay-diﬁerential equations can be described by operatorsemigroups. The state of such a system is an element in an inﬂnite-dimensionalnormed space, whence the name \inﬂnite-dimensional linear system".The study of operator semigroups is a mature area of functional analysis, which isstill very active. The study of observation and control operators for such semigroupsis relatively more recent. These operators are needed to model the interactionof a system with the surrounding world via outputs or inputs. The main topicsof interest about observation and control operators are admissibility, observability,controllability, stabilizability and detectability. Observation and control operatorsare an essential ingredient of well-posed linear systems (or more generally systemnodes). Inthisbookwedealonlywithadmissibility, observabilityandcontrollability.We deal only with operator semigroups acting on Hilbert spaces.This book is meant to be an elementary introduction into the topics mentionedabove. By \elementary" we mean that we assume no prior knowledge of ﬂnite-dimensional control theory, and no prior knowledge of operator semigroups or ofunbounded operators. We introduce everything needed from these areas. We doassume that the reader has a basic understanding of bounded operators on Hilbertspaces, diﬁerential equations, Fourier and Laplace transforms, distributions andSobolev spaces on

Observation and Control for Operator Semigroups

This book presents methods to study the controllability and the stabilization of nonlinear control systems in finite and infinite dimensions. The emphasis is put on specific phenomena due to nonlinearities. In particular, many examples are given where nonlinearities turn out to be essential to get controllability or stabilization. Various methods are presented to study the controllability or to construct stabilizing feedback laws. The power of these methods is illustrated by numerous examples coming from such areas as celestial mechanics, fluid mechanics, and quantum mechanics. The book is addressed to graduate students in mathematics or control theory, and to mathematicians or engineers with an interest in nonlinear control systems governed by ordinary or partial differential equations.

/pdf/control-and-nonlinearity-4tu8aduvuy.pdf

Control and Nonlinearity

https://math.berkeley.edu/~tataru/papers/nas.pdf

Well-posedness for the Navier–Stokes Equations

A semilinear model of the wave equation with nonlinear boundary conditions and nonlinear boundary velocity feedback is considered. Under the assumption that the velocity boundary feedback is dissipative and that the other nonlinear terms are conservative, uniform decay rates for the solutions are established.

/pdf/uniform-boundary-stabilization-of-semilinear-wave-equations-2sbyvtgfuc.pdf

Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping

We prove the existence of equivariant finite time blow-up solutions for the wave map problem from ℝ2+1→S
 2 of the form $u(t,r)=Q(\lambda(t)r)+\mathcal{R}(t,r)$
 where u is the polar angle on the sphere, $Q(r)=2\arctan r$
 is the ground state harmonic map, λ(t)=t
 -1-ν, and $\mathcal{R}(t,r)$
 is a radiative error with local energy going to zero as t→0. The number $\nu>\frac{1}{2}$
 can be prescribed arbitrarily. This is accomplished by first “renormalizing” the blow-up profile, followed by a perturbative analysis.

Renormalization and blow up for charge one equivariant critical wave maps

We prove Strichartz type estimates for the Schrodinger equation corresponding to a second order elliptic operator with variable coefficients. We assume that the coefficients are a C 2 compactly supported perturbation of the identity, satisfying a nontrapping condition.

/pdf/strichartz-estimates-for-a-schrodinger-operator-with-3tf4s2gczc.pdf

Strichartz estimates for a schrödinger operator with nonsmooth coefficients

The aim of this article is twofold. First we consider the wave equation in the hyperbolic space HI and obtain the counterparts of the Strichartz type estimates in this context. Next we examine the relationship between semilinear hyperbolic equations in the Minkowski space and in the hyperbolic space. This leads to a simple proof of the recent result of Georgiev, Lindblad and Sogge on global existence for solutions to semilinear hyperbolic problems with small data. Shifting the space-time Strichartz estimates from the hyperbolic space to the Minkowski space yields weighted Strichartz estimates in Rn x 1R which extend the ones of Georgiev, Lindblad, and Sogge.

/pdf/strichartz-estimates-in-the-hyperbolic-space-and-global-1q2ulfegek.pdf

Daniel Tataru

Papers

Well-posedness for the Navier–Stokes Equations

Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping

Renormalization and blow up for charge one equivariant critical wave maps

Strichartz estimates for a schrödinger operator with nonsmooth coefficients

Strichartz estimates in the hyperbolic space and global existence for the semilinear wave equation