Convergence of the Ostrovsky equation to the Ostrovsky–Hunter one

We consider the Ostrovsky equation, which contains nonlinear dispersive effects. We prove that as the diffusion parameter tends to zero, the solutions of the dispersive equation converge to discontinuous weak solutions of the Ostrovsky–Hunter equation. The proof relies on deriving suitable a priori estimates together with an application of the compensated compactness method in the L^p$L^p$ setting.     

Oleinik type estimates for the Ostrovsky–Hunter equation

The Ostrovsky–Hunter equation provides a model of small-amplitude long waves in a rotating fluid of finite depth. This is a nonlinear evolution equation. In this study, we consider the well-posedness of the Cauchy problem associated with this equation within a class of bounded discontinuous solutions. We show that we can replace the Kruzkov-type entropy inequalities with an Oleinik-type estimate and we prove the uniqueness via a nonlocal adjoint problem. This implies that a shock wave in an entropy weak solution to the Ostrovsky–Hunter equation is admissible only if it jumps down in value (similar to the inviscid Burgers' equation).     

Well-posedness for the system of the Hamilton–Jacobi and the continuity equations

Let H (t, x, p) be a Hamiltonian function that is convex in p. Let the associated Lagrangian satisfy the nonstandard minorization condition where m > 0, ω > 0, and C ≥ 0 are constants. Under some additional conditions, we prove that the associated value function is the unique viscosity solution of S _t + H(t, x, ∇S) = 0 in , without any conditions at infinity on the solution. Here ωT < π/2. To the Hamilton–Jacobi equation corresponding to the classical action integrand in mechanics, we adjoin the continuity equation and establish the existence and uniqueness of a viscosity–measure solution (S, ρ) of

Symmetry and Classification of the Dirac–Fock Equation

We consider the properties of the Dirac–Fock equation with differential operators of the first-order symmetry. For a relativistic particle in an electromagnetic field, we describe the covariant properties of the Dirac equation in an arbitrary Riemannian space V₄ with the signature (?1,?1,?1, 1). We present a general form of the differential operator with a first-order symmetry and characterize the pair of such commuting operators. We list the spaces where the free Dirac equation admits at least one differential operator with a first-order symmetry. We perform a symmetry classification of electromagnetic field tensors and construct complete sets of symmetry operators.     

Well-posedness of the fractional Ginzburg–Landau equation

In this paper, we investigate the well-posedness of the real fractional Ginzburg–Landau equation in several different function spaces, which have been used to deal with the Burgers’ equation, the semilinear heat equation, the Navier–Stokes equations, etc. The long time asymptotic behavior of the nonnegative global solutions is also studied in details.     

Well-posedness of the stochastic KdV–Burgers equation

We are interested in rigorously proving the invariance of white noise under the flow of a stochastic KdV–Burgers equation. This paper establishes a result in this direction. After smoothing the additive noise (by a fractional spatial derivative), we establish (almost sure) local well-posedness of the stochastic KdV–Burgers equation with white noise as initial data. Next we observe that spatial white noise is invariant under the projection of this system to the first N>0$N>0$ modes of the trigonometric basis. Finally, we prove a global well-posedness result under an additional smoothing of the noise.     

Emergent dynamics of Cucker–Smale particles under the effects of random communication and incompressible fluids

We study the dynamics of infinitely many Cucker–Smale (C–S) flocking particles under the interplay of random communication and incompressible fluids. For the dynamics of an ensemble of flocking particles, we use the kinetic Cucker–Smale–Fokker–Planck (CS–FP) equation with a degenerate diffusion, whereas for the fluid component, we use the incompressible Navier–Stokes (N–S) equations. These two subsystems are coupled via the drag force. For this coupled model, we present the global existence of weak and strong solutions in ${\mathbb{R}}^d$$(d=2,3)$. Under the extra regularity assumptions of the initial data, the unique solvability of strong solutions is also established in ${\mathbb{R}}^2$. In a large coupling regime and periodic spatial domain ${\mathbb{T}}^2:={\mathbb{R}}^2/{\mathbb{Z}}^2$, we show that the velocities of C–S particles and fluids are asymptotically aligned to two constant velocities which may be different.     

Well-posedness and analytic solutions of the two-component Euler–Poincaré system

We first establish the local well-posedness for the Cauchy problem of the two-component Euler–Poincaré system in nonhomogeneous Besov spaces. Then, we derive a blow-up criterion for strong solutions to the system. Finally, we prove the existence of analytic solutions to the system.     

Integrability aspects of NLS–MB system with variable dispersion and nonlinear effects

In this paper, we analyze the integrability aspects of the NLS–MB system with variable dispersion and nonlinear effects. We obtain the constraints for which the above system becomes integrable by using the Painlevé singularity analysis. Obtained results are in agreement with the known results.     

On the Generalized Hurwitz Equation and the Baragar–Umeda Equation

We consider the generalized Hurwitz equation \({a_1x_1^2 + \cdots + a_nx_n^2 = dx_1 \cdots x_n - k}\) and the Baragar–Umeda equation \({ax^2 + by^2 + cz^2 = dxyz + e}\) for solvability in integers.     

On the short-wave nature of Richtmyer–Meshkov instability

In the case of a variable period (wavelength) of a perturbed interface, the instability and stability of Richtmyer–Meshkov vortices in perfect gas and incompressible perfect fluid, respectively, are investigated numerically and analytically. Taking into account available experiments, the instability of the interface between the argon and xenon in the case of a relatively small period is modeled. An estimate of the magnitude of the critical period is given. The nonlinear (for arbitrary initial conditions) stability of the corresponding steady-state vortex flow of perfect fluid in a strip (vertical periodic channel) in the case of a fairly large period is shown.     

Stability of smooth solitary waves for the generalized Korteweg–de-Vries equation with combined dispersion

The problem of orbital stability of smooth solitary waves in the generalized Korteweg–de-Vries equation with combined dispersion is considered. The results show that the smooth solitary waves are stable for any speed of wave propagation.     

Well-posedness of weak and strong solutions to the kinetic Cucker–Smale model

In this paper, we develop a unified framework that can be used to establish the well-posedness of kinetic Cucker–Smale model with or without noise, for general initial data regardless of the supports; meanwhile we rigorously justify the vanishing noise limit. Our proof is based on weighted energy estimates and the velocity averaging lemma in kinetic theory.     

Numerical conservative solutions of the Hunter–Saxton equation

BIT Numerical Mathematics - In the article a convergent numerical method for conservative solutions of the Hunter–Saxton equation is derived. The method is based on piecewise linear...     

Non-existence and existence of localized solitary waves for the two-dimensional long-wave–short-wave interaction equations

In this study, we establish the non-existence and existence results for the localized solitary waves of the two-dimensional long-wave–short-wave interaction equations. Both the non-existence and existence results are based on Pohozaev-type identities. We prove the existence of solitary waves by showing that the solitary waves are the minimizers of an associated variational problem.     

Hydrodynamic Coherence and Vortex Solutions of the Euler–Helmholtz Equation

The form of the general solution of the steady-state Euler–Helmholtz equation (reducible to the Joyce–Montgomery one) in arbitrary domains on the plane is considered. This equation describes the dynamics of vortex hydrodynamic structures.     

The Thomas–Fermi Problem and Solutions of the Emden–Fowler Equation

Computational Mathematics and Mathematical Physics - A two-point boundary value problem is considered for the Emden–Fowler equation, which is a singular nonlinear ordinary differential...