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Register a new userOne sided Holder regularity of global weak solutions of negative order dispersive equations
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We prove global existence, uniqueness and stability of entropy solutions with $L^2cap L^infty$ initial data for a general family of negative order dispersive equations. It is further demonstrated that this solution concept extends in a unique continuous manner to all $L^2$ initial data. These weak solutions are found to satisfy one sided Holder conditions whose coefficients decay in time. The latter result controls the height of solutions and further provides a way to bound the maximal lifespan of classical solutions from their initial data.
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We consider a class of nonautonomous elliptic operators ${mathscr A}$ with unbounded coefficients defined in $[0,T]timesR^N$ and we prove optimal Schauder estimates for the solution to the parabolic Cauchy problem $D_tu={mathscr A}u+f$, $u(0,cdot)=g$.
This paper proves Holder continuity of viscosity solutions to certain nonlocal parabolic equations that involve a generalized fractional time derivative of Marchaud or Caputo type. As a necessary and preliminary result, this paper first shows that viscosity solutions to certain nonlinear ordinary differential equations involving the generalized fractional time derivative are Holder continuous.
This paper is concerned with a class of nonlocal dispersive models -- the $theta$-equation proposed by H. Liu [ On discreteness of the Hopf equation, {it Acta Math. Appl. Sin.} Engl. Ser. {bf 24}(3)(2008)423--440]: $$ (1-partial_x^2)u_t+(1-thetapartial_x^2)(frac{u^2}{2})_x =(1-4theta)(frac{u_x^2}{2})_x, $$ including integrable equations such as the Camassa-Holm equation, $theta=1/3$, and the Degasperis-Procesi equation, $theta=1/4$, as special models. We investigate both global regularity of solutions and wave breaking phenomena for $theta in mathbb{R}$. It is shown that as $theta$ increases regularity of solutions improves: (i) $0 <theta < 1/4$, the solution will blow up when the momentum of initial data satisfies certain sign conditions; (ii) $1/4 leq theta < 1/2$, the solution will blow up when the slope of initial data is negative at one point; (iii) ${1/2} leq theta leq 1$ and $theta=frac{2n}{2n-1}, nin mathbb{N}$, global existence of strong solutions is ensured. Moreover, if the momentum of initial data has a definite sign, then for any $thetain mathbb{R}$ global smoothness of the corresponding solution is proved. Proofs are either based on the use of some global invariants or based on exploration of favorable sign conditions of quantities involving solution derivatives. Existence and uniqueness results of global weak solutions for any $theta in mathbb{R}$ are also presented. For some restricted range of parameters results here are equivalent to those known for the $b-$equations [e.g. J. Escher and Z. Yin, Well-posedness, blow-up phenomena, and global solutions for the b-equation, {it J. reine angew. Math.}, {bf 624} (2008)51--80.]
This paper studies the Sobolev regularity estimates of weak solutions of a class of singular quasi-linear elliptic problems of the form $u_t - mbox{div}[mathbb{A}(x,t,u, abla u)]= mbox{div}[{mathbf F}]$ with homogeneous Dirichlet boundary conditions over bounded spatial domains. Our main focus is on the case that the vector coefficients $mathbb{A}$ are discontinuous and singular in $(x,t)$-variables, and dependent on the solution $u$. Global and interior weighted $W^{1,p}(Omega, omega)$-regularity estimates are established for weak solutions of these equations, where $omega$ is a weight function in some Muckenhoupt class of weights. The results obtained are even new for linear equations, and for $omega =1$, because of the singularity of the coefficients in $(x,t)$-variables
We continue to study regularity results for weak solutions of the large class of second order degenerate quasilinear equations of the form begin{eqnarray} text{div}big(A(x,u, abla u)big) = B(x,u, abla u)text{ for }xinOmega onumber end{eqnarray} as considered in our previous paper giving local boundedness of weak solutions. Here we derive a version of Harnacks inequality as well as local Holder continuity for weak solutions. The possible degeneracy of an equation in the class is expressed in terms of a nonnegative definite quadratic form associated with its principal part. No smoothness is required of either the quadratic form or the coefficients of the equation. Our results extend ones obtained by J. Serrin and N. Trudinger for quasilinear equations, as well as ones for subelliptic linear equations obtained by Sawyer and Wheeden in their 2006 AMS memoir article.
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