The Existence of Strong Solution for Generalized Navier-Stokes Equations with p x -Power Law under Dirichlet Boundary Conditions

In this note, in 2D and 3D smooth bounded domain, we show the existence of strong solution for generalized Navier-Stokes equation modeling by p x -power law with Dirichlet boundary condition under the restriction 3 n / n + 2 n + 2 < p x < 2 n + 1 / n − 1 . In particular, if we neglect the convective term, we get a unique strong solution of the problem under the restriction 2 n + 1 / n + 3 < p x < 2 n + 1 / n − 1 , which arises from the nonflatness of domain.

On blow up for the energy super critical defocusing nonlinear Schrödinger equations

We consider the energy supercritical defocusing nonlinear Schrödinger equation $$\begin{aligned} i\partial _tu+\Delta u-u|u|^{p-1}=0 \end{aligned}$$ i ∂ t u + Δ u - u | u | p - 1 = 0 in dimension $$d\ge 5$$ d ≥ 5 . In a suitable range of energy supercritical parameters (d, p), we prove the existence of $${\mathcal {C}}^\infty $$ C ∞ well localized spherically symmetric initial data such that the corresponding unique strong solution blows up in finite time. Unlike other known blow up mechanisms, the singularity formation does not occur by concentration of a soliton or through a self similar solution, which are unknown in the defocusing case, but via a front mechanism. Blow up is achieved by compression for the associated hydrodynamical flow which in turn produces a highly oscillatory singularity. The front blow up profile is chosen among the countable family of $${\mathcal {C}}^\infty $$ C ∞ spherically symmetric self similar solutions to the compressible Euler equation whose existence and properties in a suitable range of parameters are established in the companion paper (Merle et al. in Preprint (2019)) under a non degeneracy condition which is checked numerically.

The stochastic tamed MHD equations: existence, uniqueness and invariant measures

We study the tamed magnetohydrodynamics equations, introduced recently in a paper by the author, perturbed by multiplicative Wiener noise of transport type on the whole space $${\mathbb {R}}^{3}$$ R 3 and on the torus $${\mathbb {T}}^{3}$$ T 3 . In a first step, existence of a unique strong solution are established by constructing a weak solution, proving that pathwise uniqueness holds and using the Yamada–Watanabe theorem. We then study the associated Markov semigroup and prove that it has the Feller property. Finally, existence of an invariant measure of the equation is shown for the case of the torus.

The primitive equations in the scaling-invariant space $$L^{\infty }(L^1)$$

Consider the primitive equations on $$\mathbb {R}^2\times (z_0,z_1)$$ R 2 × ( z 0 , z 1 ) with initial data a of the form $$a=a_1+a_2$$ a = a 1 + a 2 , where $$a_1 \in \mathrm{BUC}_\sigma (\mathbb {R}^2;L^1(z_0,z_1))$$ a 1 ∈ BUC σ ( R 2 ; L 1 ( z 0 , z 1 ) ) and $$a_2 \in L^\infty _\sigma (\mathbb {R}^2;L^1(z_0,z_1))$$ a 2 ∈ L σ ∞ ( R 2 ; L 1 ( z 0 , z 1 ) ) . These spaces are scaling-invariant and represent the anisotropic character of these equations. It is shown that for $$a_1$$ a 1 arbitrary large and $$a_2$$ a 2 sufficiently small, this set of equations admits a unique strong solution which extends to a global one and is thus strongly globally well posed for these data provided a is periodic in the horizontal variables. The approach presented depends crucially on mapping properties of the hydrostatic Stokes semigroup in the $$L^\infty (L^1)$$ L ∞ ( L 1 ) -setting. It can be seen as the counterpart of the classical iteration schemes for the Navier–Stokes equations, now for the primitive equations in the $$L^\infty (L^1)$$ L ∞ ( L 1 ) -setting.

Stability and prevalence of McKean–Vlasov stochastic differential equations with non-Lipschitz coefficients

We consider various approximation properties for systems driven by a McKean–Vlasov stochastic differential equations (MVSDEs) with continuous coefficients, for which pathwise uniqueness holds. We prove that the solution of such equations is stable with respect to small perturbation of initial conditions, parameters and driving processes. Moreover, the unique strong solutions may be constructed by an effective approximation procedure. Finally, we show that the set of bounded uniformly continuous coefficients for which the corresponding MVSDE have a unique strong solution is a set of second category in the sense of Baire.

Stochastic mathematical model for the spread and control of Corona virus

This work is devoted to a stochastic model on the spread and control of corona virus (COVID-19), in which the total population of a corona infected area is divided into susceptible, infected, and recovered classes. In reality, the number of individuals who get disease, the number of deaths due to corona virus, and the number of recovered are stochastic, because nobody can tell the exact value of these numbers in the future. The models containing these terms must be stochastic. Such numbers are estimated and counted by a random process called a Poisson process (or birth process). We construct an SIR-type model in which the above numbers are stochastic and counted by a Poisson process. To understand the spread and control of corona virus in a better way, we first study the stability of the corresponding deterministic model, investigate the unique nonnegative strong solution and an inequality managing of which leads to control of the virus. After this, we pass to the stochastic model and show the existence of a unique strong solution. Next, we use the supermartingale approach to investigate a bound managing of which also leads to decrease of the number of infected individuals. Finally, we use the data of the COVOD-19 in USA to calculate the intensity of Poisson processes and verify our results.

Stochastic Tamed Navier–Stokes Equations on $${\mathbb {R}}^3$$: The Existence and the Uniqueness of Solutions and the Existence of an Invariant Measure

Röckner and Zhang (Probab Theory Relat Fields 145, 211–267, 2009) proved the existence of a unique strong solution to a stochastic tamed 3D Navier–Stokes equation in the whole space and for the periodic boundary case using a result from Stroock and Varadhan (Multidimensional diffusion processes, Springer, Berlin, 1979). In the latter case, they also proved the existence of an invariant measure. In this paper, we improve their results (but for a slightly simplified system) using a self-contained approach. In particular, we generalise their result about an estimate on the $$L^4$$ L 4 -norm of the solution from the torus to $${\mathbb {R}}^3$$ R 3 , see Lemma 5.1 and thus establish the existence of an invariant measure on $${\mathbb {R}}^3$$ R 3 for a time-homogeneous damped tamed 3D Navier–Stokes equation, given by (6.1).

Fatigue and phase transition in an oscillating elastoplastic beam

We study a model of fatigue accumulation in an oscillating elastoplastic beam under the hypothesis that the material can partially recover by the effect of melting. The model is based on the idea that the fatigue accumulation is proportional to the dissipated energy. We prove that the system consisting of the momentum and energy balance equations, an evolution equation for the fatigue rate, and a differential inclusion for the phase dynamics admits a unique strong solution.

An existence and uniqueness theorem for the dynamics of flexural shells

In this paper, we define, a priori, a natural two-dimensional model for a time-dependent flexural shell. As expected, this model takes the form of a set of hyperbolic variational equations posed over the space of admissible linearized inextensional displacements, and a set of initial conditions. Using a classical argument, we prove that the model under consideration admits a unique strong solution. However, the latter strategy makes use of function spaces, which are not amenable for numerically approximating the solution. We thus provide an alternate formulation of the studied problem using a suitable penalty scheme, which is more suitable in the context of numerical approximations. For the sake of completeness, in the final part of the paper, we also provide an existence and uniqueness theorem for the case where the linearly elastic shell under consideration is an elliptic membrane shell.

The global Cauchy problem for compressible Euler equations with a nonlocal dissipation

This paper studies the global existence and uniqueness of strong solutions and its large-time behavior for the compressible isothermal Euler equations with a nonlocal dissipation. The system is rigorously derived from the kinetic Cucker–Smale flocking equation with strong local alignment forces and diffusions through the hydrodynamic limit based on the relative entropy argument. In a perturbation framework, we establish the global existence of a unique strong solution for the system under suitable smallness and regularity assumptions on the initial data. We also provide the large-time behavior of solutions showing the fluid density and the velocity converge to its averages exponentially fast as time goes to infinity.