Classical Solutions to the Boltzmann Equation for Molecules with an Angular Cutoff

An important class of collision kernels in the Boltzmann theory are governed by the inverse power law, in which the intermolecular potential between two particles is an inverse power of their distance. Under the Grad angular cutoff assumption, global-in-time classical solutions near Maxwellians are constructed in a periodic box for all soft potentials with –3<<0.     

Cauchy Problem and Exponential Stability for the Inhomogeneous Landau Equation

This work deals with the inhomogeneous Landau equation on the torus in the cases of hard, Maxwellian and moderately soft potentials. We first investigate the linearized equation and we prove exponential decay estimates for the associated semigroup. We then turn to the nonlinear equation and we use the linearized semigroup decay in order to construct solutions in a close-to-equilibrium setting. Finally, we prove an exponential stability for such a solution, with a rate as close as we want to the optimal rate given by the semigroup decay.     

Global Classical Solutions for MHD System

In this paper we study the equations of magneto-hydrodynamics for a 2D incompressible ideal fluid in the exterior domain and in the half-plane. We prove the existence of a global classical solution in Hölder spaces, by applying Shauder fixed point theorem.     

Self-Similar Solutions for the Foam Drainage Equation

We provide travelling wave solutions of the equation for foam drainage in porous media, taking into account an additional symmetry requirement. The method of solution used is reminescent of the approach developed to treat the Rapoport–Leas equation for two-phase flow. Numerical solutions are also presented and compared to the analytical ones.     

Soliton Solutions for the Inverse Korteweg-De Vries Equation

We construct soliton solutions of the inverse Korteweg-de Vries equation by developing the tanh-function method and symbolic-computation techniques.     

Time-Periodic Solutions of the Burgers Equation

We investigate the time periodic solutions to the viscous Burgers equation u_t − μu_xx + uu_x = f for irregular forcing terms. We prove that the corresponding Burgers operator is a diffeomorphism between appropriate function spaces.      

A Note on the Removal of Further Breakdowns in Classical Solutions of Laplace"s Equation on Sectorial Regions

This note considers solutions to Laplace"s equation on sectors with varying vertex angles. Under Neumann conditions on the radial boundaries, there are two critical vertex angles for which classical separable solutions break down. These breakdowns have been noted in the literature and resolved. Here Dirichlet and mixed conditions are also treated. For all three boundary conditions in combination, nine further critical angles are identified and valid corresponding solutions found. This revised version was published online in August 2006 with corrections to the Cover Date.     

A Note on the Two-Dimensional Ginzburg–Landau Equation for Directional, Nearly Monochromatic Waves

In this paper we deal with the two-dimensional Ginzburg–Landauequation. First we simply expand the one-dimensional Ginzburg–Landauequation to the two-dimensional one. Then the concept of thedirectionality is imported into the two-dimensional Ginzburg–Landauequation. Directional, nearly monochromatic waves have a fixedwavenumber but spread over some propagation area in propagatingdirections. Moreover, most of the energy of waves is concentrated in asingle propagating direction. In slightly unstable, directional, nearlymonochromatic waves, the fact that the envelope surface created by theamplitude modulation is presented by the product of the solution of theSchrödinger–Nohara equation and the time function is shown. In thenonlinear case, the time function depends on space.     

Pointwise Bounds for the Solutions of the Smoluchowski Equation with Diffusion

We prove various decay bounds on solutions (f _n : n > 0) of the discrete and continuous Smoluchowski equations with diffusion. More precisely, we establish pointwise upper bounds on n ^? f _n in terms of a suitable average of the moments of the initial data for every positive ?. As a consequence, we can formulate sufficient conditions on the initial data to guarantee the finiteness of ${L^p(\mathbb{R}^d \times [0, T])}$ norms of the moments ${X_a(x, t) := \sum_{m\in\mathbb{N}}m^a f_m(x, t)}$ , ( ${\int_0^{\infty} m^a f_m(x, t)dm}$ in the case of continuous Smoluchowski’s equation) for every ${p \in [1, \infty]}$ . In previous papers [11] and [5] we proved similar results for all weak solutions to the Smoluchowski’s equation provided that the diffusion coefficient d(n) is non-increasing as a function of the mass. In this paper we apply a new method to treat general diffusion coefficients and our bounds are expressed in terms of an auxiliary function ${\phi(n)}$ that is closely related to the total increase of the diffusion coefficient in the interval (0, n].     

Smoothing Estimates for Boltzmann Equation with Full-range Interactions: Spatially Inhomogeneous Case

In this paper, we study the regularity of the solution to the Boltzmann equation with full-range interactions but for the spatially inhomogeneous case. Under the initial regularity assumption on the solution itself, we show that the solution will become immediately smooth for all the variables as long as the time is far way from zero. Our strategy relies upon the new upper and lower bounds for the collision operator established in Chen and He (Arch Ration Mech Anal 201(2):501–548, 2011), a hypo-elliptic estimate for the transport equation and the element energy method.     

Smoothing Estimates for Boltzmann Equation with Full-Range Interactions: Spatially Homogeneous Case

In this work, we are concerned with the regularities of the solutions to the Boltzmann equation with physical collision kernels for the full range of intermolecular repulsive potentials, r ^−(p−1) with p > 2. We give new and constructive upper and lower bounds for the collision operator in terms of standard weighted fractional Sobolev norms. As an application, we get the global entropy dissipation estimate which is a little stronger than that described by Alexandre et al. (Arch Rational Mech Anal 152(4):327–355, 2000). As another application, we prove the smoothing effects for the strong solutions constructed by Desvillettes and Mouhot (Arch Rational Mech Anal 193(2):227–253, 2009) of the spatially homogeneous Boltzmann equation with “true” hard potential and “true” moderately soft potential.     

Classical Solutions and the Glassey-Strauss Theorem for the 3D Vlasov-Maxwell System

R. Glassey and W. Strauss have proved in [Arch. Rational Mech. Anal. 92 (1986), 59–90] that C ¹ solutions to the relativistic Vlasov-Maxwell system in three space dimensions do not develop singularities as long as the support of the distribution function in the momentum variable remains bounded. The present paper simplifies their proof.     

Existence of Periodic Solutions for the Generalized Form of Mathieu Equation

The generalized form of the well-known Mathieu differential equation, which consists of two driving force terms, including the quadratic and cubic nonlinearities, has been analyzed in this paper. The two-dimensional Lindstedt–Poincarés perturbation technique has been considered in order to obtain the analytical solutions. The transition curves in some special cases have been presented. It is shown that the periodic solution does indeed exist and in general they are dependent on the initial conditions. Results of this analytical approach were compared with those obtained from the numerical methods and it is found that they are in a good agreement.     

Irregular Behavior of Solutions for Fisher’s Equation

This paper is concerned with the irregular behavior of solutions for Fisher’s equation when initial data do not decay in a regular way at the spatial infinity. In the one-dimensional case, we show the existence of a solution whose profile and average speed are not convergent. In the higher-dimensional case, we show the existence of expanding fronts with arbitrarily prescribed profiles. We also show the existence of irregularly expanding fronts whose profile varies in time. Proofs are based on some estimate of the difference of two distinct solutions and a comparison technique. Dedicated to Professor Pavol Brunovsky on his 70th birthday.     

Global Classical Solutions for Partially Dissipative Hyperbolic System of Balance Laws

The basic existence theory of Kato and Majda enables us to obtain local-in-time classical solutions to generally quasilinear hyperbolic systems in the framework of Sobolev spaces (in x) with higher regularity. However, it remains a challenging open problem whether classical solutions still preserve well-posedness in the case of critical regularity. This paper is concerned with partially dissipative hyperbolic system of balance laws. Under the entropy dissipative assumption, we establish the local well-posedness and blow-up criterion of classical solutions in the framework of Besov spaces with critical regularity with the aid of the standard iteration argument and Friedrichs’ regularization method. Then we explore the theory of function spaces and develop an elementary fact that indicates the relation between homogeneous and inhomogeneous Chemin–Lerner spaces (mixed space-time Besov spaces). This fact allows us to capture the dissipation rates generated from the partial dissipative source term and further obtain the global well-posedness and stability by assuming at all times the Shizuta–Kawashima algebraic condition. As a direct application, the corresponding well-posedness and stability of classical solutions to the compressible Euler equations with damping are also obtained.     

Quasi-Periodic Solutions for the Generalized Ginzburg-Landau Equation with Derivatives in the Nonlinearity

In this paper, we discuss the existence of time quasi-periodic solutions for the generalized Ginzburg-Landau equation under periodic boundary conditions. By constructing a KAM theorem for dissipative systems with unbounded perturbations and multiple normal frequencies, we obtain a Cantorian branch of 2-dimensional invariant tori for the generalized Ginzburg-Landau equation.     

Lack of Uniqueness for Weak Solutions of the Incompressible Porous Media Equation

In this work we consider weak solutions of the incompressible two-dimensional porous media (IPM) equation. By using the approach of De Lellis–Székelyhidi, we prove non-uniqueness for solutions in L ^∞ in space and time.     

Some Analytical Solutions for Groundwater Flow and Transport Equation

This paper presents the use of symmetry reduction method resulting in new exact solutions for the groundwater flow and transport equation. It is assumed that the radionuclides are transported by advection-diffusion in a single fracture and diffusion in the surrounding rock-matrix. The application of one-parameter group reduces the number of independent variables, and consequently the governing PDE of (1+2)-dimension reduces to set of ODEs which are solved analytically. This enables us to present some new exact time-dependent solutions of the advection-diffusion equation.