We consider an abstract second order linear equation with a strong dissipation, namely a friction term which depends on a power of the "elastic" operator.
In the homogeneous case, we investigate the phase spaces in which the initial value problem gives rise to a semigroup, and the further regularity of solutions. In the non-homogeneous case, we study how the regularity of solutions depends on the regularity of forcing terms, and we characterize the spaces where a bounded forcing term yields a bounded solution.
What we discover is a variety of different regimes, with completely different behaviors, depending on the exponent in the friction term.
We also provide counterexamples in order to show the optimality of our results.
We consider nonlocal initial boundary value problems with integral boundary conditions for integro-differential first order hyperbolic systems. We prove a general regularity result stating that the $L^2$-generalized solutions become eventually continuous.
We establish global-in-time frequency localized local smoothing estimates for Schrödinger equations on hyperbolic space $\mathbb{H}^d$. In the presence of symmetric first and zeroth order potentials, which are possibly time-dependent, possibly large, and have sufficiently fast polynomial decay, these estimates are proved up to a localized lower order error. Then in the time-independent case, we show that a spectral condition (namely, absence of threshold resonances) implies the full local smoothing estimates (without any error), after projecting to the continuous spectrum. In the process, as a…
We establish the Kato-type smoothing property, i.e., global-in-time smoothing estimates with homogeneous weights, for the Schrödinger equation on Riemannian symmetric spaces of non-compact type and general rank. These form a rich class of manifolds with nonpositive sectional curvature and exponential volume growth at infinity, e.g., hyperbolic spaces. We achieve it by proving the Stein-Weiss inequality and the resolvent estimate of the corresponding Fourier multiplier, which are of independent interest. Moreover, we extend the comparison principles to symmetric spaces and deduce different type…
We consider the $2$D dissipative quasi-geostrophic equation with the time periodic external force and prove the existence of a unique time periodic solution in the case of the supercritical dissipation. In this case, the smoothing effect of the semigroup generated by the dissipation term is too weak to control the nonlinearity in the Duhamel term of the correponding integral equation. In this paper, we give a new approach which does not depend on the contraction mapping principle for the integral equation.
In this paper, we prove that Kato smoothing effects for magnetic Schrödinger operators can yield the endpoint Strichartz estimates for linear wave equation with magnetic potential on two dimensional hyperbolic spaces. This result serves as a cornerstone for the author's work \cite{Lize} and collaborative work \cite{LMZ} in the study of asymptotic stability of harmonic maps for wave maps from $\Bbb R\times\Bbb H^2$ to $\Bbb H^2$.
In the context of hyperbolic systems of balance laws, the Shizuta-Kawashima coupling condition guarantees that all the variables of the system are dissipative even though the system is not totally dissipative. Hence it plays a crucial role in terms of sufficient conditions for the global in time existence of classical solutions. However, it is easy to find physically based models that do not satisfy this condition, especially in several space dimensions. In this paper, we consider two simple examples of partially dissipative hyperbolic systems violating the Shizuta-Kawashima condition ([SK]) i…
In this paper, for the 3-D Navier-Stokes-Boussinesq system with horizontal dissipation, where there is no smoothing effect on the vertical derivatives, we prove a uniqueness result of solutions $ (u,ρ)\in L^{\infty}_T\big( H^{0,s}\times H^{0,1-s}\big)$ with $ (\nabla_h u,\nabla_hρ)\in L^{2}_T\big( H^{0,s}\times H^{0,1-s}\big)$ and $s\in [1/2,1]$. As a consequence, we improve the conditions stated in the paper \cite{Miao} in order to obtain a global well-posedness result in the case of axisymmetric initial data.
We study the time-asymptotic behavior of linear hyperbolic systems under partial dissipation which is localized in suitable subsets of the domain. More precisely, we recover the classical decay rates of partially dissipative systems satisfying the stability condition (SK) with a time-delay depending only on the velocity of each component and the size of the undamped region. To quantify this delay, we assume that the undamped region is a bounded space-interval and that the system without space-restriction on the dissipation satisfies the stability condition (SK). The former assumption ensures t…
We consider a multi-dimensional scalar wave equation with memory corresponding to the viscoelastic material described by a generalized Zener model. We deduce that this relaxation system is an example of a non-strictly hyperbolic system satisfying Majda's block structure condition. Well-posedness of the associated Cauchy problem is established by showing that the symbol of the spatial derivatives is uniformly diagonalizable with real eigenvalues. A long-time stability result is obtained by plane-wave analysis when the memory term allows for dissipation of energy.
We study the vanishing dissipation limit of the three-dimensional (3D) compressible Navier-Stokes-Fourier equations to the corresponding 3D full Euler equations. Our results are twofold. First, we prove that the 3D compressible Navier-Stokes-Fourier equations admit a family of smooth solutions that converge to the planar rarefaction wave solution of the 3D compressible Euler equations with arbitrary strength. Second, we obtain a uniform convergence rate in terms of the viscosity and heat-conductivity coefficients. For this multi-dimensional problem, we first need to introduce the hyperbolic wa…
This paper studies the dissipative generalized surface quasi-geostrophic equations in a supercritical regime where the order of the dissipation is small relative to order of the velocity, and the velocities are less regular than the advected scalar by up to one order of derivative. We also consider a non-degenerate modification of the endpoint case in which the velocity is less smooth than the advected scalar by slightly more than one order. The existence and uniqueness theory of these equations in the borderline Sobolev spaces is addressed, as well as the instantaneous smoothing effect of the…
In this paper, we shall study the parabolic-elliptic Keller-Segel system on the Poincar{é} disk model of the 2D-hyperbolic space. We shall investigate how the negative curvature of this Riemannian manifold influences the solutions of this system. As in the 2D-Euclidean case, under the sub-critical condition $χ$M < 8$π$, we shall prove global well-posedness results with any initial L 1-data. More precisely, by using dispersive and smoothing estimates we shall prove Fujita-Kato type theorems for local well-posedness. We shall then use the logarithmic Hardy-Littlewood-Sobolev estimates on the hyp…
This paper is concerned with quasilinear systems of partial differential equations consisting of two hyperbolic operators interacting dissipatively. Its main theorem establishes global-in-time existence and asymptotic stability of strong solutions to the Cauchy problem close to homogeneous reference states. Notably, the operators are not required to be symmetric hyperbolic, instead merely the existence of symbolic symmetrizers is assumed. The dissipation is characterized by conditions equivalent to the uniform decay of all Fourier modes at the reference state. On a technical level, the theory …
The global dynamics and regularity of parabolic-hyperbolic systems is an interesting topic in PDEs due to the coupling of competing dissipation and hyperbolic effects. This paper is concerned with the Cauchy problem of a parabolic-hyperbolic system derived from a chemotaxis model describing the dynamics of the initiation of tumor angiogenesis. It is shown that, as time tends to infinity, the Cauchy problem with large-amplitude discontinuous data admit global weak solutions which converge to a constant state (resp. a viscous shock wave) if the asymptotic states of initial values at far field ar…
In this paper, we establish the global well-posedness of stochastic 3D Leray-$α$ model with general fractional dissipation driven by multiplicative noise. This model is the stochastic 3D Navier-Stokes equation regularized through a smoothing kernel of order $θ_1$ in the nonlinear term and a $θ_2$-fractional Laplacian. In the case of $θ_1 \ge 0, θ_2 > 0$ and $θ_1+θ_2 \geq\frac{5}{4}$, we prove the global existence and uniqueness of strong solutions. The main results cover many existing works in the deterministic cases, and also generalize some known results of stochastic models as our special c…
We consider the nonlinear degenerate parabolic equation of porous medium type, whose diffusion is driven by the (spectral) fractional Laplacian on the hyperbolic space. We provide existence results for solutions, in an appropriate weak sense, for data belonging either to the usual $L^p$ spaces or to larger (weighted) spaces determined either in terms of a ground state of $Δ_{\mathbb{H}^n}$, or of the (fractional) Green's function. For such solutions, we also prove different kind of smoothing effects, in the form of quantitative $L^1-L^\infty$ estimates. To the best of our knowledge, this seems…
Here we develop a method for investigating global strong solutions of partially dissipative hyperbolic systems in the critical regularity setting. Compared to the recent works by Kawashima and Xu, we use hybrid Besov spaces with different regularity exponent in low and high frequency. This allows to consider more general data and to track the exact dependency on the dissipation parameter for the solution. Our approach enables us to go beyond the L^2 framework in the treatment of the low frequencies of the solution, which is totally new, to the best of our knowledge. Focus is on the one-dimensi…
