Lopatinski Condition Research Articles

On weak stability of shock waves in 2D compressible elastodynamics

By using an equivalent form of the uniform Lopatinski condition for 1-shocks, we prove that the stability condition found by the energy method in [A. Morando, Y. Trakhinin and P. Trebeschi, Structural stability of shock waves in 2D compressible elastodynamics, Math. Ann. 378 (2020) 1471–1504] for the rectilinear shock waves in two-dimensional flows of compressible isentropic inviscid elastic materials is not only sufficient but also necessary for uniform stability (implying structural nonlinear stability of corresponding curved shock waves). The key point of our spectral analysis is a delicate study of the transition between uniform and weak stability. Moreover, we prove that the rectilinear shock waves are never violently unstable, i.e. they are always either uniformly or weakly stable.

Boundary stabilization of the linear MGT equation with partially absorbing boundary data and degenerate viscoelasticity

<p style='text-indent:20px;'>The Jordan–Moore–Gibson–Thompson (JMGT) equation is a well-established and recently widely studied model for nonlinear acoustics (NLA). It is a third–order (in time) semilinear Partial Differential Equation (PDE) with a distinctive feature of predicting the propagation of ultrasound waves at <i>finite</i> speed. This is due to the heat phenomenon known as <i>second sound</i> which leads to hyperbolic heat-wave propagation. In this paper, we consider the problem in the so called "critical" case, where free dynamics is unstable. In order to stabilize, we shall use boundary feedback controls supported on a portion of the boundary only. Since the remaining part of the boundary is not "controlled", and the imposed boundary conditions of Neumann type fail to saitsfy Lopatinski condition, several mathematical issues typical for mixed problems within the context o boundary stabilizability arise. To resolve these, special geometric constructs along with sharp trace estimates will be developed. The imposed geometric conditions are motivated by the geometry that is suitable for modeling the problem of controlling (from the boundary) the acoustic pressure involved in medical treatments such as lithotripsy, thermotherapy, sonochemistry, or any other procedure involving High Intensity Focused Ultrasound (HIFU).</p>

Boundary feedback stabilization of a critical nonlinear JMGT equation with Neumann-undissipated part of the boundary

<p style='text-indent:20px;'>Boundary feedback stabilization of a <i>critical</i> third–order (in time) semilinear Jordan–Moore–Gibson–Thompson (JMGT) is considered. The word <i>critical</i> here refers to the usual case where media–damping effects are non–existent or non–measurable and therefore cannot be relied upon for stabilization purposes. Motivated by modeling aspects in high-intensity focused ultrasound (HIFU) technology, the boundary feedback under consideration is supported only on a portion of the boundary. At the same time, the remaining part is undissipated and subject to Neumann/Robin boundary conditions. As such, unlike Dirichlet, it fails to satisfy the Lopatinski condition, a fact which compromises tangential regularity on the boundary [<xref ref-type="bibr" rid="b37">37</xref>]. In such a configuration, the analysis of uniform stabilization from the boundary becomes subtle and requires careful geometric considerations and microlocal analysis estimates. The nonlinear effects in the model demand construction of suitably small solutions which are invariant under the dynamics. The assumed smallness of the initial data is required only at the lowest energy level topology, which is sufficient to construct sufficiently smooth solutions to the nonlinear model.</p>

On Fredholm solvability and on the index of the generalized Neumann problem for an elliptic equation

In this paper, we investigate the Fredholm solvability of the generalized Neumann problem for a high-order elliptic equation in the plane. The equivalence of the solvability conditions of the generalized Neumann problem to the complementarity condition (Shapiro–Lopatinsky condition) is proved. The formula for the index of the specified problem in the class is calculated.

The Riemann – Hilbert boundary value problem for elliptic systems of the orthogonal type in R<sup>3</sup>

In this paper, a class of elliptic systems of four 1st order differential equations of the orthogonal type in R3 is considered. For such systems we study the issue of regularizability of the Riemann – Hilbert boundary value problem in an arbitrary limited simply-connected region with a smooth boundary in R3. Using the coefficients of the elliptic system and the matrix of the boundary operator, a special vector field is constructed, and its not entering the tangent plane in any point of the boundary provides the Lopatinski condition of the regularizability of the boundary value problem. The obtained condition permits to prove that the set of regularizable Riemann – Hilbert boundary value problems for the considered class of systems has two components of homotopic connectedness, and the index of an arbitrary regularizable problem equals to minus one.

Structural stability of shock waves in 2D compressible elastodynamics

We study the two-dimensional structural stability of shock waves in a compressible isentropic inviscid elastic material in the sense of the local-in-time existence and uniqueness of discontinuous shock front solutions of the equations of compressible elastodynamics in two space dimensions. By the energy method based on a symmetrization of the wave equation and giving an a priori estimate without loss of derivatives for solutions of the constant coefficients linearized problem we find a condition sufficient for the uniform stability of rectilinear shock waves. Comparing this condition with that for the uniform stability of shock waves in isentropic gas dynamics, we make the conclusion that the elastic force plays stabilizing role. In particular, we show that, as in isentropic gas dynamics, all compressive shock waves are uniformly stable for convex equations of state. Moreover, for some particular deformations (and general equations of state), by the direct test of the uniform Kreiss–Lopatinski condition we show that the stability condition found by the energy method is not only sufficient but also necessary for uniform stability. As is known, uniform stability implies structural stability of corresponding curved shock waves.

Parameter dependent differential operators on graphs and their applications

The paper is devoted to parameter dependent differential operators on graphs with general vertex conditions. We prove that if the differential operators are elliptic with parameter and an analogue of the well-known Lopatinsky condition at the vertices is satisfied, then the parameter dependent problem on the graph is invertible for large values of parameters. We consider applications of these results to the invertibility of general parabolic initial-vertex value problems on graphs, in particular case, for the heat equations on graphs with variable coefficients and general vertex conditions.

On the L^2 well posedness of Hyperbolic Initial Boundary Value Problems

In this paper we give a class of hyperbolic systems, which includes systems with constant multiplicity but significantly wider, for which the initial boundary value problem (IBVP) with source term and initial and boundary data in L 2 , is well posed in L 2 , provided that the necessary uniform Lopatinski condition is satisfied. Moreover, the speed of propagation is the speed of the interior problem. In the opposite direction, we show on an example that, even for symmetric systems in the sense of Friedrichs, with variable coefficients and variable multiplicities, the uniform Lopatinski condition is not sufficient to ensure the well posedness of the IBVP in Sobolev spaces.

Stability condition for strong shocks in flows over an infinite planar wedge satisfying the Lopatinski condition

We study the classical problem for a flow of uniform inviscid non-heat-conducting gas in thermodynamical equilibrium moving onto a planar infinite wedge. As it is known, theoretically this problem has solutions of two types. Solutions of the first type correspond to a strong shock when the gas velocity behind the shock front is less than the sound speed whereas solutions of the second type correspond to the case of a weak shock when the gas velocity behind the shock front is greater than the sound speed. The shock fronts are attached to the wedge vertex. We consider the linear problem for a strong shock wave provided that the Lopatinski condition holds (in a weak sense) on the shock front and the initial data are compactly, i.e. supports of the initial data are separated from the coordinate axes. Under some additional conditions on the initial data we find a solution of the generalized problem. In contrast to the previously studied case when the uniform Lopatinski condition holds on the shock front, this solution contains plane waves. The stability of the found solution is thus justified on the linear level what gives a principle possibility to realize the flow regime containing a strong shock as time increases.

Stability of a supersonic flow over a wedge containing a weak shock wave satisfying the Lopatinski condition

We study the stability problem for a stationary supersonic flow of inviscid non-heat-conducting gas in thermodynamical equilibrium moving onto a planar infinite wedge. As it is known, this problem has two solutions: a solution with a strong shock wave (when the velocity behind the front of the shock wave is subsonic) and a solution with a weak shock wave (when the velocity behind the front of the shock wave is supersonic). We consider the case of a weak shock wave and we prove that if the Lopatinski condition for the shock wave holds (in a weak sense), then the corresponding linearized initial boundary-value problem is well-posed. We thus find a classical solution to this problem. Unlike the case when the uniform Lopatinski condition holds, additional plane waves appear. For compactly supported initial data we show that the solution of the linearized problem converges to the zero solution as time tends to infinity. Therefore, for the case of a weak shock wave and when the Lopatinski condition holds in a weak sense, these results complete the proof of the well-known Courant–Friedrichs' conjecture that the strong shock wave solution is unstable whereas the weak shock wave solution is stable.

Courant-Friedrichs' Hypothesis and Stability of the Weak Shock Wave Satisfying the Lopatinski Condition

We are studying the problem of a stationary supersonic flow of an inviscid non-heat-conducting gas in thermodynamical equilibrium onto a planar infinite wedge. It is known that theoretically this problem has two solutions: the solution with a strong shock wave (when the velocity behind the front of the shock wave is subsonic) and the solution with a weak shock wave (when, generally speaking, the velocity behind the front of the shock wave is supersonic). In the present paper, the case of a weak shock wave is studied. It is proved that if the Lopatinski condition for the shock wave is satisfied (in a weak sense), then the corresponding linearized initial boundary-value problem is well-posed, and its classical solution is found. In this case, unlike the case when the uniform Lopatinski condition holds, additional plane waves appear. It is shown that for compactly supported initial data the solution of the linearized problem converges in finite time to the zero solution. Therefore, for the case of a weak shock wave and when the Lopatinski condition holds in a weak sense these results complete the verification of the well-known Courant-Friedrichs' conjecture that the strong shock wave solution is unstable whereas the weak shock wave solution is stable.

On a class of generalized functions arising in elasticity

We consider a class of generalized functions arising in the theory of thin elastic shells. Such problems were studied in the works by Sanchez-Palencia and his collaborators. Moreover, such classes arise in many other problems of mathematical physics. The principal feature of these problems is the necessity of consideration of the elliptic boundary value problems with the violated Shapiro–Lopatinsky condition. It implies unsolvability of the problems in the distribution spaces. To solve such problems it is necessary to consider the generalized functions with exponentially growing Fourier transforms.

Remarks on certain singular perturbations with ill-posed limit in shell theory and elasticity

Some problems of elasticity and shell theory are considered. Thecommon feature of these problems is the presence of a smallparameter $\varepsilon$. If $\varepsilon>0$ the correspondingequations are elliptic and the boundary conditions satisfy theShapiro - Lopatinsky condition. When $\varepsilon=0$, this conditionis violated and the problem can be non-solvable in the distributionspaces. The rather difficult passing to the limit is studied usingthe related Cauchy problem for elliptic equations. This approachallows to show that the most important is the transition zone wherethe frequencies $|\xi|\asymp \log (\varepsilon^{-1})$.

REGULARITY OF SOLUTIONS TO CHARACTERISTIC INITIAL-BOUNDARY VALUE PROBLEMS FOR SYMMETRIZABLE SYSTEMS

We consider the initial-boundary value problem for linear Friedrichs symmetrizable systems with characteristic boundary of constant rank. We assume the existence of the strong L2 solution satisfying a suitable energy estimate, but we do not assume any structural assumption sufficient for existence, such as the fact that the boundary conditions are maximally dissipative or the Kreiss–Lopatinski condition. We show that this is enough in order to get the regularity of solutions, in the natural framework of weighted anisotropic Sobolev spaces, provided the data are sufficiently smooth.

Local and global wellposedness of weak solutions for the wave equation with nonlinear boundary and interior sources of supercritical exponents and damping

We consider the wave equation with interior and boundary nonlinear sources and damping and we are interested in local and global wellposedness of finite energy solutions. The main difficulty is represented by the fact that the Lopatinski condition fails to hold (unless the dim ( Ω ) = 1 ), and thus the analysis of the boundary nonlinearities becomes a subtle issue.

A Kreiss' Symmetrizer Domain Penalization Method

In this paper, we describe a new way to approximate solutions of mixed hyperbolic systems with constant coefficients satisfying a Uniform Lopatinski Condition via a domain penalization method. The construction of the penalization operator is based on a Kreiss' symmetrizer. The obtained method has the advantage of not generating any boundary layers.

Stability of incompressible current-vortex sheets

We revisit the study in [Y. Trakhinin, On the existence of incompressible current-vortex sheets: study of a linearized free boundary value problem, Math. Methods Appl. Sci. 28 (2005) 917–945] where an energy a priori estimate for the linearized free boundary value problem for planar current-vortex sheets in ideal incompressible magnetohydrodynamics was proved for a part of the whole stability domain found a long time ago in [S.I. Syrovatskij, The stability of tangential discontinuities in a magnetohydrodynamic medium, Zh. Eksper. Teor. Fiz. 24 (1953) 622–629 (in Russian); W.I. Axford, Note on a problem of magnetohydrodynamic stability, Canad. J. Phys. 40 (1962) 654–655]. In this paper we derive an a priori estimate in the whole stability domain. The crucial point in deriving this estimate is the construction of a symbolic symmetrizer for a nonstandard elliptic problem for the small perturbation of total pressure. This symmetrizer is an analogue of Kreiss' type symmetrizers. As in hyperbolic theory, the failure of the uniform Lopatinski condition, i.e., the fact that current-vortex sheets are only weakly (neutrally) stable yields loss of derivatives in the energy estimate. The result of this paper is a necessary step to prove the local-in-time existence of stable nonplanar incompressible current-vortex sheets by a suitable Nash–Moser type iteration scheme.

The Existence of Current-Vortex Sheets in Ideal Compressible Magnetohydrodynamics

We prove the local-in-time existence of solutions with a surface of current-vortex sheet (tangential discontinuity) of the equations of ideal compressible magnetohydrodynamics in three space dimensions provided that a stability condition is satisfied at each point of the initial discontinuity. This paper is a natural completion of our previous analysis (Trakhinin in Arch Ration Mech Anal 177:331–366, 2005) where a sufficient condition for the weak stability of planar current-vortex sheets was found and a basic a priori estimate was proved for the linearized variable coefficients problem for nonplanar discontinuities. The original nonlinear problem is a free boundary hyperbolic problem. Since the free boundary is characteristic, the functional setting is provided by the anisotropic weighted Sobolev spaces \({H^m_*}\) . The fact that the Kreiss–Lopatinski condition is satisfied only in a weak sense yields losses of derivatives in a priori estimates. Therefore, we prove our existence theorem by a suitable Nash–Moser-type iteration scheme.

The Cauchy problem for 1-D linear nonconservative hyperbolic systems with possibly expansive discontinuity of the coefficient: A viscous approach

In this paper, we consider linear nonconservative Cauchy systems with discontinuous coefficients across the noncharacteristic hypersurface { x = 0 } . For the sake of simplicity, we restrict ourselves to piecewise constant hyperbolic operators of the form ∂ t + A ( x ) ∂ x with A ( x ) = A + 1 x > 0 + A − 1 x < 0 , where A ± ∈ M N ( R ) . Under assumptions, incorporating a sharp spectral stability assumption, we prove that a unique solution is successfully singled out by a vanishing viscosity approach. Due to our framework, which includes systems with expansive discontinuities of the coefficient, the selected small viscosity solution satisfies an unusual hyperbolic problem, which is well-posed even though it does not satisfy, in general, a Uniform Lopatinski Condition. In addition, based on a detailed analysis of our stability assumption, explicit examples of 2 × 2 systems checking our assumptions are given. Our result is new and contains both a stability result and a description of the boundary layers forming, at any order. Two kinds of boundary layers form, each polarized on specific linear subspaces in direct sum.

Uniqueness of weak solutions for the semilinear wave equations with supercritical boundary/interior sources and damping

We consider finite energy solutions of a wave equation with supercritical nonlinear sources and nonlinear damping. A distinct feature of the model under consideration is the presence of nonlinear sources on the boundary driven by Neumann boundary conditions. Since Lopatinski condition fails to hold (unless the $\text{dim} (\Omega) = 1$), the analysis of the nonlinearities supported on the boundary, within the framework of weak solutions, is a rather subtle issue and involves the strong interaction between the source and the damping. Thus, it is not surprising that existence theory for this class of problems has been established only recently. However, the uniqueness of weak solutions was declared an open problem. The main result in this work is <em> uniqueness</em> of weak solutions. This result is proved for the same (even larger) class of data for which existence theory holds. In addition, we prove that weak solutions are continuously depending on initial data and that the flow corresponding to weak and global solutions is a dynamical system on the finite energy space.