Exclusion of boundary blowup for 2D chemotaxis system provided with Dirichlet boundary condition for the Poisson part

We study a chemotaxis system on bounded domain in two dimensions where the formation of chemical potential is subject to the Dirichlet boundary condition. For such a system the solution is kept bounded near the boundary and hence the blowup set is composed of a finite number of interior points. If the initial total mass is 8π and the domain is close to a disc then the solution exhibits a collapse in infinite time of which movement is subject to a gradient flow associated with the Robin function.     

Steady subsonic potential flows through infinite multi-dimensional largely-open nozzles

We proved existence, regularity and uniqueness of steady subsonic potential flows in n-dimensional (n ≥ 3) infinite nozzles with largely-open convergent and divergent parts when the total mass flux is less than a certain value. Such nozzles consist of two cones with arbitrary open angles and an arbitrary smooth bounded tubular part. The existence of a weak solution is proved by applying the direct method of calculus of variation to a carefully chosen functional defined on a Hilbert space based upon Hardy inequality. Hölder gradient regularity of weak solution is shown by using Moser iteration to quasilinear elliptic equations in divergence form. Also, the obtained solution is unique in the class of functions with finite kinetic energy by modulo a constant.     

Soliton Splitting by External Delta Potentials

We show that in the dynamics of the nonlinear Schrodinger equation a soliton scattered by an external delta potential splits into two solitons and a radiation term. Theoretical analysis gives the amplitudes and phases of the reflected and transmitted solitons with errors going to zero as the velocity of the incoming soliton tends to infinity. Numerical analysis shows that this asymptotic relation is valid for all but very slow solitons. We also show that the total transmitted mass, that is, the square of the L² norm of the solution restricted on the transmitted side of the delta potential, is in good agreement with the quantum transmission rate of the delta potential.     

On optimal design of vibrating structures

Optimal design problems of linearly elastic vibrating structural members have been formulated in two ways. One is to minimize the total mass holding the frequency fixed; the other is to maximize the fundamental frequency holding the total mass fixed. Generally, these two formulations are equivalent and lead to the same solution. It is shown in this work that the equivalence is lost when the design variable (the specific stiffness) appears linearly in Rayleigh's quotient and when there is no nonstructural mass. The maximum-frequency formulation then is a normal Lagrange problem, whereas the minimum-mass problem is abnormal. The lack of recognition of this can lead to incorrect conclusions, particularly concerning existence of solutions. It is shown that existence depends directly on the boundary conditions and, when a sandwich beam has a free end, a solution to the maximum-frequency problem does not exist.     

Regularity of inverse mean curvature flow in asymptotically hyperbolic manifolds with dimension 3

By using the nice behavior of the Hawking mass of the slices of a weak solution of inverse mean curvature flow in three-dimensional asymptotically hyperbolic manifolds, we are able to show that each slice of the flow is star-shaped after a long time, and then we get the regularity of the weak solution of inverse mean curvature flow in asymptotically hyperbolic manifolds. As an application, we prove that the limit of the Hawking mass of the slices of a weak solution of inverse mean curvature flow with any connected C~2-smooth surface as initial data in asymptotically anti-de Sitter-Schwarzschild manifolds with positive mass is greater than or equal to the total mass, which is completely different from the situation in the asymptotically flat case.     

On the Cauchy problem for nonlinear dissipative systems

The large-time asymptotic behavior of solutions of the Cauchy problem for a system of nonlinear evolutionary equations with dissipation is studied. The approach used in the case of small initial data is based on the construction of solutions by the method of contracting mappings. In the case of large initial data, we will obtain the large-time asymptotics of solutions with a certain symmmetry of a nonlinear term taken into account. In the critical case, it is proved that if the initial data has a nonzero total mass, then the principal term of the large-time asymptotics of a solution is given by the self-similar solution uniquely determined by the total mass of the initial data. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 29, Voronezh Conference-1, 2005.     

Initial conditions for the cylindrical Korteweg‐de Vries equation

In this paper, we find suitable initial conditions for the cylindrical Korteweg‐de Vries equation by first solving exactly the initial‐value problem for localized solutions of the underlying axisymmetric linear long‐wave equation. The far‐field limit of the solution of this linear problem then provides, through matching, an initial condition for the cylindrical Korteweg‐de Vries equation. This initial condition is associated only with the leading wave front of the far‐field limit of the linear solution. The main motivation is to resolve the discrepancy between the exact mass conservation law, and the “mass” conservation law for the cylindrical Korteweg‐de Vries equation. The outcome is that in the linear initial‐value problem all the mass is carried behind the wave front, and then the “mass” in the initial condition for the cylindrical Korteweg‐de Vries equation is zero. Hence, the evolving solution in the cylindrical Korteweg‐de Vries equation has zero “mass.” This situation arises because, unlike the well‐known unidirectional Korteweg‐de Vries equation, the solution of the initial‐value problem for the axisymmetric linear long‐wave problem contains both outgoing and ingoing waves, but in the cylindrical geometry, the latter are reflected at the origin into outgoing waves, and eventually the total outgoing solution is a combination of these and those initially generated.     

Global subsonic compressible flows through a two-dimensional periodic duct

We deal with two-dimensional compressible potential subsonic flows in an infinitely long duct with periodic walls. It is shown that there exists a critical value of mass flux: If the incoming mass flux is less than the critical value, then the flow is also periodic. Existence, uniqueness and regularity of the periodic solution are obtained by techniques of elliptic equations.     

The rate of concentration for the radially symmetric solution to a degenerate drift-diffusion equation with the mass critical exponent

We consider the concentration rate of the total mass for radially symmetric blow-up solutions to the Cauchy problem of a degenerate drift-diffusion system with the mass critical exponent. We proved that the radially symmetric solution blows up in finite time when the initial data has negative free energy. We show that the mass concentration phenomenon occurs with the sharp lower constant related to the best constant of the Hardy–Littlewood–Sobolev inequality and the concentration rate of the total mass.     

Functions of a complex variable in problems in the theory of elasticity with mass forces

The general equations of the theory of elasticity are reduced to an inhomogeneous fourth-order equation assuming that there is a linear dependence of the third component of the displacement vector on the third coordinate and that a mass force potential exists. The solution of this equation is presented, in particular, using two complex Kolosov–Muskhelishvili potentials. A third complex potential is introduced in addition to these. Using the three complex potentials, expressions are obtained for the components of the displacement vector and the stress and strain tensors that take account of mass forces. The application of the three potentials is analysed in problems in the theory of elasticity, and analytical solutions of several plane strain problems are presented.     

Reproducing-kernel-based splines for the regularization of the inverse ellipsoidal gravimetric problem

To solve the inverse gravimetric problem, i.e. to reconstruct the Earth's mass density distribution by using the gravitational potential, we introduce a spline interpolation method for the ellipsoidal Earth model, where the ellipsoid has a rotational symmetry. This problem is ill-posed in the sense of Hadamard as the solution may not exist, it is not unique and it is not stable. Since the anharmonic part (orthogonal complement) of the density function produces a zero potential, we restrict our attention only to reconstruct the harmonic part of the density function by using the gravitational potential. This spline interpolation method gives the existence and uniqueness of the unknown solution. Moreover, this method represents a regularization, i.e. every spline continuously depends on the given gravitational potential. These splines are also combined with a multiresolution concept, i.e. we get closer and closer to the unknown solution by increasing the scale and adding more and more data at each step.     

AN EXPLICIT MULTI-CONSERVATION FINITE-DIFFERENCE SCHEME FOR SHALLOW-WATER-WAVE EQUATION

An explicit multi-conservation finite-difference scheme for solving the spherical shallow-water-wave equation set of barotropic atmosphere has been proposed. The numerical scheme is based on a special semi-discrete form of the equations that conserves four basic physical integrals including the total energy, total mass, total potential vorticity and total enstrophy. Numerical tests show that the new scheme performs closely like but is much more time-saving than the implicit multi-conservation scheme.     

A stable and conservative finite difference scheme for the Cahn-Hilliard equation

Summary. We propose a stable and conservative finite difference scheme to solve numerically the Cahn-Hilliard equation which describes a phase separation phenomenon. Numerical solutions to the equation is hard to obtain because it is a nonlinear and nearly ill-posed problem. We design a new difference scheme based on a general strategy proposed recently by Furihata and Mori. The new scheme inherits characteristic properties, the conservation of mass and the decrease of the total energy, from the equation. The decrease of the total energy implies boundedness of discretized Sobolev norm of the solution. This in turn implies, by discretized Sobolev's lemma, boundedness of max norm of the solution, and hence the stability of the solution. An error estimate for the solution is obtained and the order is . Numerical examples demonstrate the effectiveness of the proposed scheme. Received July 22, 1997 / Revised version received October 19, 1999 / Published online August 2, 2000     

Singular perturbed problems in the zero mass case: asymptotic behavior of spikes

We discuss the asymptotic behavior of the least energy solution of a Dirichlet problem in the zero mass case. If Q is a uniformly positive potential having k isolated local minima, then we prove the existence of a positive multi-spike solutions having k peaks concentrating at each local minima of the potential.     

A concept of a robust solution of a multicriterial linear programming problem

A new concept of a robust solution of a multicriterial linear programming problem is proposed. The robust solution is understood here as the best starting point, prepared while the preferences of the decision maker with respect to the criteria are still unknown, for the adaptation of the solution to the preferences of the decision maker, once they are finally known. The objective is the total cost of the initial preparation and of the later potential adaptation of the solution. In the starting robust solution the decision variables may have interval values. The problem can be solved by means of the simplex algorithm. A numerical example illustrates the approach.     

Relativistic theory of gravity and a Brans-Dicke scalar field

The natural generalization of the relativistic theory of gravity (RTG) by incorporating a Brans-Dicke scalar field is discussed. The equation for a scalar-tensor gravitational field in Minkowski space and the expression for the total energy-momentum metric tensor of a gravitational field and nongravitational matter is derived from the variational principle with a gravitational Lagrangian quadratic in the first derivatives of the scalar and tensor gravitational potentials. The two-parameter spherically symmetrical static solution for vacuum equations with a zero mass tensor graviton was obtained. This solution has a true singular Schwarzschild surface. In the case of a nonzero mass graviton, an approximate nonsingular solution for the beginning of the universe was obtained. It is noted that in the frame of the scalar-tensor generalization of RTG, a nonsingular homogeneous isotropic cosmology can be represented, not only by cyclic models, but also by models with an infinitely expanding universe and a simultaneously decreasing gravitational scalar.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 106, No. 2, pp. 325–332, February, 1996.     

GLOBAL SOLUTION TO THE CAUCHY PROBLEM ON A UNIVERSE FIREWORKS MODEL

We prove existence and uniqueness of the global solution to the Cauchy problem on a universe fireworks model with finite total mass at the initial state when the ratio of the mass surviving the explosion, the probability of the explosion of fragments and the probability function of the velocity change of a surviving particle satisfy the corresponding physical conditions. Although the nonrelativistic Boltzmann-like equation modeling the universe fireworks is mathematically easy, this article leads rather theoretically to an understanding of how to construct contractive mappings in a Banach space for the proof of the existence and uniqueness of the solution by means of methods taken from the famous work by DiPerna ＆ Lions about the Boltzmann equation. We also show both the regularity and the time-asymptotic behavior of solution to the Cauchy problem.     

Minimal Blow-Up Solutions to the Mass-Critical Inhomogeneous NLS Equation

We consider the mass-critical focusing nonlinear Schrödinger equation in the presence of an external potential, when the nonlinearity is inhomogeneous. We show that if the inhomogeneous factor in front of the nonlinearity is sufficiently flat at a critical point, then there exists a solution which blows up in finite time with the maximal (unstable) rate at this point. In the case where the critical point is a maximum, this solution has minimal mass among the blow-up solutions. As a corollary, we also obtain unstable blow-up solutions of the mass-critical Schrödinger equation on some surfaces. The proof is based on properties of the linearized operator around the ground state, and on a full use of the invariances of the equation with an homogeneous nonlinearity and no potential, via time-dependent modulations.     

Enforcing the continuity equation in numerical models of geophysical fluid flows

A method is described for modifying the velocity field in geophysical fluid models so as to enforce the continuity equation. A corrective mass flux is introduced, which derives from a scalar potential. The latter is the solution of a Poisson problem which is formulated in such a way that a suitable norm of the corrective velocity be minimum. It is seen that a generalised vertical coordinate may be used. Finally, an elementary, one-dimensional illustration of the functioning of the method suggested is provided.