Continuity Of Weak Solutions Research Articles

Weak periodic solutions of xẍ + 1 = 0 and the Harmonic Balance Method

We prove that the differential equation xẍ + 1 = 0 has continuous weak periodic solutions and compute their periods. Then, we use the Harmonic Balance Method until order six to approximate these periods and to illustrate how the accuracy of the method increases with the order. Our computations rely on the Gröbner basis approach.

Liouville and Calabi-Yau type theorems for complex Hessian equations

We prove a Liouville type theorem for entire maximal $m$-subharmonic functions in $\mathbb C^n$ with bounded gradient. This result, coupled with a standard blow-up argument, yields a (non-explicit) a priori gradient estimate for the complex Hessian equation on a compact K\ahler manifold. This terminates the program, initiated by Hou, Ma and Wu, of solving the non-degenerate Hessian equation on such manifolds in full generality. We also obtain, using our previous work, continuous weak solutions in the degenerate case for the right hand side in some $L^p, $ with sharp bound on $p$.

Regularity of Hölder continuous solutions of a modified critical porous media equation

In this paper, we study the regularity of weak solutions of a modified critical dissipative porous media equation. by using Besov space techniques combined with energy estimates, we prove that Holder continuous weak solutions to this system are smooth solutions. The porous media equation is a dissipative transport-diffusion equation with non-local divergence-free velocity field. It shares many similarities with the surface quasi-geostrophic equation. From a mathematical view, the porous media equation is somewhat a generalization of the surface quasi-geostrophic equation. In this paper, we study the regularity of weak solutions of a 3-D modified critical dissipative porous media equation. By using Besov space techniques combined with classical energy estimates, we prove that Holder continuous weak solutions to this system are smooth solutions. In particular, by applying the same method, we obtain that this result holds true for the 3-D modified critical surface quasi-geostrophic equation.

Green matrices and continuity of the weak solutions for the elliptic systems with lower order terms

In this paper, we consider the second-order elliptic systems with lower order terms coefficients belonging to Kato–Stummel type classes in a bounded domain [Formula: see text], where [Formula: see text]. We establish the existence, global pointwise estimates and [Formula: see text] estimates for Green matrices. Based on these estimates, we are able to show the continuity of weak solutions for the general elliptic systems by using method of potentials. We also obtain the optimal Schauder estimates for weak solutions.

Dissipative Euler Flows with Onsager‐Critical Spatial Regularity

AbstractFor any ɛ > 0 we show the existence of continuous periodic weak solutions v of the Euler equations that do not conserve the kinetic energy and belong to the space ; namely, x ↦ v (x,t) is ⅓−ε‐Hölder continuous in space at a.e. time t and the integral is finite. A well‐known open conjecture of L. Onsager claims that such solutions exist even in the class .© 2016 Wiley Periodicals, Inc.

A Degenerate Fourth-Order Parabolic Equation Modeling Bose–Einstein Condensation. Part I: Local Existence of Solutions

A degenerate fourth-order parabolic equation modeling condensation phenomena related to Bose-Einstein particles is analyzed. The model is a Fokker-Planck-type approximation of the Boltzmann-Nordheim equation, only keeping the leading order term. It maintains some of the main features of the kinetic model, namely mass and energy conservation and condensation at zero energy. The existence of a local-in-time nonnegative continuous weak solution is proven. If the solution is not global, it blows up with respect to the $L^\infty$ norm in finite time. The proof is based on approximation arguments, interpolation inequalities in weighted Sobolev spaces, and suitable a priori estimates for a weighted gradient $L^2$ norm.

Intrinsic Lipschitz graphs in Heisenberg groups and continuous solutions of a balance equation

We provide a characterization of intrinsic Lipschitz graphs in the sub-Riemannian Heisenberg groups in terms of their distributional gradients. Moreover, we prove the equivalence of different notions of continuous weak solutions to the equation ∂ϕ∂z+∂∂t[ϕ2/2]=w, where w is a bounded measurable function.

On the regular set of BMO weak solutions to $p$-Laplacian strongly coupled nonregular elliptic systems

This paper studies Holder continuity of weak solutions to strongly coupled elliptic systems. We do not assume that the solutions are bounded but BMO and the ellipticity constants can be unbounded.

A weak Harnack inequality for fractional evolution equations with discontinuous coefficients

We study linear time fractional diffusion equations in divergence form of time order less than one. It is merely assumed that the coefficients are measurable and bounded, and that they satisfy a uniform parabolicity condition. As the main result we establish for nonnegative weak supersolutions of such problems a weak Harnack inequality with optimal critical exponent. The proof relies on new a priori estimates for time fractional problems and uses Moser’s iteration technique and an abstract lemma of Bombieri and Giusti, the latter allowing to avoid the rather technically involved approach via BMO. As applications of the weak Harnack inequality we establish the strong maximum principle, continuity of weak solutions at t = 0, and a Liouville type theorem.

Interior W 1,p Regularity and Hölder Continuity of Weak Solutions to a Class of Divergence Kolmogorov Equations with Discontinuous Coefficients

We consider a class of Kolmogorov equation $$Lu={\sum^{p_0}_{i,j=1}{\partial_{x_i}}(a_{ij}(z){\partial_{x_j}}u)}+{\sum^{N}_{i,j=1}b_{ij}x_{i}{\partial_{x_j}}u-{\partial_t}u}={\sum^{p_0}_{j=1}{\partial_{x_j}}F_{j}(z)}$$ in a bounded open domain \({\Omega \subset \mathbb{R}^{N+1}}\), where the coefficients matrix (aij (z)) is symmetric uniformly positive definite on \({\mathbb{R}^{p_0} (1 \leq p_0 < N)}\). We obtain interior W1,p (1 < p < ∞) regularity and Holder continuity of weak solutions to the equation under the assumption that coefficients aij (z) belong to the \({VMO_L\cap L^\infty}\) and \({({b_{ij}})_{N \times N}}\) is a constant matrix such that the frozen operator \({L_{z_0}}\) is hypoelliptic.

Hypoellipticity for infinitely degenerate quasilinear equations and the dirichlet problem

In [10], we considered a class of infinitely degenerate quasilinear equations of the form div $$A(x,w)\nabla w + \overrightarrow r (x,w) \cdot \nabla w + f(x,w) = 0$$ and derived a priori bounds for high order derivatives D a w of their solutions in terms of w and ▿w. We now show that it is possible to obtain bounds in terms of just w for a further subclass of such equations, and we apply the resulting estimates to prove that continuous weak solutions are necessarily smooth. We also obtain existence, uniqueness, and interior $${\varrho ^\infty }$$ -regularity of solutions for the corresponding Dirichlet problem with continuous boundary data.

Gradient estimates for the p-Laplace heat equation under the Ricci flow

Abstract We establish space-only gradient estimates for positive continuous weak solutions to the p-Laplace heat equation on some complete manifolds evolving under the Ricci flow. As applications, we get Harnack inequalities to compare solutions at different points

HÖLDER CONTINUITY OF WEAK SOLUTIONS FOR DEGENERATE QUASILINEAR SUBELLIPTIC SYSTEMS WITH DIFFERENT WEIGHTS

We establish Hölder continuity of weak solutions for degenerate quasilinear subelliptic systems with different weights, with weak Harnack inequalities obtained here and a generalization of Caffarelli's idea.

Estimates in Morrey Spaces and Hölder Continuity for Weak Solutions to Degenerate Elliptic Systems

In this paper, we establish gradient estimates in Morrey spaces and Holder continuity for weak solutions of the following degenerate elliptic system $$-X_{\alpha}^{\ast}(a_{ij}^{\alpha\beta}(x)X_{\beta}u^{j})=g_{i}-X_{\alpha}^{\ast}f_{i}^{\alpha}(x),$$ where X1, . . . , Xq are real smooth vector fields satisfying Hormander’s condition, coefficients \({a_{ij}^{\alpha \beta }\in VMO_X \cap L^\infty (\Omega ), \alpha,\beta=1,2, \,.\,.\,.\, ,q, i,j=1,2, \,.\,.\,.\, ,N, X_{\alpha}^{\ast}}\) is the transposed vector field of Xα.

Higher Regularity of Hölder Continuous Solutions of Parabolic Equations with Singular Drift Velocities

Motivated by an equation arising in magnetohydrodynamics, we prove that Hölder continuous weak solutions of a nonlinear parabolic equation with singular drift velocity are classical solutions. The result is proved using the space–time Besov spaces introduced by Chemin and Lerner (J Differ Equ 121(2):314–328, 1995), combined with energy estimates, without any minimality assumption on the Hölder exponent of the weak solutions.

Global entropy solutions to a class of quasi-linear wave equations with large time-oscillating sources

This research explores the Cauchy problem for a class of quasi-linear wave equations with time dependent sources. It can be transformed into the Cauchy problem of hyperbolic integro-differential systems of nonlinear balance laws. We introduce the generalized Glimm scheme in new version and study its stability which is proved by Glimm-type interaction estimates in a dissipativity assumption. The generalized solutions to the perturbed Riemann problems, the building blocks of generalized Glimm scheme, are constructed by Riemann problem method modeled on the source free equations. The global existence for the Lipschitz continuous solutions and weak solutions to the systems is established by the consistency of scheme and the weak convergence of source. Finally, the weak solutions are also the entropy solutions which satisfy the entropy inequality.

Uniqueness and continuity of solution for the initial data in the scaling invariant class of the degenerate Keller–Segel system

We consider the degenerate Keller–Segel system (KS)m below. We find the functional space \({L^s(0,T;L^p(\mathbb{R}^{N}))}\) with some p,s for the uniqueness and continuity of weak solutions with respect to the initial data. Our space is discussed from a viewpoint of the scaling invariant class associated with (KS)m for γ = 0. The technique is based on the L1-contraction principle for the porous medium equation.

Notes on the regularity of harmonic map systems

In this note, we provide an alternative proof of C-regularity of continuous weak solutions to the system of harmonic map or heat flow of harmonic maps by Riesz potential estimates between Morrey spaces.

1D Vlasov-Poisson equations with electron sheet initial data

We construct global weak solutions for both one-component andtwo-component Vlasov-Poisson equations in a single space dimensionwith electron sheet initial data. We give an explicit formula of theweak solution of the one-component Vlasov-Poisson equation providedthe electron sheet remains a graph in the $x$-$v$ plane, and we givesharp conditions on whether the moment of this explicit weaksolution will blow up or not. We introduce new parameters, which wecall 'charge indexes', to construct the global weak solution. Themoment of the weak solution corresponds to a multi-valued solutionto the Euler-Poisson system. Our method guarantees that even ifconcentration in charge develops, it will disappear immediately. Weextend our method to more singular initial data, where charge canconcentrate on points at time $t=0$. Examples show that forone-component Vlasov-Poisson equation our weak solution agrees withthe continuous fission weak solution, which is the zero diffusionlimit of the Fokker-Planck equation. Finally, we propose a novelnumerical method to compute solutions of both one-component andtwo-component Vlasov-Poisson equations and the multi-valued solutionof the one-dimensional Euler-Poisson equation.

Long time existence of a class of contact discontinuities for second

The study of long time existence of classical smooth solutions for second order quasilinear wave equations has received much attention (see e.g. [1] and the references given there). Results were also obtained for continuous semilinear waves with gradient jumps on a characteristic hypersurface in [2] and for C1 quasilinear waves with second order derivatives jumps on a characteristic hypersurface in [3], [4]. In this paper we show how the methods and results of [2] can be extended to a class of continuous weak solutions, with gradient jumps on a characteristic hypersurface, for some second order quasilinear balance laws.