Strong Uniqueness Research Articles

On weak–strong uniqueness for the compressible Navier–Stokes system with non-monotone pressure law

We show the weak–strong uniqueness property for the compressible Navier–Stokes system with general non-monotone pressure law. A weak solution coincides with the strong solution emanating from the same initial data as long as the latter solution exists.

Two-dimensional nonlocal Cahn–Hilliard–Navier–Stokes systems with variable viscosity, degenerate mobility and singular potential

We consider a nonlinear system which consists of the incompressible Navier–Stokes equations coupled with a convective nonlocal Cahn–Hilliard equation. This is a diffuse interface model which describes the motion of an incompressible isothermal mixture of two (partially) immiscible fluids having the same density. We assume that both the viscosity and mobility functions depend smoothly on the order parameter. Moreover, we assume that the mobility degenerates at the pure phases and that the potential is singular (e.g. of logarithmic type). This system is endowed with a no-slip boundary condition for the (average) velocity and a homogeneous Neumann boundary condition for the chemical potential. Thus the total mass is conserved. In the two-dimensional case, this problem was already analyzed in some joint papers of the first three authors. However, in the present general case, only the existence of a global weak solution, the (conditional) weak–strong uniqueness and the existence of the global attractor were proven. Here we are able to establish the existence of a (unique) strong solution through an approximation procedure based on time discretization. As a consequence, we can prove suitable uniform estimates which allow us to show some smoothness of the global attractor. Finally, we discuss the existence of strong solutions for the convective nonlocal Cahn–Hilliard equation, with a given velocity field, in the three dimensional case as well.

Shift toward Emerging Engineering and Application-Oriented Education <br/>—Taking Architecture Education in Huizhou University as an Example

Under the circumstances of shift toward emerging engineering and application-oriented education, the Undergraduate Architecture Program in Huizhou University has been exploring how to fit into the future-oriented, region-oriented and industries-oriented teaching. We strive for a breakthrough in teaching reform by innovation, integration, collaboration and practice to cultivate talents in architecture education. Since 2012, we have been committed to teaching approach exploration with a strong characteristic uniqueness, innovation and application. It proves to be a success that our teaching approach has nurtured graduates with uniqueness, innovation and application.

Optimal relaxed control of stochastic hereditary evolution equations with Lévy noise

Existence theory of optimal relaxed control problem for a class of stochastic hereditary evolution equations driven by Lévy noise has been studied. We formulate the problem in the martingale sense of Stroock and Varadhan to establish existence of optimal controls. The construction of the solution is based on the classical Faedo–Galerkin approximation, the compactness method and the Jakubowski version of the Skorokhod theorem for nonmetric spaces, and certain compactness properties of the class of Young measures on Suslin metrizable control sets. As application of the abstract theory, Oldroyd and Jeffreys fluids have been studied and existence of optimal relaxed control is established. Existence and uniqueness of a strong solution and uniqueness in law for the two-dimensional Oldroyd and Jeffreys fluids are also shown.

Measure-valued solutions for the equations of polyconvex adiabatic thermoelasticity

For the system of polyconvex adiabatic thermoelasticity, we define a notion of dissipative measure-valued solution, which can be considered as the limit of a viscosity approximation. We embed the system into a symmetrizable hyperbolic one in order to derive the relative entropy. Exploiting the weak-stability properties of the transport and stretching identities, we base our analysis in the original variables, instead of the symmetric ones (in which the entropy is convex) and we prove measure-valued weak versus strong uniqueness using the averaged relative entropy inequality.

Stochastic ODEs and stochastic linear PDEs with critical drift: regularity, duality and uniqueness

In this paper linear stochastic transport and continuity equations with drift in critical $L^{p}$ spaces are considered. In this situation noise prevents shocks for the transport equation and singularities in the density for the continuity equation, starting from smooth initial conditions. Specifically, we first prove a result of Sobolev regularity of solutions, which is false for the corresponding deterministic equation. The technique needed to reach the critical case is new and based on parabolic equations satisfied by moments of first derivatives of the solution, opposite to previous works based on stochastic flows. The approach extends to higher order derivatives under more regularity of the drift term. By a duality approach, these regularity results are then applied to prove uniqueness of weak solutions to linear stochastic continuity and transport equations and certain well-posedness results for the associated stochastic differential equation (sDE) (roughly speaking, existence and uniqueness of flows and their $C^\alpha$ regularity, strong uniqueness for the sDE when the initial datum has diffuse law). Finally, we show two types of examples: on the one hand, we present well-posed sDEs, when the corresponding ODEs are ill-posed, and on the other hand, we give a counterexample in the supercritical case.

Strong time-periodic solutions to the bidomain equations with arbitrary large forces

We prove the existence of strong time-periodic solutions to the bidomain equations with arbitrary large forces. We construct weak time-periodic solutions by a Galerkin method combined with Brouwer’s fixed point theorem and a priori estimate independent of approximation. We then show their regularity so that our solution is a strong time-periodic solution in L2 spaces. Our strategy is based on the weak–strong uniqueness method.

Simultaneously recovering potentials and embedded obstacles for anisotropic fractional Schrödinger operators

Let \begin{document}$A∈{\rm{Sym}}(n× n)$\end{document} be an elliptic 2-tensor. Consider the anisotropic fractional Schrodinger operator \begin{document}$\mathscr{L}_A^s+q$\end{document} , where \begin{document}$\mathscr{L}_A^s: = (-\nabla·(A(x)\nabla))^s$\end{document} , \begin{document}$s∈ (0, 1)$\end{document} and \begin{document}$q∈ L^∞$\end{document} . We are concerned with the simultaneous recovery of \begin{document}$q$\end{document} and possibly embedded soft or hard obstacles inside \begin{document}$q$\end{document} by the exterior Dirichlet-to-Neumann (DtN) map outside a bounded domain \begin{document}$Ω$\end{document} associated with \begin{document}$\mathscr{L}_A^s+q$\end{document} . It is shown that a single measurement can uniquely determine the embedded obstacle, independent of the surrounding potential \begin{document}$q$\end{document} . If multiple measurements are allowed, then the surrounding potential \begin{document}$q$\end{document} can also be uniquely recovered. These are surprising findings since in the local case, namely \begin{document}$s = 1$\end{document} , both the obstacle recovery by a single measurement and the simultaneous recovery of the surrounding potential by multiple measurements are long-standing problems and still remain open in the literature. Our argument for the nonlocal inverse problem is mainly based on the strong uniqueness property and Runge approximation property for anisotropic fractional Schrodinger operators.

Perturbation analysis of the Moore–Penrose metric generalized inverse with applications

In this article, based on some geometric properties of Banach spaces and one feature of the metric projection, we introduce a new class of bounded linear operators satisfying the so-called (α,β)-USU (uniformly strong uniqueness) property. This new convenient property allows us to take the study of the stability problem of the Moore–Penrose metric generalized inverse a step further. As a result, we obtain various perturbation bounds of the Moore–Penrose metric generalized inverse of the perturbed operator. They offer the advantage that we do not need the quasiadditivity assumption, and the results obtained appear to be the most general case found to date. Closely connected to the main perturbation results, one application, the error estimate for projecting a point onto a linear manifold problem, is also investigated.

Davie’s type uniqueness for a class of SDEs with jumps

Un résultat de A.M. Davie (Int. Math. Res. Not. 24 (2007) rnm124) établit qu’une équation stochastique multi-dimensionnelle $dX_{t}=b(t,X_{t})\,dt+dW_{t}$, $X_{0}=x$, dirigée par un processus de Wiener $W=(W_{t})$ avec un coefficient $b$ qui est seulement borné et mesurable admet une unique solution pour presque tout choix de la trajectoire du processus $W$ la dirigeant. Nous considérons un problème similaire lorsque $W$ est remplacé par un processus de Lévy $L=(L_{t})$ et $b$ est $\beta$-Hölder continu en espace. Nous supposons que $L_{1}$ a un moment fini d’ordre $\theta$ pour un certain $\theta>0$. En utilisant un nouveau résultat de régularité càdlàg, nous prouvons que l’existence et unicité forte pour l’EDS, associées à une $L^{p}$-Lipschitz continuité de la solution forte par rapport à $x$, impliquent une unicité de type Davie pour presque tout choix de la trajectoire de Lévy. Nous appliquons ce résultat à une classe d’EDS dirigées par un processus de Lévy $\alpha$-stable non dégénéré pour $\alpha\in(0,2)$ et $\beta>1-\alpha/2$.

Strong uniqueness polynomials of degree 6 and unique range sets for powers of meromorphic functions

We investigate the uniqueness problems for meromorphic functions with higher multiplicities of zeros and poles. As a consequence, we present a class of strong uniqueness polynomials for meromorphic functions (SUPM) of degree [Formula: see text]. We also proved that there exist the sets [Formula: see text] of seven elements such that for arbitrary two meromorphic functions [Formula: see text] and [Formula: see text], the condition [Formula: see text] with [Formula: see text] implies [Formula: see text] where [Formula: see text] is a root of unity of degree [Formula: see text]

Localised relative energy and finite speed of propagation for compressible flows

For the incompressible and the isentropic compressible Euler equations in arbitrary space dimension, we establish the principle of localised relative energy, thus generalising the well-known relative energy method. To this end, we adapt classical arguments of C. Dafermos to the Euler equations. We give several applications to the behaviour of weak solutions, like local weak–strong uniqueness, local preservation of smoothness, and finite speed of propagation for the isentropic system.

Tangents to subsolutions: existence and uniqueness, Part I

There is an interesting potential theory associated to each degenerate elliptic, fully nonlinear equation $f(D^2u) = 0$. These include all the potential theories attached to calibrated geometries. This paper begins the study of tangents to the subsolutions in these theories, a topic inspired by the results of Kiselman in the classical plurisubharmonic case. Fundamental to this study is a new invariant of the equation, called the characteristic, which governs asymptotic structures. The existence of tangents to subsolutions is established in general, as is the existence of an upper semi-continuous density function. Two theorems establishing the strong uniqueness of tangents (which means every tangent is a Riesz kernel) are proved. They cover all O(n)-invariant convex cone equations and their complex and quaternionic analogues, with the exception of the homogeneous Monge-Ampere equations, where uniqueness fails. They also cover a large class of geometrically defined subequations which includes those coming from calibrations. A discreteness result for the sets where the density is $\geq c > 0$ is also established in any case where strong uniqueness holds. A further result (which is sharp) asserts the Holder continuity of subsolutions when the Riesz characteristic p satisfies $1 \leq p < 2$. Many explicit examples are examined. The second part of this paper is devoted to the geometric cases. A Homogeneity Theorem and a Second Strong Uniqueness Theorem are proved, and the tangents in the Monge-Ampere case are completely classified.

Absolute continuity of semimartingales

We derive equivalent conditions for the (local) absolute continuity of two laws of semimartingales on random sets. Our result generalizes previous results for classical semimartingales by replacing a strong uniqueness assumption by a weaker uniqueness assumption. The main tool is a generalized Girsanov’s theorem, which relates laws of two possibly explosive semimartingales to a candidate density process. Its proof is based on an extension theorem for consistent families of probability measures. Moreover, we show that in a one-dimensional Ito-diffusion setting our result reproduces the known deterministic characterizations for (local) absolute continuity. Finally, we give a Khasminskii-type test for the absolute continuity of multidimensional Ito-diffusions and derive linear growth conditions for the martingale property of stochastic exponentials.

Relative Entropy for Hyperbolic–Parabolic Systems and Application to the Constitutive Theory of Thermoviscoelasticity

We extend the relative entropy identity to the class of hyperbolic-parabolic systems whose hyperbolic part is symmetrizable. The resulting identity, in the general theory, is useful to provide stability of viscous solutions and yields a convergence result in the zero-viscosity limit to smooth solutions in an $L^p$ framework. Also it provides measure valued weak versus strong uniqueness theorems for the hyperbolic problem. The relative entropy identity is also developed for the system of gas dynamics for viscous and heat conducting gases, and for the system of thermoviscoelasticity with viscosity and heat-conduction. Existing differences in applying the relative entropy method between the general hyperbolic-parabolic theory and the examples are underlined.

Weak regularization by stochastic drift : Result and counter example

In this paper, weak uniqueness of hypoelliptic stochastic differential equation with Holder drift is proved when the Holder exponent is strictly greater than 1/3. This result then extends to a weak framework the previous works [ 4 , 23 , 10 ], where strong uniqueness was proved when the regularity index of the drift is strictly greater than 2/3. Part of the result is also shown to be almost sharp thanks to a counter example when the Holder exponent of the degenerate component is just below 1/3. The approach is based on martingale problem formulation of Stroock and Varadhan and so on smoothing properties of the associated PDE which is, in the current setting, degenerate.

Pathwise uniqueness for stochastic differential equations driven by pure jump processes

Based on the weak existence and weak uniqueness, we study the pathwise uniqueness of the solutions for a class of one-dimensional stochastic differential equations driven by pure jump processes. By using Tanaka’s formula and the local time technique, we show that there is no gap between the strong uniqueness and weak uniqueness when the coefficients of the Poisson random measures satisfy a suitable condition.

Erratum to: On the Characterisations of a New Class of Strong Uniqueness Polynomials Generating Unique Range Sets

The purpose of the paper is to introduce a new class of strong uniqueness polynomials with the best possible answer to the question posed by the first author (Banerjee, Ann Acad Sci Fenn Math 40:465-474, 2015). We find the corresponding unique range set which improves a result due to Frank–Reinders (Complex Var Theory Appl 37(1):185–193, 1998). At the time of characterising the strong uniqueness polynomial, we also encounter the cases where the polynomial is not necessarily critically injective and find its degree and the corresponding unique range set. Furthermore, we find some connections between this new class of polynomials and most of the polynomials generating URSM introduced so far in the literature of value distribution theory. As an application of our results, we will show that the theorems improve many previous results. Lastly, we rectify the gap in the last section of An (Acta Math Vietnam 27(3), 251–256, 2002, p. 255, l. 11–14) and thus improve the result obtained by An (Acta Math Vietnam 27(3), 251–256, 2002) significantly.

Strong existence and uniqueness to a class of nonlinear SPDEs driven by Gaussian colored noises

In this paper, we establish the strong existence and uniqueness to the following stochastic partial differential equation: ut(x)=u0(x)+12∫0tΔus(x)ds+∫0t∫Rg(us(x),∇us(x))dF(s,y),where F(s,x) is a Gaussian noise on R+×R that is white in time and colored in space. This gives an affirmative answer to the open problem in Gomez et al. (2013) for certain non-Lipschitz condition of g.

Convergence of a Mixed Finite Element–Finite Volume Scheme for the Isentropic Navier–Stokes System via Dissipative Measure-Valued Solutions

We study convergence of a mixed finite element–finite volume numerical scheme for the isentropic Navier–Stokes system under the full range of the adiabatic exponent. We establish suitable stability and consistency estimates and show that the Young measure generated by numerical solutions represents a dissipative measure-valued solutions of the limit system. In particular, using the recently established weak–strong uniqueness principle in the class of dissipative measure-valued solutions we show that the numerical solutions converge strongly to a strong solutions of the limit system as long as the latter exists.