Strong Uniqueness Research Articles

On a Theorem by A.S. Cherny for Semilinear Stochastic Partial Differential Equations

AbstractWe consider analytically weak solutions to semilinear stochastic partial differential equations with non-anticipating coefficients driven by a cylindrical Brownian motion. The solutions are allowed to take values in Banach spaces. We show that weak uniqueness is equivalent to weak joint uniqueness, and thereby generalize a theorem by A.S. Cherny to an infinite dimensional setting. Our proof for the technical key step is different from Cherny’s and uses cylindrical martingale problems. As an application, we deduce a dual version of the Yamada–Watanabe theorem, i.e. we show that strong existence and weak uniqueness imply weak existence and strong uniqueness.

On Davie’s uniqueness for some degenerate SDEs

We consider singular SDEs like \begin{equation} \label{ss} dX_t = b(t, X_t) dt + A X_t dt + \sigma(t) d{L}_t , \;\; t \in [0,T], \;\; X_0 =x \in {\mathbb R}^n, \end{equation} where $A$ is a real $n \times n $ matrix, i.e., $A \in {\mathbb R}^n \otimes {\mathbb R}^n$, $b$ is bounded and Holder continuous, $\sigma : [0,\infty) \to {\mathbb R}^n \otimes {\mathbb R}^d $ is a locally bounded function and $L= ({L}_t)$ is an ${\mathbb R}^d$-valued Levy process, $1 \le d \le n$. We show that strong existence and uniqueness together with $L^p$-Lipschitz dependence on the initial condition $x $ imply Davie's uniqueness or path by path uniqueness. This extends a result of [E. Priola, AIHP, 2018] proved when $n=d$, $A=0$ and $\sigma(t) \equiv I $. We apply the result to some singular degenerate SDEs associated to the kinetic transport operator $ \frac{1}{2} \triangle_v f + $ ${v \cdot \partial_{x}f} $ $+F(x,v)\cdot \partial_{v}f $ when $n =2d $ and $L$ is an ${\mathbb R}^d$-valued Wiener process. For such equations strong existence and uniqueness are known under Holder type conditions on $b$. We show that in addition also Davie's uniqueness holds.

Existence and uniqueness theorems for solutions of McKean–Vlasov stochastic equations

New weak and strong existence and weak and strong uniqueness results for the solutions of multi-dimensional stochastic McKean–Vlasov equation are established under relaxed regularity conditions. Weak existence requires a non-degeneracy of diffusion and no more than a linear growth of both coefficients in the state variable. Weak and strong uniqueness are established under the restricted assumption of diffusion, yet without any regularity of the drift; this part is based on the analysis of the total variation metric.

Existence and weak–strong uniqueness of solutions to the Cahn–Hilliard–Navier–Stokes–Darcy system in superposed free flow and porous media

We study a diffuse interface model for two-phase flows of similar densities in superposed free flow and porous media. The model consists of the Navier–Stokes–Cahn–Hilliard system in free flow and the Darcy–Cahn–Hilliard system in porous media coupled through a set of domain interface boundary conditions. These domain interface boundary conditions include the nonlinear Lions interface condition and the linear Beavers–Joseph–Saffman–Jones interface condition. We establish global existence of weak solutions in three dimension. We also show that the strong solution if exists agrees with the weak solutions.

Measure-Valued Solutions and Weak–Strong Uniqueness for the Incompressible Inviscid Fluid–Rigid Body Interaction

We consider a coupled system of partial and ordinary differential equations describing the interaction between an incompressible inviscid fluid and a rigid body moving freely inside the fluid. We prove the existence of measure-valued solutions which is generated by the vanishing viscosity limit of incompressible fluid–rigid body interaction system under some physically constitutive relations. Moreover, we show that the measure-valued solution coincides with strong solution on the interval of its existence. This relies on the weak–strong uniqueness analysis. This is the first result of an existence of measure-valued solution and weak–strong uniqueness in measure-valued sense in the case of inviscid fluid–structure interaction.

Navier–Stokes–Fourier System with General Boundary Conditions

We consider the Navier--Stokes--Fourier system in a bounded domain $\Omega \subset R^d$, $d=2,3$, with physically realistic in/out flow boundary conditions. We develop a new concept of weak solutions satisfying a general form of relative energy inequality. The weak solutions exist globally in time for any finite energy initial data and comply with the weak--strong uniqueness principle.

Практико-ориентированная модель формирования абнотивности студентов колледжа искусств

The article discusses a current issue of college of arts students' readiness for the professional work with gifted children. The article outlines a range of problems of gifted students in the field of music education, including difficulties in adapting to traditional teaching methods, emotional vulnerability, difficulties in communication and mutual understanding of the child with peers, and other peculiarities of the psyche. The author highlights the concept of "abnotivity" as a necessary quality of the teacher’s personality, motivated to work with gifted children, able to arrange special conditions for this and achieve productive results: to educate the personality of a musician with strong subjectivity, creativity, and uniqueness. The structure of the teacher’s abnotivity is defined. It contains motivational, cognitive, and reflexive components, as well as such complex psychological formations as empathy, social intelligence, actual creativity, and supra-situational activity. The advantages of various practice-oriented learning models used in colleges are revealed. The author presents a practice-oriented model of forming abnotivity in students of the college of arts. A brief description of the proposed model components (target, diagnostic, procedural and performance-evaluation blocks), the features of its implementation in Saratov Regional College of Arts, and the results obtained during the experimental study are given.

Stochastic partial differential equations with gradient driven by space-time fractional noises

We establish a class of stochastic partial differential equations (SPDEs) driven by space-time fractional noises, where we suppose that the drfit term contains a gradient and satisfies certain non-Lipschitz condition. We prove the strong existence and uniqueness and joint Holder continuity of the solution to the SPDEs.

Dissipative Solutions to Compressible Navier–Stokes Equations with General Inflow–Outflow Data: Existence, Stability and Weak Strong Uniqueness

So far existence of dissipative weak solutions for the compressible Navier–Stokes equations (i.e. weak solutions satisfying the relative energy inequality) is known only in the case of boundary conditions with non zero inflow/outflow (i.e., in particular, when the normal component of the velocity on the boundary of the flow domain is equal to zero). Most of physical applications (as flows in wind tunnels, pipes, reactors of jet engines) requires to consider non-zero inflow–outflow boundary condtions. We prove existence of dissipative weak solutions to the compressible Navier–Stokes equations in barotropic regime (adiabatic coefficient \(\gamma >3/2\), in three dimensions, \(\gamma >1\) in two dimensions) with large velocity prescribed at the boundary and large density prescribed at the inflow boundary of a bounded piecewise regular Lipschitz domain, without any restriction neither on the shape of the inflow/outflow boundaries nor on the shape of the domain. It is well known that the relative energy inequality has many applications, e.g., to investigation of incompressible or inviscid limits, to the dimension reduction of flows, to the error estimates of numerical schemes. In this paper we deal with one of its basic applications, namely weak–strong uniqueness principle.

Uniqueness properties of the KAM curve

<p style='text-indent:20px;'>Classical KAM theory guarantees the existence of a positive measure set of invariant tori for sufficiently smooth non-degenerate near-integrable systems. When seen as a function of the frequency this invariant collection of tori is called the KAM curve of the system. Restricted to analytic regularity, we obtain strong uniqueness properties for these objects. In particular, we prove that KAM curves completely characterize the underlying systems. We also show some of the dynamical implications on systems whose KAM curves share certain common features.</p>

Revisiting a Cutting-Plane Method for Perfect Matchings

In 2016, Chandrasekaran, V\'egh, and Vempala published a method to solve the minimum-cost perfect matching problem on an arbitrary graph by solving a strictly polynomial number of linear programs. However, their method requires a strong uniqueness condition, which they imposed by using perturbations of the form $c(i)=c_0(i)+2^{-i}$. On large graphs (roughly $m>100$), these perturbations lead to cost values that exceed the precision of floating-point formats used by typical linear programming solvers for numerical calculations. We demonstrate, by a sequence of counterexamples, that perturbations are required for the algorithm to work, motivating our formulation of a general method that arrives at the same solution to the problem as Chandrasekaran et al. but overcomes the limitations described above by solving multiple linear programs without using perturbations. We then give an explicit algorithm that exploits are method, and show that this new algorithm still runs in strongly polynomial time.

Constants of strong uniqueness of minimal projections onto some n-dimensional subspaces of [formula omitted

In this paper, we find strong uniqueness constants for a certain class of operators with a unit norm from a space of dimension 2n onto its subspace of codimension n, which is formed by using hyperplanes in l∞2n(n≥2).

Weak solutions and invariant measures of stochastic Oldroyd-B type model driven by jump noise

In this work, we consider sub-critical and critical models for viscoelastic flows driven by pure jump Lévy noise. Due to the elastic property, the noise in the equation for the stress tensor is considered in the Marcus canonical form. We investigate existence of a weak martingale solution for stochastic Oldroyd-B models, with full dissipation in whole of Rd,d=2,3. The key ingredients of the proof are classical Faedo-Galerkin approximation, the compactness method and the Jakubowski version of the Skorokhod Theorem for nonmetric spaces. Pathwise uniqueness, existence of a strong solution and uniqueness in law for the two-dimensional model are also shown. We also prove, in a Poincaré domain in two-dimensions, existence of an invariant measure using bw-Feller property of the associated Markov semigroup.

Generalized solutions to models of compressible viscous fluids

<p style='text-indent:20px;'>We propose a new approach to models of general compressible viscous fluids based on the concept of dissipative solutions. These are weak solutions satisfying the underlying equations modulo a defect measure. A dissipative solution coincides with the strong solution as long as the latter exists (weak–strong uniqueness) and they solve the problem in the classical sense as soon as they are smooth (compatibility). We consider general models of compressible viscous fluids with non–linear viscosity tensor and non–homogeneous boundary conditions, for which the problem of existence of global–in–time weak/strong solutions is largely open.

Parallel iterative stabilized finite element algorithms for the Navier–Stokes equations with nonlinear slip boundary conditions

AbstractBased on full domain partition technique, some parallel iterative pressure projection stabilized finite element algorithms for the Navier–Stokes equations with nonlinear slip boundary conditions are designed and analyzed. In these algorithms, the lowest equal‐order P1 − P1 elements are used for finite element discretization and a local pressure projection stabilized method is used to counteract the invalidness of the discrete inf‐sup condition. Each subproblem is solved on a global composite mesh with the vast majority of the degrees of freedom associated with the particular subdomain that it is responsible for and hence can be solved in parallel with other subproblems by using an existing sequential solver without extensive recoding. All of the subproblems are nonlinear and are independently solved by three kinds of iterative methods. We estimate the optimal error bounds of the approximate solutions with the use of some (strong) uniqueness conditions. Numerical results are also given to demonstrate the effectiveness of the parallel algorithms.

Strong backward uniqueness for sublinear parabolic equations

In this paper, we establish strong backward uniqueness for solutions to sublinear parabolic equations of the type (1.1). The proof of our main result Theorem 1.3 is achieved by means of a new Carleman estimate and a Weiss type monotonicity formula that are tailored for such parabolic sublinear operators.

Parallel iterative stabilized finite element methods based on the quadratic equal-order elements for incompressible flows

Combining the quadratic equal-order stabilized method with the approach of local and parallel finite element computations and classical iterative methods for the discretization of the steady-state Navier–Stokes equations, three parallel iterative stabilized finite element methods based on fully overlapping domain decomposition are proposed and compared in this paper. In these methods, each processor independently computes an approximate solution in its own subdomain using a global composite mesh that is fine around its own subdomain and coarse elsewhere, making the methods be easy to implement based on existing codes and have low communication complexity. Under some (strong) uniqueness conditions, stability and convergence theory of the parallel iterative stabilized methods are derived. Numerical tests are also performed to demonstrate the stability, convergence orders and high efficiency of the proposed methods.

Inverting the Markovian projection, with an application to local stochastic volatility models

We study two-dimensional stochastic differential equations (SDEs) of McKean–Vlasov type in which the conditional distribution of the second component of the solution given the first enters the equation for the first component of the solution. Such SDEs arise when one tries to invert the Markovian projection developed in (Probab. Theory Related Fields 71 (1986) 501–516), typically to produce an Itô process with the fixed-time marginal distributions of a given one-dimensional diffusion but richer dynamical features. We prove the strong existence of stationary solutions for these SDEs as well as their strong uniqueness in an important special case. Variants of the SDEs discussed in this paper enjoy frequent application in the calibration of local stochastic volatility models in finance, despite the very limited theoretical understanding.

The relative entropy method for inhomogeneous systems of balance laws

General hyperbolic systems of balance laws with inhomogeneity in space and time in all constitutive functions are studied in the context of relative entropy. A framework is developed in this setting that contributes to a measure-valued weak vs. strong uniqueness theorem, a stability theorem of viscous solutions and a convergence theorem as the viscosity parameter tends to zero. The main goal of this paper is to develop hypotheses under which the relative entropy framework can still be applied. Examples of systems with inhomogeneity that have different characteristics are presented and the hypotheses are discussed in the setting of each example.

Strong uniqueness results for first-order planar equations

We discuss strong uniqueness results in the forward Cauchy Problem for a class of complex fist-order equations with Hölder coefficients defined in the plane. We use an appropriate variant of the similarity principle in order to reduce the original question to a local version of Riesz's uniqueness theorem for holomorphic functions.