Unique Smooth Solution Research Articles

Localization of Beltrami fields: Global smooth solutions and vortex reconnection for the Navier-Stokes equations

We introduce a class of divergence-free vector fields on R3 obtained after a suitable localization of Beltrami fields. First, we use them as initial data to construct unique global smooth solutions of the three dimensional Navier-Stokes equations. The relevant fact here is that these initial data can be chosen to be large in any critical space for the Navier–Stokes problem, however they satisfy the nonlinear smallness assumption introduced in [10]. As a further application of the method, we use these vector fields to provide analytical example of vortex-reconnection for the three-dimensional Navier-Stokes equations on R3. To do so, we exploit the ideas developed in [13] but differently from this latter we cannot rely on the non-trivial homotopy of the three-dimensional torus. To overcome this obstacle we use a different topological invariant, i.e. the number of hyperbolic zeros of the vorticity field.

Sharp non-uniqueness for the 3D hyperdissipative Navier-Stokes equations: Beyond the Lions exponent

We study the 3D hyperdissipative Navier-Stokes equations on the torus, where the viscosity exponent α can be larger than the Lions exponent 5/4. It is well-known that, due to Lions [1], for any L2 divergence-free initial data, there exist unique smooth Leray-Hopf solutions when α≥5/4. We prove that even in this high dissipative regime, the uniqueness would fail in the supercritical spaces LtγWxs,p, in view of the Ladyženskaja-Prodi-Serrin criteria. The non-uniqueness is proved in the strong sense and, in particular, yields the sharpness at two endpoints (3/p+1−2α,∞,p) and (2α/γ+1−2α,γ,∞). Moreover, the constructed solutions are allowed to coincide with the unique Leray-Hopf solutions near the initial time and, more delicately, admit the partial regularity outside a fractal set of singular times with zero Hausdorff Hη⁎ measure, where η⁎>0 is any given small positive constant. These results also provide the sharp non-uniqueness in the supercritical Lebesgue and Besov spaces. Furthermore, we prove the strong vanishing viscosity result for the hyperdissipative Navier-Stokes equations.

Duality in optimal consumption–investment problems with alternative data

This study investigates an optimal consumption–investment problem in which the unobserved stock trend is modulated by a hidden Markov chain that represents different economic regimes. In the classic approach, the hidden state is estimated using historical asset prices, but recent technological advances now enable investors to consider alternative data in their decision-making. These data, such as social media commentary, expert opinions, COVID-19 pandemic data and GPS data, come from sources other than standard market data sources but are useful for predicting stock trends. We develop a novel duality theory for this problem and consider a jump-diffusion process for alternative data series. This theory helps investors identify “useful” alternative data for dynamic decision-making by providing conditions for the filter equation that enable the use of a control approach based on the dynamic programming principle. We apply our theory to provide a unique smooth solution for an agent with constant relative risk aversion once the distributions of the signals generated from alternative data satisfy a bounded likelihood ratio condition. In doing so, we obtain an explicit consumption–investment strategy that takes advantage of different types of alternative data that have not been addressed in the literature.

Well-posedness for a molecular beam epitaxy model

We study a general molecular beam epitaxy (MBE) equation modeling the epitaxial growth of thin films. We show that, in the deterministic case, the associated Cauchy problem admits a unique smooth solution for all time, given initial data in the space X0=L2(Rd)∩W˙1,4(Rd) with d=1,2. This improves a recent result by Agélas [1], who established global existence in H3(Rd). Moreover, we investigate the local existence and uniqueness of solutions in the space X0 for the stochastic MBE equation, with an additive noise that is white in time and regular in the space variable.

Long time existence of the non-isentropic slightly compressible Navier-Stokes equations with boundary conditions

We investigate the long time existence of smooth solutions to the initial boundary value problem for the non-isentropic slightly compressible Navier–Stokes equations with slip or non-slip boundary conditions on the velocity. We verify that the compressible Navier–Stokes equations with boundary conditions admit a unique smooth solution on the time interval where the smooth solution of the incompressible Navier–Stokes equations exists, when the Mach number is sufficiently small. Moreover, we obtain the uniform convergence of smooth solutions for the compressible system toward those for the corresponding incompressible system on that time interval.

Remarks on the global smooth solution of the 3D generalized magneto‐micropolar equations

This paper is concerned with the global smooth solution of the 3D generalized magneto‐micropolar equations. When the velocity dissipation is logarithmically hyperdissipative and the magnetic diffusion is fractional Laplacian, based on some new observations on the nonlinear structure for the magneto‐micropolar equations, it is examined the system has a unique global smooth solution .

Global smooth solutions for triangular reaction-cross diffusion systems

For a class of reaction cross-diffusion systems of two equations with a cross-diffusion term in the first equation and with self-diffusion terms, we prove that the unique local smooth solution given by Amann theorem is actually global. This class of systems arises in Population dynamics, and extends the triangular Shigesada-Kawasaki-Teramoto system when general power-laws growth are considered in the reaction and diffusion rates.

Global solutions of 2D isentropic compressible Navier-Stokes equations with one slow variable

Motivated by [2,3], we prove the global existence of solutions to the two-dimensional isentropic compressible Navier-Stokes equations with smooth initial data which are slowly varying in one direction and with initial density being away from vacuum. In particular, we present examples of initial data which generate unique global smooth solutions to 2D compressible Navier-Stokes equations with constant viscosity and with initial data which are neither small perturbation of some constant equilibrium state nor of small energy.

Existence and uniqueness of steady incompressible impinging flow with finite inlet

In this paper, we discuss the well‐posedness of impinging flow through a semi‐infinitely long nozzle and impacting on a plane. More precisely, given the semi‐infinitely long nozzle with finite inlet, the plane, and the positive incoming horizontal velocity, then there exists a unique smooth solution to the impinging flow. Moreover, there exists a smooth free interface that separates the fluid such that it flows out of different outlets in the downstream.

Subsonic time-periodic solution to the compressible Euler equations triggered by boundary conditions

In this paper, we consider the one-dimensional isentropic compressible Euler equations with source term β(t,x)ρ|u|αu in a bounded domain, which can be used to describe gas transmission in a nozzle. The model is imposed a subsonic time-periodic boundary condition. The main results reveal that the time-periodic boundary can trigger an unique subsonic time-periodic smooth solution and this unique periodic solution is stable under small perturbations on initial and boundary data. To get the existence of subsonic time-periodic solution, we use the linear iterative skill and transfer the boundary value problem into two initial value ones by using the hyperbolic property of the system, then the corresponding linearized system can be decoupled. The uniqueness is a direct by-product of the stability. There is no small assumptions on coefficient β(t,x).

Positive solutions for singular (p,q)-Laplacian equations with negative perturbation

We consider a nonlinear Dirichlet problem driven by the -Laplacian and with a reaction consisting of a singular term plus a negative perturbation. Using regularization of the singular term and truncation and comparison techniques, we show that the problem has a unique positive smooth solution.

Hyper-elastic Ricci flow: Gradient flow, local existence-uniqueness, and a Perelman energy functional

The equation of hyper-elastic Ricci flow amends classical Ricci flow by the addition of the Cauchy stress tensor which itself is derived from the a free energy. In this paper hyper-elastic Ricci flow is shown to possess three properties derived by G. Perelman for classical Ricci flow, specifically it is diffeomorphically equivalent to a gradient flow, unique smooth solutions exist locally in time, and the system possesses a non-decreasing energy function.

The Dirichlet problem for fully nonlinear elliptic equations on Riemannian manifolds

We study the Dirichlet problem for a broad class of fully nonlinear elliptic equations in Euclidean space as well as on general Riemannian manifolds. Under a set of fundamental structure conditions which have become standard since the pioneering work of Cafferalli, Nirenberg and Spruck, we prove that the Dirichlet problem admits a (unique) smooth solution provided that there exists a subsolution. The conditions are essentially optimal, especially with no geometric restrictions to the boundary of the underlying manifold, which is important in applications. We search new ideas and techniques to make use of the concavity condition and subsolution in order to overcome difficulties in deriving key a priori estimates. Along the way we discover some interesting properties of concave functions which should be useful in other fields. Our methods can be adopted to treat other types of fully nonlinear elliptic and parabolic equations on real or complex manifolds. We shall also solve some new equations, which were not covered by previous results even in Rn, with interesting properties.

Subsonic time-periodic solution to compressible Euler equations with damping in a bounded domain

In this paper, we consider the one-dimensional isentropic compressible Euler equations with linear damping β(t,x)ρu in a bounded domain, which can be used to describe the process of compressible flows through a porous medium. And the model is imposed a dissipative subsonic time-periodic boundary condition. Our main results reveal that the time-periodic boundary can trigger a unique subsonic time-periodic smooth solution which is stable under small perturbations on initial data. Moreover, the time-periodic solution possesses higher regularity and stability provided a higher regular boundary condition.

Global regularity for the 3D axisymmetric incompressible Hall-MHD system with partial dissipation and diffusion

In this paper, we study the Cauchy problem for the 3D incompressible axisymmetric Hall-MHD system with horizontal velocity dissipation and vertical magnetic diffusion.We obtain a unique global smooth solution of which in the cylindrical coordinate system the swirl velocity fields, the radial and the vertical components of the magnetic fields are trivial. This type of solution has been studied for the MHD system in [17][16] and [15] and for the Hall-MHD system with total dissipation and diffusion in [11]. Some new and fine estimates are obtained in this paper to overcome the difficulties raised from the Hall term and the loss of vertical velocity dissipation and horizontal magnetic diffusion. Finally we can show that the estimates $\int_0^T \ \nabla u(t)\ _{L^\infty} dt$ and $\int_0^T \ \nabla b(t)\ _{L^\infty} dt$ are finite in a priori way and hence obtain the global well-posedness to the system under considered.

Non-isothermal viscoelastic flows with conservation laws and relaxation

We propose a system of conservation laws with relaxation source terms (i.e. balance laws) for non-isothermal viscoelastic flows of Maxwell fluids. The system is an extension of the polyconvex elastodynamics of hyperelastic bodies using additional structure variables. It is obtained by writing the Helmholtz free energy as the sum of a volumetric energy density (function of the determinant of the deformation gradient det F and the temperature [Formula: see text] like the standard perfect-gas law or Noble–Abel stiffened-gas law) plus a polyconvex strain energy density function of F, [Formula: see text] and of symmetric positive-definite structure tensors that relax at a characteristic time scale. One feature of our model is that it unifies various ideal materials ranging from hyperelastic solids to perfect fluids, encompassing fluids with memory like Maxwell fluids. We establish a strictly convex mathematical entropy to show that the system is symmetric-hyperbolic. Another feature of the proposed model is therefore the short-time existence and uniqueness of smooth solutions, which define genuinely causal viscoelastic flows with waves propagating at finite speed. In heat-conductors, we complement the system by a Maxwell–Cattaneo equation for an energy-flux variable. The system is still symmetric-hyperbolic, and smooth evolutions with finite-speed waves remain well-defined.

Global smooth solutions for the Landau–Lifshitz–Bloch equation with helicity term

In this short paper, the global existence and uniqueness of smooth solutions are shown for the Cauchy problem of the Landau–Lifshitz–Bloch equation with helicity.

On the 3D Incompressible Boussinesq Equations in a Class of Variant Spherical Coordinates

This paper investigates the global stabilizing effects of the geometry of the domain at which the flow locates and of the geometric structure of the solution to the incompressible flows by studying the three-dimensional (3D) incompressible, viscosity, and diffusivity Boussinesq system in spherical coordinates. We establish the global existence and uniqueness of the smooth solution to the Cauchy problem for a full 3D incompressible Boussinesq system in a class of variant spherical coordinates for a class of smooth large initial data. We also construct one class of nonempty bounded domains in the three-dimensional space ℝ 3 , in which the initial boundary value problem for the full 3D Boussinesq system in a class of variant spherical coordinates with a class of large smooth initial data with swirl has a unique global strong or smooth solution with exponential decay rate in time.

Optimal convergence rate to nonlinear diffusion waves for bipolar Euler–Poisson equations with critical overdamping

This paper is concerned with the large time behaviors of smooth solutions to the Cauchy problem of the one dimensional bipolar Euler-Poisson equations with the time dependent critical overdamping. We show that in this critical overdamping case the bipolar Euler-Poisson system admits a unique global smooth solution that asymptotically converges to the nonlinear diffusion wave. In particular, the optimal convergence rate in logarithmic form is derived when the initial perturbations are L 2 sense by using the technical time-weighted energy method.

HIGHER REGULARITY FOR PARABOLIC EQUATIONS BASED ON MAXIMAL Lp-Lq SPACES

In this paper we prove higher regularity for 2m-th order parabolic equations with general boundary conditions. This is a kind of maximal L_p-L_q regularity with differentiability, i.e. the main theorem is isomorphism between the solution space and the data space using Besov and Triebel--Lizorkin spaces. The key is compatibility conditions for the initial data. We are able to get a unique smooth solution if the data satisfying compatibility conditions are smooth.