Junxiong Jia

Existence and uniqueness of mild solutions for abstract delay fractional differential equations

In this paper, by means of solution operator approach and contraction mapping theorem, the existence and uniqueness of mild solutions for a class of abstract delay fractional differential equations are obtained.

Bayesian approach to inverse problems for functions with a variable-index Besov prior

The Bayesian approach has been adopted to solve inverse problems that reconstruct a function from noisy observations. Prior measures play a key role in the Bayesian method. Hence, many probability measures have been proposed, among which total variation (TV) is a well-known prior measure that can preserve sharp edges. However, it has two drawbacks, the staircasing effect and a lack of the discretization-invariant property. The variable-index TV prior has been proposed and analyzed in the area of image analysis for the former, and the Besov prior has been employed recently for the latter. To overcome both issues together, in this paper, we present a variable-index Besov prior measure, which is a non-Gaussian measure. Some useful properties of this new prior measure have been proven for functions defined on a torus. We have also generalized Bayesian inverse theory in infinite dimensions for our new setting. Finally, this theory has been applied to integer- and fractional-order backward diffusion problems. To the best of our knowledge, this is the first time that the Bayesian approach has been used for the fractional-order backward diffusion problem, which provides an opportunity to quantify its uncertainties.

Maximum principles for a time–space fractional diffusion equation

In this paper, we focus on maximum principles of a time-space fractional diffusion equation. Maximum principles for classical solution and weak solution are all obtained by using properties of the time fractional derivative operator and the fractional Laplace operator. We deduce maximum principles for a full fractional diffusion equation, other than time-fractional and spatial-integer order diffusion equations.

Optimal Time Decay of Navier–Stokes Equations with Low Regularity Initial Data

In this paper, we study the optimal time decay rate of isentropic Navier–Stokes equations under the low regularity assumptions about initial data. In the previous works about optimal time decay rate, the initial data need to be small in H[N/2]+2(ℝN). Our work combined negative Besov space estimates and the conventional energy estimates in Besov space framework which is developed by Danchin. Through our methods, we can get optimal time decay rate with initial data just small in B̃N/2−1,N/2+1 ∩ B̃N/2−1,N/2 and belong to some negative Besov space (need not to be small). Finally, combining the recent results in [25] with our methods, we only need the initial data to be small in homogeneous Besov space B̃N/2−2,N/2 ∩ B̃N/2−1 to get the optimal time decay rate in space L2.

Optimal time decay rate for compressible viscoelastic equations in critical spaces

In this paper, we are concerned with the optimal time convergence rate of the global strong solution to some constant equilibrium states for the compressible viscoelastic fluids in the whole space. Green’s matrix method and energy estimate method are used to obtain the optimal time decay rate under the critical Besov space framework. Our result implies the optimal -time decay rate and only need the initial datum to be small in some critical Besov space which have very low regularity compared with the classical Sobolev space.

Well-posedness for compressible MHD systems with highly oscillating initial data

In this paper, a unique local solution for compressible magnetohydrodynamics systems has been constructed in the critical Besov space framework by converting the system in Euler coordinates to a system in Lagrange coordinates. Our results improve the range of the Lebesgue exponent in the Besov space from [2, N) to [2, 2N), where N denotes the space dimension. Then, we give a lower bound for the maximal existence time, which is important for our construction of global solutions. Based on the lower bound, we use the effective viscous flux and Hoff’s energy method to obtain the unique global solution, which allows the initial velocity field and the magnetic field to have large energies and allows the initial density to exhibit large oscillations on a set of small measure.

Iterative thresholding algorithm based on non-convex method for modified lp-norm regularization minimization

Recently, the

A Carleman estimate of some anisotropic space-fractional diffusion equations

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Cauchy problems for fractional differential equations with Riemann-Liouville fractional derivatives ✩

-norm regularization minimization problem

WELL-POSEDNESS OF ABSTRACT DISTRIBUTED-ORDER FRACTIONAL DIFFUSION EQUATIONS

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