Unique Smooth Solution Research Articles

Global existence and lifespan of smooth solutions of biwave maps

We show that if f:R1+m→N is a biwave map from a Minkowski spacetime into a Riemannian manifold N with initial data such that the spatial dimension m > 6 and the Riemannian curvature of N satisfies certain conditions, then there exists a small constant ϵ0 such that the biwave map f has a unique global smooth solution for ϵ ∈ [0, ϵ0) (where the parameter ϵ indicates the size of the initial data). In case 1 ≤ m ≤ 6, we obtain the lifespan of the smooth solution of the above biwave map. Furthermore, if f:M→N is a biwave map on a Friedmann–Lemaître–Robertson–Walker spacetime under certain circumstances, we obtain the lifespan of the smooth solution of the biwave map for spatial dimension m ≥ 1.

Large Ericksen number limit for the 2D general Ericksen–Leslie system

In this paper, we address the issue of the so-called large Ericksen number limit for the general Ericksen–Leslie system of liquid crystals. By the standard nondimensionalization, we first prove that there exists a unique local smooth solution to the limiting system. Then we provide a qualitative estimate for the difference between the solutions of the general Ericksen–Leslie system and the limiting system, which corresponds to the partial decoupling effect in the limit of large Ericksen number.

A class of curvature flows expanded by support function and curvature function in the Euclidean space and hyperbolic space

In this paper, we first consider a class of expanding flows of closed, smooth, star-shaped hypersurface in Euclidean space Rn+1 with speed uαf−β, where u is the support function of the hypersurface, f is a smooth, symmetric, homogenous of degree one, positive function of the principal curvatures of the hypersurface on a convex cone. For α⩽0<β⩽1−α, we prove that the flow has a unique smooth solution for all time, and converges smoothly after normalization, to a sphere centered at the origin. In particular, the results of Gerhardt [16] and Urbas [40] can be recovered by putting α=0 and β=1 in our first result. If the initial hypersurface is convex, this is our previous work [11]. If α⩽0<β<1−α and the ambient space is hyperbolic space Hn+1, we prove that the flow ∂X∂t=(uαf−β−ηu)ν has a longtime existence and smooth convergence to a coordinate slice. The flow in Hn+1 is equivalent (up to an isomorphism) to a re-parametrization of the original flow in Rn+1 case. Finally, we find a family of monotone quantities along the flows in Rn+1. As applications, we give a new proof of a family of inequalities involving the weighted integral of kth elementary symmetric function for k-convex, star-shaped hypersurfaces, which is an extension of the quermassintegral inequalities in [20].

Stability of Transonic Contact Discontinuity for Two-Dimensional Steady Compressible Euler Flows in a Finitely Long Nozzle

We consider the stability of transonic contact discontinuity for the two-dimensional steady compressible Euler flows in a finitely long nozzle. This is the first work on the mixed-type problem of transonic flows across a contact discontinuity as a free boundary in nozzles. We start with the Euler-Lagrangian transformation to straighten the contact discontinuity in the new coordinates. However, the upper nozzle wall in the subsonic region depending on the mass flux becomes a free boundary after the transformation. Then we develop new ideas and techniques to solve the free-boundary problem in three steps: (1) we fix the free boundary and generate a new iteration scheme to solve the corresponding fixed boundary value problem of the hyperbolic-elliptic mixed type by building some powerful estimates for both the first-order hyperbolic equation and a second-order nonlinear elliptic equation in a Lipschitz domain; (2) we update the new free boundary by constructing a mapping that has a fixed point; (3) we establish via the inverse Lagrangian coordinate transformation that the original free interface problem admits a unique piecewise smooth transonic solution near the background state, which consists of a smooth subsonic flow and a smooth supersonic flow with a contact discontinuity.

Global solutions of a surface quasigeostrophic front equation

We consider a nonlinear, spatially-nonlocal initial value problem in one space dimension on $\mathbb{R}$ that describes the motion of surface quasi-geostrophic (SQG) fronts. We prove that the initial value problem has a unique local smooth solution under a convergence condition on the multilinear expansion of the nonlinear term in the equation, and, for sufficiently smooth and small initial data, we prove that the solution is global.

Global smooth solution of 2D temperature-dependent tropical climate model

This paper focuses on a two-dimensional tropical climate model with temperature-dependent viscosity and thermal diffusivity. We show that there is a unique global smooth solution to this model with general initial data in the Sobolev class H s for any s > 1.

Finite state mean field games with Wright–Fisher common noise

We force uniqueness in finite state mean field games by adding a Wright–Fisher common noise. We achieve this by analyzing the master equation of this game, which is a degenerate parabolic second-order partial differential equation set on the simplex whose characteristics solve the stochastic forward-backward system associated with the mean field game; see Cardaliaguet et al. [10]. We show that this equation, which is a non-linear version of the Kimura type equation studied in Epstein and Mazzeo [28], has a unique smooth solution whenever the normal component of the drift at the boundary is strong enough. Among others, this requires a priori estimates of Hölder type for the corresponding Kimura operator when the drift therein is merely continuous.

Local Existence and Uniqueness of Navier–Stokes–Schrödinger System

In this article, we prove that there exists a unique local smooth solution for the Cauchy problem of the Navier–Stokes–Schrodinger system. Our methods rely upon approximating the system with a sequence of perturbed system and parallel transport and are closer to the one in Ding and Wang (Sci China 44(11):1446–1464, 2001) and McGahagan (Commun Partial Differ Equ 32(1–3):375–400, 2007).

Coriolis effect on temporal decay rates of global solutions to the fractional Navier–Stokes equations

The incompressible fractional Navier–Stokes equations in the rotational framework is considered. We establish the global existence result and temporal decay estimate for a unique smooth solution when the speed of rotation is sufficiently rapid. It is found that the strong rotational effect enhances the temporal decay rate of a certain norm of the velocity.

Global Smooth Solutions With Large Data for a System Modeling Aurora Type Phenomena in the 2-Torus

We prove the existence and uniqueness of smooth solutions with large initial data for a system of equations modeling the interaction of short waves, governed by a nonlinear Schrodinger equation, an...

Stability of equilibria uniformly in the inviscid limit for the Navier-Stokes-Poisson system

We prove a stability result of constant equilibria for the three dimensional Navier-Stokes-Poisson system uniform in the inviscid limit. We allow the initial density to be close to a constant and the potential part of the initial velocity to be small independently of the rescaled viscosity parameter ε while the incompressible part of the initial velocity is assumed to be small compared to ε. We then get a unique global smooth solution. We also prove a uniform in ε time decay rate for these solutions. Our approach allows to combine the parabolic energy estimates that are efficient for the viscous equation at ε fixed and the dispersive techniques (dispersive estimates and normal forms) that are useful for the inviscid irrotational system.

Global regularity for the hyperdissipative Navier-Stokes equation below the critical order

We consider solutions of the Navier-Stokes equation with fractional dissipation of order α≥1. We show that for any divergence-free initial datum u0 such that ‖u0‖Hδ≤M, where M is arbitrarily large and δ is arbitrarily small, there exists an explicit ε=ε(M,δ)>0 such that the Navier-Stokes equations with fractional order α have a unique global smooth solution for α∈(54−ε,54]. This is related to a new stability result on smooth solutions of the Navier-Stokes equations with fractional dissipation showing that the set of initial data and fractional orders giving rise to smooth solutions is open in H5/4×(34,54].

Global existence and time decay rates for the 3D bipolar compressible Navier-Stokes-Poisson system with unequal viscosities

We consider the Cauchy problem for the three-dimensional bipolar compressible Navier-Stokes-Poisson system with unequal viscosities. Under the assumption that the H3 norm of the initial data is small but its higher order derivatives can be arbitrarily large, the global existence and uniqueness of smooth solutions are obtained by an ingenious energy method. Moreover, if additionally, the $${\dot H^{- s}}\left({{1 \over 2} \leqslant s < {3 \over 2}} \right)\,\,{\rm{or}}\,\,\dot B_{2,\infty }^{- s}\left({{1 \over 2} < s \leqslant {3 \over 2}} \right)$$ norm of the initial data is small, the optimal decay rates of solutions are also established by a regularity interpolation trick and delicate energy methods.

Global well-posedness for a model of 2D temperature-dependent Boussinesq equations without diffusivity

In this paper we consider the Cauchy problem of a model of the two-dimensional zero diffusivity Boussinesq equations with temperature-dependent viscosity. We show that there is a unique global smooth solution to this system for arbitrarily large initial data in Sobolev spaces. Our key argument is the De Giorgi-Nash-Moser estimates for the vorticity equation.

Mathematical modeling and qualitative analysis of viscoelastic conductive fluids

We use an energetic variational approach to model the transport of compressible viscoelastic conductive fluids. Such a model can be called the three-dimensional compressible viscoelastic Navier–Stokes–Poisson equations. The global unique smooth solution to the Cauchy problem is obtained. In particular, we obtain the optimal time-decay rates of the solution and its higher-order spatial derivatives by using a pure energy method.

Zero Mach number limit of the compressible Euler\u2013Korteweg equations

In this paper, we investigate the zero Mach number limit for the three-dimensional compressible Euler–Korteweg equations in the regime of smooth solutions. Based on the local existence theory of the compressible Euler–Korteweg equations, we establish a convergence-stability principle. Then we show that when the Mach number is sufficiently small, the initial-value problem of the compressible Euler–Korteweg equations has a unique smooth solution in the time interval where the corresponding incompressible Euler equations have a smooth solution. It is important to remark that when the incompressible Euler equations have a global smooth solution, the existence time of the solution for the compressible Euler–Korteweg equations tends to infinity as the Mach number goes to zero. Moreover, we obtain the convergence of smooth solutions for the compressible Euler–Korteweg equations towards those for the incompressible Euler equations with a convergence rate.

Global smoothness for a 1D supercritical transport model with nonlocal velocity

We are concerned with a nonlocal transport 1D-model with supercritical dissipation γ ∈ ( 0 , 1 ) \gamma \in (0,1) in which the velocity is coupled via the Hilbert transform, namely the so-called CCF model. This model arises as a lower dimensional model for the well-known 2D dissipative quasi-geostrophic equation and in connection with vortex-sheet problems. It is known that its solutions can blow up in finite time when γ ∈ ( 0 , 1 / 2 ) \gamma \in (0,1/2) . On the other hand, as stated by Kiselev (2010), in the supercritical subrange γ ∈ [ 1 / 2 , 1 ) \gamma \in \lbrack 1/2,1) it is an open problem to know whether its solutions are globally regular. We show global existence of nonnegative H 3 / 2 H^{3/2} -strong solutions in a supercritical subrange (close to 1) that depends on the initial data norm. Then, for each arbitrary smooth nonnegative initial data, the model has a unique global smooth solution provided that γ ∈ [ γ 1 , 1 ) \gamma \in \lbrack \gamma _{1},1) where γ 1 \gamma _{1} depends on the H 3 / 2 H^{3/2} -initial data norm. Our approach is inspired by that of Coti Zelati and Vicol (IUMJ, 2016).

The Burgers\u2019 equation with stochastic transport: shock formation, local and global existence of smooth solutions

In this work, we examine the solution properties of the Burgers’ equation with stochastic transport. First, we prove results on the formation of shocks in the stochastic equation and then obtain a stochastic Rankine–Hugoniot condition that the shocks satisfy. Next, we establish the local existence and uniqueness of smooth solutions in the inviscid case and construct a blow-up criterion. Finally, in the viscous case, we prove global existence and uniqueness of smooth solutions.

Low Mach number limit of the three-dimensional full compressible Navier–Stokes–Korteweg equations

In this paper, we justify the low Mach number limit for the three-dimensional full compressible Navier–Stokes–Korteweg equations rigorously within the framework of smooth solution. Under the assumptions of small density and temperature perturbation, we show that for sufficiently small Mach number, the initial-value problem of the three-dimensional full compressible Navier–Stokes–Korteweg equations admits a unique smooth solution on the time interval where the smooth solution of the corresponding incompressible Navier–Stokes equations exists. Moreover, we obtain the convergence of smooth solutions for the full compressible Navier–Stokes–Korteweg equations toward those for the incompressible Navier–Stokes equations with a convergence rate.

Well-posedness of the free boundary problem in incompressible elastodynamics under the mixed type stability condition

We consider the free boundary problem for the flow of an incompressible inviscid elastic fluid. The columns of the deformation gradient are tangent and the pressure vanishes along the free interface. We prove the local existence of a unique smooth solution to the free boundary problem under the mixed type stability condition, provided that the Rayleigh-Taylor sign condition is satisfied at all the points of the initial interface where the non-collinearity condition for the deformation gradient (among three columns of the deformation gradient there are two non-collinear vectors) fails. In particular, we solve an open problem proposed by Y. Trakhinin in the paper [29] under the incompressible setting.