Moser Iteration Research Articles

Nonlinear Stability of MHD Contact Discontinuities with Surface Tension

We consider the motion of two inviscid, compressible, and electrically conducting fluids separated by an interface across which there is no fluid flow in the presence of surface tension. The magnetic field is supposed to be nowhere tangential to the interface. This leads to the characteristic free boundary problem for contact discontinuities with surface tension in three-dimensional ideal compressible magnetohydrodynamics (MHD). We prove the nonlinear structural stability of MHD contact discontinuities with surface tension in Sobolev spaces by a modified Nash--Moser iteration scheme. The main ingredient of our proof is deriving the resolution and tame estimate of the linearized problem in usual Sobolev spaces of sufficiently large regularity. In particular, for solving the linearized problem, we introduce a suitable regularization that preserves the transport-type structure for the linearized entropy and divergence of the magnetic field.

C1,α-regularity for quasilinear degenerate elliptic equation with a drift term on the Heisenberg group

The aim of the paper is to derive the C1,α-regularity of weak solutions to a quasilinear degenerate elliptic equation with a drift term on the Heisenberg group. Establishing a Caccioppoli inequality (i.e. energy estimate) for the horizontal gradient of weak solutions, we use it and the Sobolev embedding theorem on the Heisenberg group and the Moser iteration method to prove the local boundedness of the horizontal gradient of weak solutions. Then we prove an oscillation estimate for the horizontal gradient of weak solutions and combine the iteration Lemma to show that the horizontal gradient of weak solutions is Hölder continuous.

Harnack inequality for a p-Laplacian equation with a source reaction term involving the product of the function and its gradient

<abstract><p>A p-Laplacian type problem with a source reaction term involving the product of the function and its gradient is considered in this paper. A Harnack inequality is proved, and the main idea is based on de Giorgi-Nash-Moser iteration and Moser's iteration technique. As a consequence, $ H\ddot{o}lder $ continuity and boundness for the solution of this problem also are obtained.</p></abstract>

Predator-prey systems with defense switching and density-suppressed dispersal strategy.

In this paper, we consider the following predator-prey system with defense switching mechanism and density-suppressed dispersal strategy $ \begin{equation*} \begin{cases} u_t = \Delta(d_1(w)u)+\frac{\beta_1 uvw}{u+v}-\alpha_1 u, & x\in \Omega, \; \; t>0, \\ v_t = \Delta(d_2(w)v)+\frac{\beta_2 uvw}{u+v}-\alpha_2 v, & x\in \Omega, \; \; t>0, \\ w_t = \Delta w-\frac{\beta_3 uvw}{u+v}+\sigma w\left(1-\frac{w}{K}\right), & x\in \Omega, \; \; t>0, \\ \frac{\partial u}{\partial \nu} = \frac{\partial v}{\partial \nu} = \frac{\partial w}{\partial \nu} = 0, & x\in\partial\Omega, \; \; t>0, \\ (u, v, w)(x, 0) = (u_0, v_0, w_0)(x), & x\in\Omega, \ \end{cases} \end{equation*} $ where $ \Omega\subset{\mathbb{R}}^2 $ is a bounded domain with smooth boundary. Based on the method of energy estimates and Moser iteration, we establish the existence of global classical solutions with uniform-in-time boundedness. We further prove the global stability of co-existence equilibrium by using the Lyapunov functionals and LaSalle's invariant principle. Finally we conduct linear stability analysis and perform numerical simulations to illustrate that the density-suppressed dispersal may trigger the pattern formation.

Boundedness and stabilization of a predator-prey model with attraction- repulsion taxis in all dimensions.

This paper establishes the existence of globally bounded classical solutions to a predator-prey model with attraction-repulsion taxis in a smooth bounded domain of any dimensions with Neumann boundary conditions. Moreover, the global stabilization of solutions with convergence rates to constant steady states is obtained. Using the local time integrability of the L2-norm of solutions, we build up the basic energy estimates and derive the global boundedness of solutions by the Moser iteration. The global stability of constant steady states is established based on the Lyapunov functional method.

Regularity Theory for Mixed Local and Nonlocal Parabolic p-Laplace Equations

We investigate the mixed local and nonlocal parabolic p-Laplace equation $$\begin{aligned} \partial _t u(x,t)-\Delta _p u(x,t)+\mathcal {L}u(x,t)=0, \end{aligned}$$where \(\Delta _p\) is the usual local p-Laplace operator and \(\mathcal {L}\) is the nonlocal p-Laplace type operator. Based on the combination of suitable Caccioppoli-type inequality and Logarithmic Lemma with a De Giorgi–Nash–Moser iteration, we establish the local boundedness and Hölder continuity of weak solutions for such equations.

RD‐systems with chemotaxis: Global existence, boundedness, and blow‐up of solutions

We study the finite time blow‐up of solutions to general RD‐systems with chemotaxis for multi‐species. Our result shows that the blow‐up is equivalent to the blow‐up of the Lr‐norms of the solutions for r exceeding some critical value rc. Under very loose conditions, we give the estimation of rc, relying on a variant of Gagliardo–Nirenberg inequality, and a kind of bootstrap method which is very similar to the Alikakos–Moser iteration procedure.

Quantitative spectral gaps for hypoelliptic stochastic differential equations with small noise

We study the convergence rate to equilibrium for a family of Markov semigroups $\{\mathcal{P}_t^{\epsilon}\}_{\epsilon > 0}$ generated by a class of hypoelliptic stochastic differential equations on $\mathbb{R}^d$, including Galerkin truncations of the incompressible Navier-Stokes equations, Lorenz-96, and the shell model SABRA. In the regime of vanishing, balanced noise and dissipation, we obtain a sharp (in terms of scaling) quantitative estimate on the exponential convergence in terms of the small parameter $\epsilon$. By scaling, this regime implies corresponding optimal results both for fixed dissipation and large noise limits or fixed noise and vanishing dissipation limits. As part of the proof, and of independent interest, we obtain uniform-in-$\epsilon$ upper and lower bounds on the density of the stationary measure. Upper bounds are obtained by a hypoelliptic Moser iteration, the lower bounds by a de Giorgi-type iteration (both uniform in $\epsilon$). The spectral gap estimate on the semigroup is obtained by a weak Poincar\'e inequality argument combined with quantitative hypoelliptic regularization of the time-dependent problem.

On sign-changing solutions for quasilinear Schrödinger-Poisson system with critical growth

In this paper, we consider the following quasilinear Schrödinger-Poisson system with critical growth: where , is a smooth potential function and g is a appropriate nonlinear function. For the sake of overcoming the technical difficulties caused by the quasilinear term, we shall apply the perturbation method by adding a 4-Laplacian operator to consider the perturbation problem. Moreover, when g satisfying suitable assumptions and sufficiently large μ, we take advantage of constraint variational method, the quantitative deformation lemma, Moser iteration and approximation technique to obtain a least-energy sign-changing solution , which has precisely two nodal domains.

Existence of solutions for a quasilinear elliptic system with local nonlinearity on ℝN

In this paper, we investigate the existence of solutions for a class of quasilinear elliptic system. By developing the Moser iteration technique, we obtain that system has a nontrivial solution (uλ, vλ) with ‖(uλ, vλ)‖∞ ≤ 2 for every λ large enough when the nonlinear term F satisfies some growth conditions only in a circle with center 0 and radius 4, and the families of solutions {(uλ, vλ)} satisfy that ‖(uλ, vλ) ‖ → 0 as λ → ∞. Moreover, because the interaction of u and v in elliptic system causes that the estimate of ‖u‖∞ cannot vary with ‖u‖, the conclusion for the elliptic system is weaker than the corresponding result for the quasilinear elliptic equation, which is given in the end as a comparison.

Response Solutions for Wave Equations with Variable Wave Speed and Periodic Forcing

We consider a model of nonlinear wave equations with periodically varying wave speed and periodic external forcing. By imposing non-resonance conditions on the frequency, we establish the existence of response solutions (i.e., periodic solutions with the same frequency as the forcing) for such a model in a Cantor set of asymptotically full measure. The proof relies on a Lyapunov–Schmidt reduction together with the Nash–Moser iteration.

Well-posedness for the free-boundary ideal compressible magnetohydrodynamic equations with surface tension

We establish the local existence and uniqueness of solutions to the free-boundary ideal compressible magnetohydrodynamic equations with surface tension in three spatial dimensions by a suitable modification of the Nash–Moser iteration scheme. The main ingredients in proving the convergence of the scheme are the tame estimates and unique solvability of the linearized problem in the anisotropic Sobolev spaces $$H_*^m$$ for m large enough. In order to derive the tame estimates, we make full use of the boundary regularity enhanced from the surface tension. The unique solution of the linearized problem is constructed by designing some suitable $$\varepsilon $$ –regularization and passing to the limit $$\varepsilon \rightarrow 0$$ .

An existence result for singular nonlocal fractional Kirchhoff–Schrödinger–Poisson system

In this paper, we study the existence of infinitely many weak solutions to a fractional Kirchhoff–Schrödinger–Poisson system involving a weak singularity, i.e. when . Further, we obtain the existence of a solution with a strong singularity, i.e. when . We employ variational techniques to prove the existence and multiplicity results. Moreover, an estimate is obtained by using the Moser iteration method.

Sign-Changing Solutions for Fractional Kirchhoff-Type Equations with Critical and Supercritical Nonlinearities

This paper concerns the existence of sign-changing solutions for the following fractional Kirchhoff-type equation with critical and supercritical nonlinearities $$\begin{aligned} \left( a+b[u]^{2}\right) (-\Delta )^{\alpha }u+V(x)u=f(x,u)+\lambda |u| ^{r-2}u,\,\, \text {in}\,\,\mathbb {R}^3, \end{aligned}$$ where $$a,b>0$$ are constants, $$\alpha \in (\frac{3}{4}, 1)$$ , $$\lambda >0$$ is a real parameter, $$r\ge 2_{\alpha }^{*}=\frac{6}{3-2\alpha }$$ , $$(-\Delta )^{\alpha }$$ is the fractional Laplace operator, the potential function V and the nonlinearity f satisfy some suitable conditions. By combining an appropriate truncation argument with Moser iteration method, we prove that the existence of sign-changing solutions for the above equation when the parameter $$\lambda $$ is sufficiently small. Our results enrich and improve the previous ones in the literature.

Conditions of solvability of the nonlocal boundary-value problem for a differential-operator equation with weak nonlinearity

We study the solvability of a nonlocal boundary-value problem for a differential equation with nonlinearity. The linear part of the equation has complex coefficients which together with the coefficient in nonlocal conditions are considered to be the parameters of the problem. The area of change of each parameter is limited by a complex circle with its center at the origin. The nonlinear part of the equation is given by a smooth function that satisfies, together with its derivatives, some conditions of growth in the Dirichlet--Fourier space scale. The proofs are based on the differentiable Nash--Moser iteration scheme, where the main difficulty is to get estimates of the interpolation type for the inverse linearized operators obtained at each step of the iteration. The estimation is connected with the problem of small denominators which is solved, by using a metric approach on the set of parameters of the problem.

Positive solutions of the prescribed mean curvature equation with exponential critical growth

In this paper, we are concerned with the problem -div∇u1+|∇u|2=f(u)inΩ,u=0on∂Ω,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} -\\text{ div } \\left( \\displaystyle \\frac{\\nabla u}{\\sqrt{1+|\\nabla u|^2}}\\right) = f(u) \\ \\text{ in } \\ \\Omega , \\ \\ u=0 \\ \\text{ on } \\ \\ \\partial \\Omega , \\end{aligned}$$\\end{document}where Omega subset {mathbb {R}}^{2} is a bounded smooth domain and f:{mathbb {R}}rightarrow {mathbb {R}} is a superlinear continuous function with critical exponential growth. We first make a truncation on the prescribed mean curvature operator and obtain an auxiliary problem. Next, we show the existence of positive solutions of this auxiliary problem by using the Nehari manifold method. Finally, we conclude that the solution of the auxiliary problem is a solution of the original problem by using the Moser iteration method and Stampacchia’s estimates.

Existence and asymptotic behavior of solutions for a class of semilinear subcritical elliptic systems

In this paper we study the asymptotic behaviour of a family of elliptic systems, as far as the existence of solutions is concerned. We give a special attention to the asymptotic behaviour of W and V as ε goes to zero in the system − ε 2 Δ u + W ( x ) u = Q u ( u , v ) in R N , − ε 2 Δ v + V ( x ) v = Q v ( u , v ) in R N , u , v ∈ H 1 ( R N ) , u ( x ) , v ( x ) > 0 for each x ∈ R N , where ε > 0, W and V are positive potentials of C 2 class and Q is a p-homogeneous function with subcritical growth. We establish the existence of a positive solution by considering two classes of potentials W and V. Our arguments are based on penalization techniques, variational methods and the Moser iteration scheme.

Further regularity and uniqueness results for a non-isothermal Cahn-Hilliard equation

The aim of this paper is to establish new regularity results for a non-isothermal Cahn-Hilliard system in the two dimensional setting. The main achievement is a crucial \begin{document}$ L^{\infty} $\end{document} estimate for the temperature, obtained by a suitable Moser iteration scheme. Our results in particular allow us to get a new simplified version of the uniqueness proof for the considered model.

Well-posedness of Free Boundary Problem in Non-relativistic and Relativistic Ideal Compressible Magnetohydrodynamics

We consider the free boundary problem for non-relativistic and relativistic ideal compressible magnetohydrodynamics in two and three spatial dimensions with the total pressure vanishing on the plasma--vacuum interface. We establish the local-in-time existence and uniqueness of solutions to this nonlinear characteristic hyperbolic problem under the Rayleigh--Taylor sign condition on the total pressure. The proof is based on certain tame estimates in anisotropic Sobolev spaces for the linearized problem and a modification of the Nash--Moser iteration scheme. Our result is uniform in the light speed and appears to be the first well-posedness result for the free boundary problem in ideal compressible magnetohydrodynamics with zero total pressure on the moving boundary.

Global solution and global orbit to reaction–diffusion equation for fractional Dirichlet‐to‐Neumann operator with subcritical exponent

We consider the reaction–diffusion equation for fractional Dirichlet‐to‐Neumann operator with subcritical exponent motivated by electrical impedance tomography (EIT) and a need to overcome the non‐locality of a fractional differential equation for modeling anomalous diffusion. We mainly deal with the asymptotic behavior of global solution and the boundedness of global orbit, which allows us to show that any global solution is classical solution using Moser iteration technique.