Intermediate Domains for Scalar Conservation Laws

Nonlinear Analysis-theory Methods & Applications, 2009

In this paper we study a class of fractional order integrodifferential equations considered in an arbitrary Banach space. Using the theory of analytic semigroups we establish the existence, uniqueness and regularity of a mild solution to these fractional order integrodifferential equations.

Nonlinear Analysis-theory Methods & Applications, 2016

We investigate quantitative properties of nonnegative solutions u(t, x) ≥ 0 to the nonlinear fractional diffusion equation, ∂ t u + LF (u) = 0 posed in a bounded domain, x ∈ Ω ⊂ R N , with appropriate homogeneous Dirichlet boundary conditions. As L we can use a quite general class of linear operators that includes the two most common versions of the fractional Laplacian (−∆) s , 0 < s < 1, in a bounded domain with zero Dirichlet boundary conditions, but it also includes many other examples since our theory only needs some basic properties that are typical of "linear heat semigroups". The nonlinearity F is assumed to be increasing and is allowed to be degenerate, the prototype is the power case F (u) = |u| m−1 u, with m > 1. In this paper we propose a suitable class of solutions of the equation, and cover the basic theory: we prove existence, uniqueness of such solutions, and we establish upper bounds of two forms (absolute bounds and smoothing effects), as well as weighted-L 1 estimates. The class of solutions is very well suited for that work. The standard Laplacian case s = 1 is included and the linear case m = 1 can be recovered in the limit. In a companion paper [12], we will complete the study with more advanced estimates, like the upper and lower boundary behaviour and Harnack inequalities, for which the results of this paper are needed.

Handbook of Mathematical Analysis in Mechanics of Viscous Fluids

In this chapter, we provide a review of results on the global well-posedness and optimal decay rate of strong solutions to the compressible Navier-Stokes equations in several type of domains: (1) whole space (Theorems 6, 7, 8, 9, 10, 11, and 12), (2) exterior domains (Theorems 13 and 14), (3) half-space (Theorem 15), (4) bounded domains (Theorem 16), and (5) infinite layers. Global well-posedness for the compressible viscous barotropic fluid motion with nonslip boundary condition was for the first time proved in the early 1980s by Matsumura and Nishida (Commun Math Phys 89:445-464, 1983) under the assumption that the H 3 norm of the initial data is small. In Theorems 1, 2, 3, and 4, we revisit the same problem as in Matsumura and Nishida (Commun Math Phys 89:445-464, 1983) under the weaker assumptions, namely, that the H 2 norm of initial data is small. This is an improvement of the result in Matsumura and Nishida (Commun Math Phys 89:445-464, 1983) in view of the regularity assumption of the initial data. To show the methods, we perform the proof of Theorems 1, 2, 3, and 4 in all essential details. In this process, the L p-L q decay properties of solutions to the linearized equations are proved by using the cutoff technique and combining the local energy decay and the result in the whole space. This result was first proved by Kobayashi and Shibata (Commun Math Phys 200:621-659, 1999) under some additional assumption, and in this chapter, this assumption is eliminated by using a bootstrap argument. In the final section of this chapter, the optimal decay rate of the H 2 norm of solution of the nonlinear problem is proved by combining the L p-L q decay properties of the linearized equations with some energy inequality of exponential decay type under the assumption that the initial data belong to the intersection space of H 2 and L 1. The main idea of this part of the proof is to combine the L p-L q decay properties of the Stokes semigroup and some Lyapunov-type energy inequality.

Nonlinear Analysis: Theory, Methods & Applications, 1984

Advances in Differential Equations

Using regularized resolvent families, we investigate the solvability of the fractional order inhomogeneous Cauchy problem D α t u(t) = Au(t) + f (t), t > 0, 0 < α ≤ 1, where D α t is the Caputo fractional derivative of order α, A a closed linear operator on some Banach space X, f : [0, ∞) → X is a given function. We define an operator family associated with this problem and study its regularity properties. When A is the generator of a β-times integrated semigroup (T β (t)) on a Banach space X, explicit representations of mild and classical solutions of the above problem in terms of the integrated semigroup are derived. The results are applied to the fractional diffusion equation with non-homogeneous, Dirichlet, Neumann and Robin boundary conditions and to the time fractional order Schrödinger equation D α t u(t, x) = e iθ ∆pu(t, x) + f (t, x), t > 0, x ∈ R N where π/2 ≤ θ < (1 − α/2)π and ∆p is a realization of the Laplace operator on L p (R N), 1 ≤ p < ∞.

Journal of Hyperbolic Differential Equations, 2014

We obtain new fine properties of entropy solutions to scalar nonlinear conservation laws. For this purpose, we study the &quot;fractional BV spaces&quot; denoted by BVs(ℝ) (for 0 &lt; s ≤ 1), which were introduced by Love and Young in 1937 and closely related to the critical Sobolev space Ws,1/s(ℝ). We investigate these spaces in connection with one-dimensional scalar conservation laws. The BVsspaces allow one to work with less regular functions than BV functions and appear to be more natural in this context. We obtain a stability result for entropy solutions with BVsinitial data. Furthermore, for the first time, we get the maximal Ws,psmoothing effect conjectured by Lions, Perthame and Tadmor for all nonlinear (possibly degenerate) convex fluxes.

Mathematical Models and Methods in Applied Sciences, 2012

We consider the Cauchy problem on nonlinear scalar conservation laws with a diffusion-type source term related to an index s ∈ ℝ over the whole space ℝn for any spatial dimension n ≥ 1. Here, the diffusion-type source term behaves as the usual diffusion term over the low frequency domain while it admits on the high frequency part a feature of regularity-gain and regularity-loss for s &lt; 1 and s &gt; 1, respectively. For all s ∈ ℝ, we not only obtain the Lp–Lq time-decay estimates on the linear solution semigroup but also establish the global existence and optimal time-decay rates of small-amplitude classical solutions to the nonlinear Cauchy problem. In the case of regularity-loss, the time-weighted energy method is introduced to overcome the weakly dissipative property of the equation. Moreover, the large-time behavior of solutions asymptotically tending to the heat diffusion waves is also studied. The current results have general applications to several concrete models arising f...

Dynamics of Partial Differential Equations, 2005

Riemann solutions for the systems of conservation laws uτ +f (u) ξ = 0 are self-similar solutions of the form u = u(ξ/τ). Using the change of variables x = ξ/τ, t = ln(τ), Riemann solutions become stationary to the system ut + (Df (u) − xI)ux = 0. For the linear variational system around the Riemann solution with n-Lax shocks, we introduce a semigroup in the Hilbert space with weighted L 2 norm. We show that (A) the region λ > −η consists of normal points only. (B) Eigenvalues of the linear system correspond to zeros of the determinant of a transcendental matrix. They lie on vertical lines in the complex plane. There can be resonance values where the response of the system to forcing terms can be arbitrarily large, see Definition 6.2. Resonance values also lie on vertical lines in the complex plane. (C) Solutions of the linear system are O(e γt) for any constant γ that is greater than the largest real parts of the eigenvalues and the coordinates of resonance lines. This work can be applied to the linear and nonlinear stability of Riemann solutions of conservation laws and the stability of nearby solutions of the Dafermos regularizations ut + (Df (u) − xI)ux = uxx.

Filomat

In this paper, we consider a one-dimensional weakly degenerate wave equation with a dynamic nonlocal boundary feedback of fractional type acting at a degenerate point. First We show well-posedness by using the semigroup theory. Next, we show that our system is not uniformly stable by spectral analysis. Hence, we look for a polynomial decay rate for a smooth initial data by using a result due Borichev and Tomilov which reduces the problem of estimating the rate of energy decay to finding a growth bound for the resolvent of the generator associated with the semigroup. This analysis proves that the degeneracy affect the energy decay rates.

Calculus of Variations and Partial Differential Equations, 2013