Uniqueness Of Strong Solutions Research Articles

Stability, uniqueness and existence of solutions to McKean–Vlasov SDEs: a multidimensional Yamada–Watanabe approach

We establish stability and pathwise uniqueness of solutions to Wiener noise driven McKean–Vlasov equations with random non-Lipschitz continuous coefficients. In the deterministic case, we also obtain the existence of unique strong solutions. By using our approach, which is based on an extension of the Yamada–Watanabe ansatz to the multidimensional setting and which does not rely on the construction of Lyapunov functions, we prove first moment and pathwise exponential stability. Furthermore, Lyapunov exponents are computed explicitly.

The BKM criterion to the 3D double-diffusive magneto convection systems involving planar components

In this paper, we investigate the BKM type blowup criterion applied to 3D double-diffusive magneto convection systems. Specifically, we demonstrate that a unique local strong solution does not experience blow-up at time T, given that ). To prove this, we employ the logarithmic Sobolev inequality in the Besov spaces with negative indices and a well-known commutator estimate established by Kato and Ponce. This result is the further improvement and extension of the previous works by O (2021) and Wu (2023).

Stability, Uniqueness and Existence of Solutions to McKean–Vlasov Stochastic Differential Equations in Arbitrary Moments

We deduce stability and pathwise uniqueness for a McKean–Vlasov equation with random coefficients and a multidimensional Brownian motion as driver. Our analysis focuses on a non-Lipschitz continuous drift and includes moment estimates for random Itô processes that are of independent interest. For deterministic coefficients, we provide unique strong solutions even if the drift fails to be of affine growth. The theory that we develop rests on Itô’s formula and leads to pth moment and pathwise exponential stability for p≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p\\ge 2$$\\end{document} with explicit Lyapunov exponents.

Pathwise uniqueness for singular stochastic Volterra equations with Hölder coefficients

AbstractPathwise uniqueness is established for a class of one-dimensional stochastic Volterra equations driven by Brownian motion with singular kernels and Hölder continuous diffusion coefficients. Consequently, the existence of unique strong solutions is obtained for this class of stochastic Volterra equations.

Quasi-stationary distribution for continuous-state branching processes with competition

We study quasi-stationary distribution of the continuous-state branching process with competition introduced by Berestycki et al. (2018). This process is defined as the unique strong solution to a stochastic integral equation with jumps. An important example is the logistic branching process proposed by Lambert (2005). We establish the strong Feller property, trajectory Feller property, Lyapunov condition, weak Feller property and irreducibility, respectively. These properties together allow us to prove that if the competition is strong enough near +∞, then there is a unique quasi-stationary distribution, which attracts all initial distributions with exponential rates.

Local existence of compressible magnetohydrodynamic equations without initial compatibility conditions

In this paper, we study the initial‐boundary value problem of three‐dimensional viscous, compressible, and heat conductive magnetohydrodynamic equations. Local existence and uniqueness of strong solutions are established with any such initial data that the initial compatibility conditions do not be required. The analysis is based on some suitable prior estimates for the strong coupling term and strong nonlinear term . Our proof of the existence and uniqueness of solutions is in the Lagrangian coordinates first and then transformed back to the Euler coordinates.

Global well-posedness for 2D non-resistive MHD equations in half-space

This paper focuses on the initial boundary value problem of two-dimensional non-resistive MHD equations in a half space. We prove that the non-resistive MHD equations have a unique global strong solution around the equilibrium state (0, e1) for Dirichlet boundary condition of velocity and modified Neumann boundary condition of magnetic.

Global well‐posedness of the 3D hydrostatic magnetohydrodynamics equations

In this paper, the three‐dimensional hydrostatic magnetohydrodynamic (HMHD) equations are considered on a thin domain. We showed the global existence and uniqueness (regularity) of strong solutions to the three‐dimensional incompressible HMHD equations without any small assumption on the initial data. More precisely, there exists a unique strong solution globally in time for any given ‐smooth initial data.

Strong solutions to McKean–Vlasov SDEs with coefficients of Nemytskii type: the time-dependent case

We consider a large class of nonlinear FPKEs with coefficients of Nemytskii type depending explicitly on time and space, for which it is known that there exists a sufficiently Sobolev-regular Schwartz-distributional solution u∈L1∩L∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$u\\in L^1\\cap L^\\infty $$\\end{document}. We show that there exists a unique strong solution to the associated McKean–Vlasov SDE with time marginal law densities u. In particular, every weak solution of this equation with time marginal law densities u can be written as a functional of the driving Brownian motion. Moreover, plugging any Brownian motion into this very functional produces a weak solution with time marginal law densities u.

Global existence and exponential decay of the 2D density‐dependent Bénard system with vacuum in bounded domain

In this paper, an initial boundary value problem of the 2D incompressible Bénard system is concerned. The global existence of a unique strong solution is established by the energy method and a key logarithmic interpolation inequality. In particular, the exponential decay rates for the gradients of velocity and temperature field are obtained.

Vanishing capillarity–viscosity limit of the incompressible Navier–Stokes–Korteweg equations with slip boundary condition

In this paper, we investigate the vanishing capillarity–viscosity limit of the incompressible Navier–Stokes–Korteweg (NSK) equations in a three-dimensional horizontally periodic strip domain, in which the velocity of the fluid is supplemented with slip boundary condition and the gradient of density with Dirichlet boundary condition on the boundary. We prove that there exists an positive constant T0 independent on the capillarity and viscosity coefficients, such that the incompressible NSK equations have a unique strong solution on [0,T0] and the solution is uniformly bounded in H3. Based on the uniform estimates, we further give the convergence rate in H1 from the solutions of the incompressible NSK equations to the solution of the inhomogeneous incompressible Euler equations as the capillarity and viscosity coefficients go to zero simultaneously.

Uniform regularity and zero capillarity-viscosity limit for an inhomogeneous incompressible fluid model of Korteweg type in half-space

In this paper, we study the uniform regularity and zero capillarity-viscosity limit for an inhomogeneous incompressible fluid model of Korteweg type in the half-space R+3 . We consider the Navier-slip boundary condition for velocity and the Dirichlet boundary condition for the gradient of density. By establishing the conormal energy estimates, we prove that there exists a unique strong solution of the model in a finite time interval [0,T0] , where T 0 is independent of the capillary and viscosity coefficients, and the solution is uniformly bounded in a conormal Sobolev space. Based on the aforementioned uniform estimates, we further show that there exists a constant 0<T1⩽T0 , such that the solutions of this model converge to the solution of the inhomogeneous incompressible Euler equations with the rates of convergence in L∞(0,T1;L2(R+3)) and L∞(0,T1;H1(R+3)) , as the capillary and viscosity coefficients tend to zero simultaneously.

Global well-posedness to the 3D incompressible magneto-micropolar Bénard system with damping and zero thermal conductivity

We are concerned with the three-dimensional (3D) Cauchy problem of the incompressible magneto-micropolar Bénard system with a nonlinear damping term α|u|β−1u(α>0andβ≥1) in the momentum equations. By energy method, we derive a unique global strong solution for such a model when β≥4. Our result extends previous related works.

On the existence and uniqueness of solution to a stochastic Chemotaxis–Navier–Stokes model

In this article, we study a mathematical system modelling the dynamic of the collective behaviour of oxygen-driven swimming bacteria in an aquatic fluid flowing in a two dimensional bounded domain under stochastic perturbation. This model can be seen as a stochastic version of Chemotaxis–Navier–Stokes model. We prove the existence of a unique (probabilistic) strong solution. In addition, we establish some properties of the strong solution. More precisely, we prove that the unique solution is positive and satisfies the mass conservation property and an energy inequality.

Weighted-minimax programming subject to the RTA max-product fuzzy relation inequality system and its optimal strong solution

Recently the max-product fuzzy relation inequalities have been applied to describe the wireless communication basic-station system. In this paper we further define the so-called random-term-absent (RTA) max-product fuzzy relation inequalities, which enable the system to bear a random communication line fault. Moreover, in order to make all basic-stations take part in the communication activity and avoid waste of resources, we investigate the strong solution to the RTA system. In general case, the strong solution set is composed by a unique maximum strong solution and a finite number of minimal strong solutions. Based on the practical application consideration, we propose the corresponding weighted minimax optimization problem. Detailed algorithm is designed for obtaining the optimal strong solution of such an optimization problem, with an illustrative example.

Global strong solutions to full compressible magnetohydrodynamic system with vacuum in 3D exterior domains

This paper investigates the full compressible magnetohydrodynamic system in three-dimensional exterior domains. For the initial-boundary-value problem of this system with slip boundary condition for the velocity, adiabatic one for the temperature, and perfect one for the magnetic field, we establish the global existence and uniqueness of strong solutions, under the condition that the initial data are of small energy but possibly large oscillations, where the initial density and temperature are both allowed to vanish. Moreover, the large-time behavior of the strong solutions is also shown.

LOG-NORMAL STOCHASTIC VOLATILITY MODEL WITH QUADRATIC DRIFT

In this paper, we introduce the log-normal stochastic volatility (SV) model with a quadratic drift to allow arbitrage-free valuation of options on assets under money-market account and inverse martingale measures. We show that the proposed volatility process has a unique strong solution, despite non-Lipschitz quadratic drift, and we establish the corresponding Feynman–Kac partial differential equation (PDE) for computation of conditional expectations under this SV model. We derive conditions for arbitrage-free valuations when return–volatility correlation is positive to preclude the “loss of martingality”, which occurs in many traditional SV models. Importantly, we develop an analytic approach to compute an affine expansion for the moment generating function of the log-price, its quadratic variance (QV) and the instantaneous volatility. Our solution allows for semi-analytic valuation of vanilla options under log-normal SV models closing a gap in existing studies. We apply our approach for solving the joint valuation problem of vanilla and inverse options, which are popular in the cryptocurrency option markets. We demonstrate the accuracy of our solution for valuation of vanilla and inverse options. a By calibrating the model to time series of options on Bitcoin over the past four years, we show that the log-normal SV model can work efficiently in different market regimes. Our model can be well applied for modeling of implied volatilities of assets with positive return–volatility correlation.

Exponential stability and stabilization of a stochastic 2D nonlocal Cahn-Hilliard-Navier-Stokes equations with multiplicative noise

In this article, we study the asymptotic stability of the unique strong solution to a stochastic version of a diffuse interface model which describes the motion of an incompressible isothermal mixture of two immiscible fluids. The model consists of the 2D Navier-Stokes equations coupled with a convective nonlocal Cahn-Hilliard equation. We establish the existence, uniqueness and regularity results for the stationary solution of the stochastic 2D nonlocal Cahn-Hilliard Navier-Stokes equations. Furthermore, we prove that under some conditions on the forcing terms, the strong solution converges exponentially in the mean square and almost surely exponentially to the stationary solution. Finally, we also prove a result related to the stabilization of the stochastic 2D nonlocal Cahn-Hilliard-Navier-Stokes equations.

Three-dimensional stochastic Navier–Stokes equations with Markov switching

A finite-state Markov chain is introduced in the noise terms of the three-dimensional stochastic Navier–Stokes equations in order to allow for transitions between two types of multiplicative noises. We call such systems as stochastic Navier–Stokes equations with Markov switching. To solve such a system, a family of regularized stochastic systems is introduced. For each such regularized system, the existence of a unique strong solution (in the sense of stochastic analysis) is established by the method of martingale problems and pathwise uniqueness. The regularization is removed in the limit by obtaining a weakly convergent sequence from the family of regularized solutions, and identifying the limit as a solution of the three-dimensional stochastic Navier–Stokes equation with Markov switching.

A hyperbolic-elliptic-parabolic PDE model of chemotactic E. coli colonies

This paper studies the hyperbolic-elliptic-parabolic system{ut=−∇⋅(u∇c)+g(u)nu−b(n)u1+θ,(x,t)∈Ω×R+,0=Δc+αu−βc,(x,t)∈Ω×R+,nt=Δn−γg(u)nu,(x,t)∈Ω×R+,∂c∂ν=∂n∂ν=0,(x,t)∈∂Ω×R+,u(x,0)=u0(x),n(x,0)=n0(x),x∈Ω in a bounded domain Ω⊂RN(N≥1) with smooth boundary, where g(s) and b(s) are nonnegative smooth functions, α,β,γ>0 and θ∈[0,∞). We prove that if θ>1 or θ=1 with lims→+∞⁡b(s)=b0>α, the system admits a unique global strong solution, while if θ∈[0,1), the system exhibits a finite-time explosion phenomenon in the sense of L∞ norm for some large ‖u0‖Lp(Ω) with p>N. Here the convexity of the domain Ω is not required.