Small Initial Data Research Articles

Large BV solution for isentropic gas dynamics system with initial data far from vacuum

We give the global existence results for the isentropic gas dynamics system with large BV initial data, assuming that the initial data are far from vacuum. The distance of the initial data from vacuum may be so large depending on the initial BV bounds. A new scaling framework is introduced, and we obtain the crucial BV bounds estimates of the vanishing viscosity solutions with the help of this framework coupled with the famous result for the corresponding small BV initial data given by Bianchini and Bressan.

On the Jordan\u2013Moore\u2013Gibson\u2013Thompson Wave Equation in Hereditary Fluids with Quadratic Gradient Nonlinearity

We prove global solvability of the third-order in time Jordan–More–Gibson–Thompson acoustic wave equation with memory in {mathbb {R}}^n, where n ge 3. This wave equation models ultrasonic propagation in relaxing hereditary fluids and incorporates both local and cumulative nonlinear effects. The proof of global existence is based on a sequence of high-order energy bounds that are uniform in time, and derived under the assumption of an exponentially decaying memory kernel and sufficiently small and regular initial data.

Global Well-Posedness for the Navier–Stokes Equations with the Coriolis Force in Function Spaces Characterized by Semigroups

We consider the initial value problem for the Navier–Stokes equations with the Coriolis force. We introduce function spaces of the Besov type characterized by the time evolution semigroup associated with the linear Stokes–Coriolis operator. Then, we show the unique existence of global in time mild solutions for small initial data belonging to our function spaces in both the scaling subcritical and critical settings.

Blow-up and global existence for solutions to the porous medium equation with reaction and fast decaying density

We are concerned with nonnegative solutions to the Cauchy problem for the porous medium equation with a variable density ρ(x) and a power-like reaction term up with p>1. The density decays fast at infinity, in the sense that ρ(x)∼|x|−q as |x|→+∞ with q≥2. In the case when q=2, if p is bigger than m, we show that, for large enough initial data, solutions blow-up in finite time and for small initial datum, solutions globally exist. On the other hand, in the case when q>2, we show that existence of global in time solutions always prevails. The case of slowly decaying density at infinity, i.e. q∈[0,2), is examined in Meglioli and Punzo (2020).

On initial-boundary value problems for one-dimension semilinear wave equations with null conditions

We consider the initial-boundary value problem on R+×R+ for one-dimension systems of semilinear wave equations with null conditions. We show that for homogeneous Dirichlet or Neumann boundary values and sufficiently small initial data, classical solutions always globally exist. Furthermore, we also prove that the global solution is asymptotically free in the energy sense.

Global dynamics of the generalized fifth-order KdV equation with quintic nonlinearity

We prove global existence and modified scattering for the solutions of the generalized fifth-order KdV equation with critical nonlinearity for small and localized initial data. The proof is undergoing by using the space-time resonance method and the stationary phase argument.

Remarks on global solutions for the semilinear diffusion equation in the de Sitter spacetime

The Cauchy problem for the semilinear diffusion equation is considered in the de Sitter spacetime with the spatial zero-curvature. Global solutions and their asymptotic behaviors for small initial data are obtained for positive and negative Hubble constants. The effects of the spatial expansion and contraction are studied on the problem.

Optimal time decay rates for the compressible Navier-Stokes system with and without Yukawa-type potential

We consider the time decay rates of smooth solutions to the Cauchy problem for the compressible Navier-Stokes system with and without a Yukawa-type potential. We prove the existence and uniqueness of global solutions by the standard energy method under small initial data assumptions. Furthermore, if the initial data belong to \(L^1(\mathbb R^3)\), we establish the optimal time decay rates of the solution as well as its higher-order spatial derivatives. In particular, we obtain the optimal decay rates of the highest-order spatial derivatives of the velocity. Finally, we derive the lower bound time decay rates for the solution and its spacial derivatives. For more information see https://ejde.math.txstate.edu/Volumes/2020/102/abstr.html

Semilinear mixed problems in exterior domains for σ-evolution equations with friction and coefficients depending on spatial variables

The main purpose of this paper is to investigate decay estimates for solutions to the Cauchy problemutt+a1(x)(−Δ)σu+aut=0,u(0,x)=u0(x),ut(0,x)=u1(x)forx∈Rn, as well as the estimates for solutions to the corresponding Cauchy-Dirichlet problem in an exterior domain Ω⊂Rn. Here a is a positive constant. The coefficient a1=a1(x) is supposed to be continuous and positive on the closure Ω‾. The parameter σ∈(0,1) brings to the model the so-called Levi-stable behavior for the corresponding diffusion stochastic process. Finally, we show the global (in time) existence of energy solutions from evolution spaces to the semilinear modelsutt+a1(x)(−Δ)σu+aut=|ut|p,u(0,x)=u0(x),ut(0,x)=u1(x), in domain (0,∞)×Ω with arbitrarily small initial data.

Strong solution for Korteweg system in bmo−1(RN) with initial density in L∞

In this paper we investigate the question of the existence of strong solution in finite time for the Korteweg system for small initial data provided that the initial momentum ρ 0 u 0 belongs to bmo −1 T (R N) for T > 0 and the initial density ρ 0 is in L ∞ (R N) with N ≥ 1 and far away from the vacuum. This result extends the so called Koch-Tataru theorem for the Korteweg system. It is also interesting to observe that any initial shock on the density is instantaneously regularized inasmuch as the density becomes Lipschitz for any ρ(t, ·) with t > 0. We also prove the existence of global strong solution for initial data (ρ 0 − 1, ρ 0 u 0) ∈ (B N 2 −1 2,∞ (R N) ∩ B N 2 2,∞ (R N)∩L ∞ (R N))×(B N 2 −1 2,∞ (R N)) N. This result allows in particular to extend the notion of Oseen solution (corresponding to particular solution of the incompressible Navier Stokes system in dimension N = 2) to the Korteweg system provided that the vorticity of the momentum ρ 0 u 0 is a Dirac mass αδ 0 with α sufficiently small. IHowever unlike the Navier Stokes equations the property of self similarity is not conserved for the Korteweg system since there is no invariance by scaling because the term of pressure.

Global existence of the three‐dimensional compressible Euler equations for generalized Chaplygin gas with damping

In this paper, we establish a global existence (GE) result for the three‐dimensional compressible Euler equations (CEE) for generalized Chaplygin gas (GCG) with damping. More precisely, by transforming the three‐dimensional CEE for GCG with damping to a symmetric hyperbolic system, one shows that the total energy in H3 is strictly decaying with time for any sufficiently small initial data. As a result, the GE result follows from the local well‐posedness of the system in H3. This gives the first GE result relating to the three‐dimensional CEE.

Global regularity for solutions of the three dimensional Navier–Stokes equation with almost two dimensional initial data

In this paper, we will prove a new result that guarantees the global existence of solutions to the Navier–Stokes equation in three dimensions when the initial data is sufficiently close to being two dimensional. This result interpolates between the global existence of smooth solutions for the two dimensional Navier–Stokes equation with arbitrarily large initial data, and the global existence of smooth solutions for the Navier–Stokes equation in three dimensions with small initial data in . This result states that the closer the initial data is to being two dimensional, the larger the initial data can be in while still guaranteeing the global existence of smooth solutions. In the whole space, this set of almost two dimensional initial data is unbounded in the critical space , but is bounded in the critical Besov spaces for all 2 < p ⩽ +∞. On the torus, however, this approach does give examples of arbitrarily large initial data in the endpoint Besov space that generate global smooth solutions to the Navier–Stokes equation. In addition to these new results, we will also sharpen the constants in a number of previously known estimates for the growth of solutions to the Navier–Stokes equation and clarify the relationship between certain component reduction type regularity criteria.

The IVP for the evolution equation of wave fronts in chemical reactions in low-regularity Sobolev spaces

In this work, we study the initial-value problem for an equation of evolution of wave fronts in chemical reactions. We show that the associated initial-value problem is locally and globally well-posed in Sobolev spaces $$H^s({\mathbb {R}})$$ , where $$s>1/2$$ . The well-posedness in critical space $$\dot{H}^{1/2}(\mathbb R)$$ for small initial data is obtained. We also show that our result is sharp, in the sense that the flow-map data solution is not $$C^2$$ at origin, for $$s<1/2$$ . Furthermore, we study the behavior of the solutions when $$\mu \downarrow 0$$ .

Low Mach number limit for the compressible inertial Qian-Sheng model of liquid crystals: Convergence for classical solutions

<p style='text-indent:20px;'>In this paper we study the incompressible limit of the compressible inertial Qian-Sheng model for liquid crystal flow. We first derive the uniform energy estimates on the Mach number <inline-formula><tex-math id="M1">\begin{document}$ \epsilon $\end{document}</tex-math></inline-formula> for both the compressible system and its differential system with respect to time under uniformly in <inline-formula><tex-math id="M2">\begin{document}$ \epsilon $\end{document}</tex-math></inline-formula> small initial data. Then, based on these uniform estimates, we pass to the limit in the compressible system as <inline-formula><tex-math id="M3">\begin{document}$ \epsilon \rightarrow 0 $\end{document}</tex-math></inline-formula>, so that we establish the global classical solution of the incompressible system by compactness arguments. We emphasize that, on global in time existence of the incompressible inertial Qian-Sheng model under small size of initial data, the range of our assumptions on the coefficients are significantly enlarged, comparing to the results of De Anna and Zarnescu's work [<xref ref-type="bibr" rid="b6">6</xref>]. Moreover, we also obtain the convergence rates associated with <inline-formula><tex-math id="M4">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula>-norm with well-prepared initial data.

Dyadic bilinear estimates and applications to the well-posedness for the 2D Zakharov–Kuznetsov equation in the endpoint space 𝐻−1/4

Abstract The Cauchy problem of the 2D Zakharov–Kuznetsov equation ∂ t ⁡ u + ∂ x ⁡ ( ∂ x ⁢ x + ∂ y ⁢ y ) ⁡ u + u ⁢ u x = 0 {\partial_{t}u+\partial_{x}(\partial_{xx}+\partial_{yy})u+uu_{x}=0} is considered. It is shown that the 2D Z-K equation is locally well-posed in the endpoint Sobolev space H - 1 / 4 {H^{-1/4}} , and it is globally well-posed in H - 1 / 4 {H^{-1/4}} with small initial data. In this paper, we mainly establish some new dyadic bilinear estimates to obtain the results, where the main novelty is to parametrize the singularity of the resonance function in terms of a univariate polynomial.

Global existence and Gevrey regularity to the Navier-Stokes-Nernst-Planck-Poisson system in critical Besov-Morrey spaces

<p style='text-indent:20px;'>The paper is concerned with the Navier-Stokes-Nernst-Planck-Poisson system arising from electrohydrodynamics in <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}^d $\end{document}</tex-math></inline-formula>. By means of the implicit function theorem, we prove the global existence of mild solutions for Cauchy problem of this system with small initial data in critical Besov-Morrey spaces. In comparison to the previous works, our existence result provides a new class of initial data, for which the problem is global solvability. Meanwhile, based on the so-called Gevrey estimates, we verify that the obtained mild solutions are analytic in the spatial variables. As a byproduct, we show the asymptotic stability of solutions as the time goes to infinity. Furthermore, decay estimates of higher-order derivatives of solutions are deduced in Morrey spaces.

Uniqueness of solutions of the KdV-hierarchy via Dubrovin-type flows

We consider the Cauchy problem for the KdV hierarchy – a family of integrable PDEs with a Lax pair representation involving one-dimensional Schrödinger operators – under a local in time boundedness assumption on the solution. For reflectionless initial data, we prove that the solution stays reflectionless. For almost periodic initial data with absolutely continuous spectrum, we prove that under Craig-type conditions on the spectrum, Dirichlet data evolve according to a Lipschitz Dubrovin-type flow, so the solution is uniquely recovered by a trace formula. This applies to algebro-geometric (finite gap) solutions; more notably, we prove that it applies to small quasiperiodic initial data with analytic sampling functions and Diophantine frequency. This also gives a uniqueness result for the Cauchy problem on the line for periodic initial data, even in the absence of Craig-type conditions.

Global existence and asymptotic behavior for some multidimensional quasilinear hyperbolic systems

For the Cauchy problem of multidimensional quasilinear hyperbolic systems of diagonal form without self-interaction, we show the global existence of classical solutions with small initial data. We also prove that the global solution will converge to a solution of the linearized system as time tends to infinity, and the convergence rate is also determined.

Formation of finite-time singularities for nonlinear hyperbolic systems with small initial disturbances

This article concerns the formation of finite-time singularities in solutions to quasilinear hyperbolic systems with small initial data. We propose a universal test function method that works for many nonlinear hyperbolic systems arising from physical applications. We first present a simpler proof of the main result in the work of Sideris [Commun. Math. Phys. 101(4), 475–485 (1985)]: the global classical solution is non-existent for compressible Euler equations even for some small initial data. Then, we apply this approach to nonlinear magnetohydrodynamics in two space dimensions. Finally, we consider second order quasilinear hyperbolic systems with quadratic nonlinearity arising from elastodynamics of isotropic hyperelastic materials by ignoring the cubic and higher order terms. Under some restriction on the coefficients of the nonlinear terms that imply genuine nonlinearity, we show that the classical solutions to these equations can still blow up in finite time even if the initial data are small enough.

Stability of small solitary waves for the one-dimensional NLS with an attractive delta potential

We consider the initial-value problem for the one-dimensional nonlinear Schr\"odinger equation in the presence of an attractive delta potential. We show that for sufficiently small initial data, the corresponding global solution decomposes into a small solitary wave plus a radiation term that decays and scatters as $t\to\infty$. In particular, we establish the asymptotic stability of the family of small solitary waves.