Global existence and asymptotic behavior of smooth solutions to a bipolar Euler–Poisson equation in a bound domain

In this paper, we present a bipolar hydrodynamic model from semiconductor devices and plasmas, which takes the form of bipolar isentropic Euler–Poisson with electric field and frictional damping added to the momentum equations. We firstly prove the existence of the stationary solutions. Next, we present the global existence and the asymptotic behavior of smooth solutions to the initial boundary value problem for a one-dimensional case in a bounded domain. The result is shown by an elementary energy method. Compared with the corresponding initial data case, we find that the asymptotic state is the stationary solution.     

Radial functions and regularity of solutions to the Schrödinger equation

Letf be a radial function and setT ^* f(x)=sup_0<t<1 |T _t f(x)|, x ∈ ?ⁿ, n≥2, where(T_t f)^ (ξ)=e ^it|ξ|a \(\hat f\) (ξ),a > 1. We show that, ifB is the ball centered at the origin, of radius 100, then \(\int\limits_B {|T^ * f(x)|} dx \leqslant c(\int {|\hat f(\xi )|^2 (l + |\xi |^s )ds} )^{1/2} \) if and only ifs≥1/4.     

On the Euler–Lagrange equation for a variational problem: the general case II

In this paper we study the existence of a solution in ${L^\infty_{\rm loc}(\Omega)}In this paper we study the existence of a solution in L^￥_loc(W){L^\infty_{\rm loc}(\Omega)} to the Euler–Lagrange equation for the variational problem

inf[`(u)] + W1,￥0(W) òW (ID(?u) + g(u)) dx,                   (0.1)\inf_{\bar u + W^{1,\infty}_0(\Omega)} \int\limits_{\Omega} ({\bf I}_D(\nabla u) + g(u)) dx,\quad \quad \quad \quad \quad(0.1)      7. On the existence,regularity and decay of solitary waves to a generalized Benjamin–Ono equation      《Nonlinear Analysis: Theory, Methods & Applications》2002,51(6):1073-1085       8. Analysis of the one-dimensional Euler–Lagrange equation of continuum mechanics with a Lagrangian of a special form      《Applied Mathematical Modelling》2020 The equations describing the flow of a one-dimensional continuum in Lagrangian coordinates are studied in this paper by the group analysis method. They are reduced to a single Euler–Lagrange equation which contains two undetermined functions (arbitrary elements). Particular choices of these arbitrary elements correspond to different forms of the shallow water equations, including those with both, a varying bottom and advective impulse transfer effect, and also some other motions of a continuum. A complete group classification of the equations with respect to the arbitrary elements is performed.One advantage of the Lagrangian coordinates consists of the presence of a Lagrangian, so that the equations studied become Euler–Lagrange equations. This allows us to apply Noether’s theorem for constructing conservation laws in Lagrangian coordinates. Not every conservation law in Lagrangian coordinates has a counterpart in Eulerian coordinates, whereas the converse is true. Using Noether’s theorem, conservation laws which can be obtained by the point symmetries are presented, and their analogs in Eulerian coordinates are given, where they exist.      9. On the regularity of solutions to variational and boundary-value Problems in Domains with Hölder boundary      I. V. Tsylin 《Mathematical Notes》2016,99(5-6):785-791       10. Minimizers of Kirchhoff's plate functional: Euler–Lagrange equations and regularity      Peter Hornung 《Comptes Rendus Mathematique》2009,347(11-12):647-650       11. Decay and stability of solutions to the Fokker–Planck–Boltzmann equation in      Xiaolong Wang  Huangping Shi 《Applicable analysis》2018,97(11):1933-1959 In this paper, we use the method of constructing the compensating function introduced by Kawashima and the standard energy method to study the global existence of solutions to the Fokker–Planck–Boltzmann equation in the whole space. The time decay and uniform stability of solutions to the global Maxwellian are also obtained.      12. Non-even positive solutions of the one-dimensional p-Laplace Emden–Fowler equation      Ryuji Kajikiya 《Applied Mathematics Letters》2012,25(11):1891-1895       13. Exact solutions of the one-dimensional modified complex Ginzburg–Landau equation      《Chaos, solitons, and fractals》2003,15(1):187-199       14. A sum operator method for the existence and uniqueness of positive solutions to Riemann–Liouville fractional differential equation boundary value problems      Chengbo Zhai  Weiping Yan  Chen Yang 《Communications in Nonlinear Science & Numerical Simulation》2013,18(4):858-866       15. Global existence and temporal decay of large solutions for the Poisson–Nernst–Planck equations in low regularity spaces      Jihong Zhao  Xilan Liu 《Mathematical Methods in the Applied Sciences》2023,46(2):1667-1686 We are concerned with the global existence and decay rates of large solutions for the Poisson–Nernst–Planck equations. Based on careful observation of algebraic structure of the equations and using the weighted Chemin–Lerner-type norm, we obtain the global existence and optimal decay rates of large solutions without requiring the summation of initial densities of a negatively and positively charged species that is small enough. Moreover, the large solution is obtained for initial densities belonging to the low regularity Besov spaces with different regularity and integral indices, which indicates more specific coupling relations between the difference and the summation of negatively and positively charged densities.      16. Maximal and minimal solutions of an Aronsson equation: L ∞ variational problems versus the game theory      Yifeng Yu 《Calculus of Variations and Partial Differential Equations》2010,37(1-2):63-74 The Dirichlet problem $$ \left\{ \begin{array}{l}\Delta _\infty u - |Du|^2 = 0 \quad {\rm on} \, \Omega \subset {{\mathbb R}^n} \\ u|\partial \Omega = g \\\end{array} \right. $$ might have many solutions, where ${\Delta_{\infty}u=\sum_{1\leq i,j\leq n}u_{x_i}u_{x_j}u_{x_ix_j}}$ . In this paper, we prove that the maximal solution is the unique absolute minimizer for ${H(p,z)={\frac{1}{2}}|p|^2-z}$ from calculus of variations in L ∞ and the minimal solution is the continuum value function from the “tug-of-war” game. We will also characterize graphes of solutions which are neither an absolute minimizer nor a value function. A remaining interesting question is how to interpret those intermediate solutions. Most of our approaches are based on an idea of Barles–Busca (Commun Partial Differ Equ 26(11–12):2323–2337, 2001).      17. On the existence of pair positive–negative solutions for resonance problems      Yavdat Il’yasov  Thomas Runst  Abdellah Youssfi 《Nonlinear Analysis: Theory, Methods & Applications》2009 This paper deals with a class of semilinear elliptic Dirichlet boundary value problems at resonance. We introduce a sufficient Landesman–Lazer condition for the existence of pair positive–negative solutions. Furthermore, developing the fibering method in the framework of the Leray–Schauder degree theory we can prove the existence of branches for positive and negative solutions.      18. Global existence of solution to the Camassa–Holm equation      《Nonlinear Analysis: Theory, Methods & Applications》2005,60(5):945-953       19. Local existence of solutions to the Euler–Poisson system,including densities without compact support      Uwe Brauer  Lavi Karp 《Journal of Differential Equations》2018,264(2):755-785 Local existence and well posedness for a class of solutions for the Euler Poisson system is shown. These solutions have a density ρ which either falls off at infinity or has compact support. The solutions have finite mass, finite energy functional and include the static spherical solutions for $\gamma =\frac{6}{5}$. The result is achieved by using weighted Sobolev spaces of fractional order and a new non-linear estimate which allows to estimate the physical density by the regularised non-linear matter variable. Gamblin also has studied this setting but using very different functional spaces. However we believe that the functional setting we use is more appropriate to describe a physical isolated body and more suitable to study the Newtonian limit.      20. Homotopy and existence of solutions for the nonlinear equation Lx∈Nx      《Nonlinear Analysis: Theory, Methods & Applications》2001,44(4):537-544

Analysis of the one-dimensional Euler–Lagrange equation of continuum mechanics with a Lagrangian of a special form

The equations describing the flow of a one-dimensional continuum in Lagrangian coordinates are studied in this paper by the group analysis method. They are reduced to a single Euler–Lagrange equation which contains two undetermined functions (arbitrary elements). Particular choices of these arbitrary elements correspond to different forms of the shallow water equations, including those with both, a varying bottom and advective impulse transfer effect, and also some other motions of a continuum. A complete group classification of the equations with respect to the arbitrary elements is performed.One advantage of the Lagrangian coordinates consists of the presence of a Lagrangian, so that the equations studied become Euler–Lagrange equations. This allows us to apply Noether’s theorem for constructing conservation laws in Lagrangian coordinates. Not every conservation law in Lagrangian coordinates has a counterpart in Eulerian coordinates, whereas the converse is true. Using Noether’s theorem, conservation laws which can be obtained by the point symmetries are presented, and their analogs in Eulerian coordinates are given, where they exist.     

Decay and stability of solutions to the Fokker–Planck–Boltzmann equation in

In this paper, we use the method of constructing the compensating function introduced by Kawashima and the standard energy method to study the global existence of solutions to the Fokker–Planck–Boltzmann equation in the whole space. The time decay and uniform stability of solutions to the global Maxwellian are also obtained.     

Global existence and temporal decay of large solutions for the Poisson–Nernst–Planck equations in low regularity spaces

We are concerned with the global existence and decay rates of large solutions for the Poisson–Nernst–Planck equations. Based on careful observation of algebraic structure of the equations and using the weighted Chemin–Lerner-type norm, we obtain the global existence and optimal decay rates of large solutions without requiring the summation of initial densities of a negatively and positively charged species that is small enough. Moreover, the large solution is obtained for initial densities belonging to the low regularity Besov spaces with different regularity and integral indices, which indicates more specific coupling relations between the difference and the summation of negatively and positively charged densities.     

Maximal and minimal solutions of an Aronsson equation: L ∞ variational problems versus the game theory

The Dirichlet problem $$ \left\{ \begin{array}{l}\Delta _\infty u - |Du|^2 = 0 \quad {\rm on} \, \Omega \subset {{\mathbb R}^n} \\ u|\partial \Omega = g \\\end{array} \right. $$ might have many solutions, where ${\Delta_{\infty}u=\sum_{1\leq i,j\leq n}u_{x_i}u_{x_j}u_{x_ix_j}}$ . In this paper, we prove that the maximal solution is the unique absolute minimizer for ${H(p,z)={\frac{1}{2}}|p|^2-z}$ from calculus of variations in L ^∞ and the minimal solution is the continuum value function from the “tug-of-war” game. We will also characterize graphes of solutions which are neither an absolute minimizer nor a value function. A remaining interesting question is how to interpret those intermediate solutions. Most of our approaches are based on an idea of Barles–Busca (Commun Partial Differ Equ 26(11–12):2323–2337, 2001).     

On the existence of pair positive–negative solutions for resonance problems

This paper deals with a class of semilinear elliptic Dirichlet boundary value problems at resonance. We introduce a sufficient Landesman–Lazer condition for the existence of pair positive–negative solutions. Furthermore, developing the fibering method in the framework of the Leray–Schauder degree theory we can prove the existence of branches for positive and negative solutions.     

Local existence of solutions to the Euler–Poisson system,including densities without compact support

Local existence and well posedness for a class of solutions for the Euler Poisson system is shown. These solutions have a density ρ which either falls off at infinity or has compact support. The solutions have finite mass, finite energy functional and include the static spherical solutions for $\gamma =\frac{6}{5}$. The result is achieved by using weighted Sobolev spaces of fractional order and a new non-linear estimate which allows to estimate the physical density by the regularised non-linear matter variable. Gamblin also has studied this setting but using very different functional spaces. However we believe that the functional setting we use is more appropriate to describe a physical isolated body and more suitable to study the Newtonian limit.