Equations In Half-space Research Articles

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.

On initial-boundary value problem of the stochastic Navier–Stokes equations in the half space

We study the initial-boundary value problem of the stochastic Navier–Stokes equations in half-space. We prove the existence of weak solutions in standard Besov space-valued random processes when the initial data belong to the critical Besov space.

The [formula omitted]-asymptotic behavior of strong solutions to the incompressible magneto-hydrodynamic equations in half-spaces

We give the L1-decay estimates on the fractional spatial derivatives of the strong solution to the incompressible magneto-hydrodynamic (MHD) equations in a half-space. Moreover, by using the Stokes solution formula and the decomposition of the nonlinear terms in MHD equations, we derive asymptotic expansion properties of the strong solution in L1(R+n).

On Well-Posed Boundary Conditions for the Linear Non-Homogeneous Moment Equations in Half-Space

On Well-Posed Boundary Conditions for the Linear Non-Homogeneous Moment Equations in Half-Space

Stationary measures for the log-gamma polymer and KPZ equation in half-space

Stationary measures for the log-gamma polymer and KPZ equation in half-space

A Liouville theorem of the 2-Hessian equation in half-space

In this paper, we establish the gradient and second derivative estimates for solutions to the 2-Hessian equation in half-space under certain boundary data and growth condition. We also use such estimates to obtain the Liouville theorem.

Generalized Liouville theorem for viscosity solutions to a singular Monge-Ampère equation

Abstract In this article, we study the asymptotic behaviour at infinity for viscosity solutions to a singular Monge-Ampère equation in half space from affine geometry. In particular, we extend the Liouville theorem for smooth solutions to the case of viscosity solutions by a completely different method from the smooth case.

Liouville type theorems of fully nonlinear elliptic equations in half spaces

In this paper, we investigate the asymptotic behavior of viscosity solutions for a class of fully nonlinear second-order elliptic equations in the half space. Moreover, the result is applied to Monge-Ampère equations and consequently we improve some existent result.

THE GROWTH OF SOLUTIONS OF MONGE–AMPÈRE EQUATIONS IN HALF SPACES AND ITS APPLICATION

AbstractWe consider the growth of the convex viscosity solution of the Monge–Ampère equation $\det D^2u=1$ outside a bounded domain of the upper half space. We show that if u is a convex quadratic polynomial on the boundary $\{x_n=0\}$ and there exists some $\varepsilon>0$ such that $u=O(|x|^{3-\varepsilon })$ at infinity, then $u=O(|x|^2)$ at infinity. As an application, we improve the asymptotic result at infinity for viscosity solutions of Monge–Ampère equations in half spaces of Jia, Li and Li [‘Asymptotic behavior at infinity of solutions of Monge–Ampère equations in half spaces’, J. Differential Equations269(1) (2020), 326–348].

Pointwise wave behavior of the non-isentropic Navier–Stokes equations in half space

Pointwise wave behavior of the non-isentropic Navier–Stokes equations in half space

‘Far interaction’ of small spectral perturbations of the Neumann boundary conditions for an elliptic system of differential equations in a three-dimensional domain

A formally selfadjoint system of second-order differential equations is considered in a three-dimensional domain on small parts of whose boundary an analogue of Steklov spectral conditions is set, while the Neumann boundary conditions are set on the rest of the boundary. Under certain algebraic and geometric conditions an asymptotic expression for the eigenvalues of this problem is presented and a limiting problem is put together, which produces the leading asymptotic terms and involves systems of integro-differential equations in half-spaces, interconnected by means of certain integral characteristics of vector-valued eigenfunctions. One example of a concrete problem in mathematical physics describes surface waves in several ice holes made in the ice cover of a water basin, and the asymptotic formula for eigenfrequencies shows that the local wave processes interact independently of the distance between the holes. Another series of applied problems relates to elastic fixings of bodies along small pieces of their surfaces. Possible generalizations are discussed; a number of related open questions are stated. Bibliography: 41 titles.

Monotonicity of positive solutions to quasilinear elliptic equations in half-spaces with a changing-sign nonlinearity

In this paper we prove the monotonicity of positive solutions to \( -\Delta _p u = f(u) \) in half-spaces under zero Dirichlet boundary conditions, for \((2N+2)/(N+2)< p < 2\) and for a general class of regular changing-sign nonlinearities f. The techniques used in the proof of the main result are based on a fine use of comparison and maximum principles and on an adaptation of the celebrated moving plane method to quasilinear elliptic equations in unbounded domains.

Boundary Layer Solution of the Boltzmann Equation for Diffusive Reflection Boundary Conditions in Half-Space

We study steady Boltzmann equation in half-space, which arises in the Knudsen boundary layer problem, with diffusive reflection boundary conditions. Under certain admissible conditions and the source term decaying exponentially, we establish the existence of boundary layer solution for both linear and nonlinear Boltzmann equation in half-space with diffusive reflection boundary condition in $L^\infty_{x,v}$ when the far-field Mach number of the Maxwellian is zero. The continuity and the spacial decay of the solution are obtained. The uniqueness is established under some constraint conditions.

Second-order diffusion limit for the phonon transport equation: asymptotics and numerics

We investigate the numerical implementation of the limiting equation for the phonon transport equation in the small Knudsen number regime. The main contribution is that we derive the limiting equation that achieves the second order convergence, and provide a numerical recipe for computing the Robin coefficients. These coefficients are obtained by solving an auxiliary half-space equation. Numerically the half-space equation is solved by a spectral method that relies on the even-odd decomposition to eliminate corner-point singularity. Numerical evidences will be presented to justify the second order asymptotic convergence rate.

Growth of subsolutions to fully nonlinear equations in halfspaces

We characterize lower growth estimates for subsolutions in halfspaces of fully nonlinear partial differential equations on the formF(x,u,Du,D2u)=0 in terms of solutions to ordinary differential equations built upon assumptions on F. Using this characterization we derive several sharp Phragmen–Lindelöf-type theorems for certain classes of well known PDEs. The equation need not be uniformly elliptic nor homogeneous and we obtain results both in case the subsolution is bounded or unbounded. Among our results we retrieve classical estimates in the halfspace for p-subharmonic functions and extend those to more general equations; we prove sharp growth estimates, in terms of k and the asymptotic behavior of ∫0RC(s)ds, for subsolutions of equations allowing for sublinear growth in the gradient of the form C(|x|)|Du|k with k≥1; we establish a Phragmen–Lindelöf theorem for weak subsolutions of the variable exponent p-Laplace equation in halfspaces, 1<p(x)<∞, p(x)∈C1, of which we conclude sharpness by finding the “slowest growing” p(x)-harmonic function together with its corresponding family of p(x)-exponents. The paper ends with a discussion of our results from the point of view of a spatially dependent diffusion problem.

Partial Fourier Transform Methods to Solve the Solution Formula of Stokes Equation in Half-Space

Fluids are a shape of a matter which have substance liquids, gases and plasmas. In our daily life, fluids become important part, such as part of our blood and also help our body getting nutrients. It is well known that fluid motion can be described in mathematical model in especially in form of partial differential equations (PDE) and called as Navier Stokes Equations (NSE). The Navier Stokes equation is derived from balance of conservation of mass and conservation of momentum. In this paper, we consider the solution formula of the linearized of the Navier Stokes Equation (NSE) with the initial boundary value (IBV) problem in half space without surface tension. The model problem under consideration covers of non-linear fluid type. We solve the solution formula of velocity and density of the model problem by using Fourier transform and partial Fourier transform method. The strategy geting the solution of the model problem is based on the analysis of some resolvent of the model problem which obtained by using Laplace transform of the Stokes equations. Therefore, In particular, the formula of velocity and density of the Stokes equation are obtained.

Partial Fourier Transform Methods to Solve the Solution Formula of Stokes Equation in Half-Space

Fluids are a shape of a matter which have substance liquids, gases and plasmas. In our daily life, fluids become important part, such as part of our blood and also help our body getting nutrients. It is well known that fluid motion can be described in mathematical model in especially in form of partial differential equations (PDE) and called as Navier Stokes Equations (NSE). The Navier Stokes equation is derived from balance of conservation of mass and conservation of momentum. In this paper, we consider the solution formula of the linearized of the Navier Stokes Equation (NSE) with the initial boundary value (IBV) problem in half space without surface tension. The model problem under consideration covers of non-linear fluid type. We solve the solution formula of velocity and density of the model problem by using Fourier transform and partial Fourier transform method. The strategy geting the solution of the model problem is based on the analysis of some resolvent of the model problem which obtained by using Laplace transform of the Stokes equations. Therefore, In particular, the formula of velocity and density of the Stokes equation are obtained.

Pointwise estimates of the solution to one dimensional compressible Naiver-Stokes equations in half space

&lt;p style='text-indent:20px;'&gt;In this paper, we study the global existence and pointwise behavior of classical solution to one dimensional isentropic Navier-Stokes equations with mixed type boundary condition in half space. Based on classical energy method for half space problem, the global existence of classical solution is established firstly. Through analyzing the quantitative relationships of Green's function between Cauchy problem and initial boundary value problem, we observe that the leading part of Green's function for the initial boundary value problem is composed of three items: delta function, diffusive heat kernel, and reflected term from the boundary. Then applying Duhamel's principle yields the explicit expression of solution. With the help of accurate estimates for nonlinear wave coupling and the elliptic structure of velocity, the pointwise behavior of the solution is obtained under some appropriate assumptions on the initial data. Our results prove that the solution converges to the equilibrium state at the optimal decay rate &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ (1+t)^{-\frac{1}{2}} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; in &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ L^\infty $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; norm.&lt;/p&gt;

On the existence and asymptotic behavior of viscosity solutions of Monge–Ampère equations in half spaces

On the existence and asymptotic behavior of viscosity solutions of Monge–Ampère equations in half spaces

Liouville type theorems for the minimal surface equation in half space

For n≥2, we obtain Liouville type theorems for the minimal surface equation in half space R+n with constant Neumann boundary value or linear Dirichlet boundary value.