Critical exponent for evolution equations in modulation spaces

The dispersion of inertial particles continuously emitted from a point source is analytically investigated in the limit of small but finite inertia. Our focus is on the evolution equation of the particle joint probability density function p(x, v, t), x and v being the particle position and velocity, respectively. For arbitrary inertia, position and velocity variables are coupled, with the result that p(x, v, t) can be determined by solving a partial differential equation in a 2d-dimensional space, d being the physical-space dimensionality. For small (but nevertheless finite) inertia, (x, v)-variables decouple and the determination of p(x, v, t) is reduced to solve a system of two standard forced advection–diffusion equations in the space variables x. The latter equations are derived here from first principles, i.e., from the well-known Lagrangian evolution equations for position and particle velocity.

Well-posedness of the modified dispersion-generalized Benjamin-Ono equations in modulation spaces

Navier–Stokes equations and nonlinear heat equations in modulation spaces with negative derivative indices

This paper is concerned with the existence of \(\alpha \)-mild solutions for a class of fractional stochastic integro-differential evolution equations with nonlocal initial conditions in a real separable Hilbert space. We assume that the linear part generates a compact, analytic and uniformly bounded semigroup, the nonlinear part satisfies some local growth conditions in Hilbert space \(\mathbb {H}\) and the nonlocal term satisfies some local growth conditions in fractional power space \(\mathbb {H}_\alpha \). The result obtained in this paper improves and extends some related conclusions on this topic. An example is also given to illustrate the feasibility of our abstract result.

Sobolev type fractional abstract evolution equations with nonlocal conditions and optimal multi-controls

In this paper, we consider a class of fractional Kirchhoff equations with L^{2} critical exponents. By using the scaling technique and concentration-compactness principle we obtain the existence and nonexistence of ground state for fractional Kirchhoff equation with L^{2} critical exponent.

The study extends models of the cw-regime of operation of diffusion-controlled discharges by incorporating the actual shape of the high-frequency signal producing the discharge as a narrow-band signal. The slow variations of the wave envelope determining slowly varying Joule heating of the electrons in the wave field leads to a low-frequency plasma-density response which influences the propagation properties of the high-frequency signal. Depending on the frequency shift of the spectral components from the carrier-wave frequency of the signal, the low-frequency plasma response appears as stationary or nonstationary. The derived nonlinear evolution equation for the wave envelope shows cross-phase modulation acting simultaneously through self-action and mutual action of coupled spectral components. A strong impact of the mutual action of the coupled spectral components is found. It removes the effect of self-action: The decay of the spectral components associated with a stationary low-frequency response transforms into an instability, and the monotonic variations of the amplitudes of the spectral components associated with a nonstationary low-frequency plasma response are replaced by space modulation. For verification of the origin of the effects, also the case of axially homogeneous plasma columns is treated in which obtaining analytical solutions of the evolution equations is possible.

Using direct integration of the Vlassov equation in configurational space, nonlinear dynamics of a model one-dimensional periodic self-gravitating system are investigated near the point of a marginal stability. The critical velocity dispersion σ, corresponding to a marginal stability of the Jeans mode with the least k, is assumed as the critical point. The peak amplitude of the Jeans mode computed in a run is assumed as the order parameter. In the neighborhood of the critical point, the dynamics can be described by a set of the power laws typical for a system undergoing a second-order phase transition. For the order parameter, this is A ∝ -θβ, where β = 1.907 ± 0.006 and θ = (σ2 - σ)/σ 0. Under this accuracy, these critical exponents satisfy the equality γ± = β(δ - 1). For a gravitating system, its existence is a direct consequence of scaling invariance of the distribution function at |θ| 1; i.e., f(λt, x, λv, λθ, λA0, λF1) = λf(t, x, v, θ, A0, F1). These critical exponents indicate a wide critical area where critical phenomena may determine macroscopic dynamics. The random external drive field of a small amplitude causes anomalous growth of the marginally stable Jeans perturbation into a spatially coherent structure.

Global attractors for gradient flows in metric spaces

In this work, we interest in the study of the wellposedness and the stability of fractional stochastic nonlinear heat equation in the Hilbert space $$ L^{2}(0,1) $$ ; perturbed by a trace-class noise and driven by the fractional Laplacian. Precisely, we use the fixed point theorem to prove the wellposedness of the problem. Moreover, we prove the $$ p^{th} $$ -moment exponential stability and the almost surely exponential stability by imposing an additional assumption. Some examples are considered in order to confirm and support the validity of our theoretical results.

Singular backward stochastic Volterra integral equations in infinite dimensional spaces

This paper studies fractional differential equations (FDEs) with mixed fractional derivatives. Existence, uniqueness, stability, and asymptotic results are derived.

A general local existence theorem for quasi-linear evolution equations in nonreflexive banach spaces

Singularly perturbed evolution equations in Banach spaces

This Chapter makes use of the vector-valued distributions and ultra-distributions of Chapter 7. From the point of view of evolution boundary value problems, we assume the knowledge of the essentials of Chapter 3 and of the beginning of Chapter 4.